interval groupoids, by w. b. vasantha kandasamy, florentin smarandche, moon kumar chetry

8/9/2019 Interval Groupoids, by W. B. Vasantha Kandasamy, Florentin Smarandche, Moon Kumar Chetry

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INTERVAL

GROUPOIDS

W. B. Vasantha KandasamyFlorentin SmarandacheMoon Kumar Chetry



This book can be ordered in a paper bound reprint from:

Books on DemandProQuest Information & Learning(University of Microfilm International)300 N. Zeeb RoadP.O. Box 1346, Ann Arbor MI 48106-1346, USATel.: 1-800-521-0600 (Customer Service)

http://wwwlib.umi.com/bod/

Peer reviewers:Prof. Zhang Wenpeng, Department of Mathematics, Northwest UniversityXi’an, Shaanxi, P.R.China.Prof. Mircea Eugen Selariu,Polytech University of Timisoara, Romania.

Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, RomaniaDr. Fu Yuhua, 13-603, LiufangbeiliLiufang Street, Chaoyang district, Beijing, 100028 P. R. China

Copyright 2010 by InfoLearnQuest and authorsCover Design and Layout by Kama Kandasamy

Many books can be downloaded from the followingDigital Library of Science:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN-10: 1-59973-125-8

ISBN-13: 978-1-59973-125-4

EAN: 9781599731254



CONTENTS

Preface

Chapter One

PRELIMINARY NOTIONS

1.1 Groupoids 1.2 Introduction to Neutrosophic Algebraic Structu

Chapter TwoNEW CLASSES OF GROUPOIDS

2.1 New Classes of Matrix Groupoids 2 2 Polynomial Groupoids



Chapter ThreeON SOME NEW CLASSES OF NEUTROSOPHICGROUPOIDS

3.1 Neutrosophic Groupoids 3.2 New Classes of Neutrosophic groupoids using Z n

3.3 Neutrosophic Polynomial Groupoids 3.4 Neutrosophic Matrix Groupoids 3.5 Neutrosophic Interval Groupoids 3.6 Neutrosophic Interval Matrix Groupoids 3.7 Neutrosophic Interval Polynomial Groupoids

Chapter FourAPPLICATION OF THESE NEW CLASSES OFGROUPOIDS AND INTERVAL GROUPOIDS

Chapter FiveSUGGESTED PROBLEMS

FURTHER READING

INDEX

ABOUT THE AUTHORS



PREFACE

This book introduces several new classes of g polynomial groupoids, matrix groupoids, interv polynomial interval groupoids, matrix interval gtheir neutrosophic analogues.

Interval groupoid happens to be the first nostructure constructed using intervals built using ZR or Z + {0} or Q + {0} and so on.

This book has five chapters. Chapter one is innature. In chapter two new classes of groupoids

id d fi d d d ib d



The authors deeply acknowledge Dr. Kandasamy for

proof reading and Meena and Kama for the formattingdesigning of the book.

W.B.VASANTHA KANFLORENTIN SMARA

MOON KUMAR



Chapter One

PRELIMINARYNOTIONS

In this chapter we just give the basic definition of which forms the first section. In section two we j

properties associated with neutrosophy.

1 1 Groupoids



D EFINITION 1.1.2:A non empty set of elements G is form a groupoid if in G is defined a binary operation called

product denoted by such that a b G for all a, b

It is important to mention here that the binary operatidefined on the set G need not be associative that is (a

a (b c) in general for all a, b, c G, so we cangroupoid (G, ) is a set on which is defined a non assoc

binary operation which is closed on G.A groupoid G is said to be a commutative groupoid i

every a, b G we have a b = b a. A groupoid G ishave an identity element e in G if a e = e a = a for al

We call the order of the groupoid G to be the number of distelements in it denoted by o(G) or |G|. If the number of elemin G is finite we say the groupoid G is of finite order or a fgroupoid otherwise we say G is an infinite groupoid.

D EFINITION 1.1.3: Let (G, ) be a groupoid a proper su G is a subgroupoid if (H, ) is itself a groupoid.

D EFINITION 1.1.4: A groupoid G is said to be a M groupoid if it satisfies the Moufang identity (xy) (zx) = (x for all x, y, z in G.

D EFINITION 1.1.5: A groupoid G is said to be a Bol grouG satisfies the Bol identity ((xy) z) y = x ((yz) y) for all x, G.



D EFINITION 1.1.8: A groupoid G is alternative if

and left alternative, simultaneously.D EFINITION 1.1.9: A groupoid G is said to be

groupoid if x 2 = x for all x G.

D EFINITION 1.1.10: Let G be a groupoid. P a no

subset of G, P is said to be a left ideal of the groupis a subgroupoid of G and 2) For all x G andOne can similarly define right ideal of the gro

called an ideal if P is simultaneously a left and a rthe groupoid G.

D EFINITION 1.1.11: Let G be a groupoid A subgis said to be a normal subgroupoid of G if

1. aV = Va2. (Vx)y = V(xy)3. y(xV) = (yx)V

for all x, y, a V.

D EFINITION 1.1.12: A groupoid G is said tohas no non trivial normal subgroupoids.

Example 1.1.1: The groupoid G given by the fis simple.

a0 a1 a2 a3 a4 a5



It is left for the reader to verify (G, ) = {a 0, a 1, a} has no normal subgroupoids. Hence, G is simple.

D EFINITION 1.1.13: A groupoid G is normal if

1. xG = Gx

2. G(xy) = (Gx)y3. y(xG) = (yx)G

for all x, y G.

D EFINITION 1.1.14: A Smarandache groupoid G is a gro

which has a proper subset S, S G such that S unoperations of G is a semigroup.

D EFINITION 1.1.15: Let G be a Smarandache groupoid (the number of elements in G is finite we say G is a finitotherwise G is said to be an infinite SG.

D EFINITION 1.1.16: Let G be a Smarandache groupoid said to be a Smarandache commutative groupoid if there proper subset, which is a semigroup, is a comm semigroup.

For more about groupoids and Smarandache groupoids pleasrefer [20]

1 2 Introduction to Neutrosophic Algebraic Structures



Here we recall the notion of neutroso

neutrosophic groups in general do not have group stD EFINITION 1.2.1: Let (G, *) be any group, th

group is generated by I and G under * denoted by I ² , *}.

Example 1.2.1: Let Z 7 = {0, 1, 2, …, 6} beaddition modulo 7. N(G) = { ¢Z7 I², ‘+’ neutrosophic group which is in fact a group. For N/ a, b Z7} is a group under ‘+’ moduloneutrosophic group is also a group.

Example 1.2.2: Consider the set G = Z 5 \ {under multiplication modulo 5. N(G) = { ¢G

binary operation, multiplication modulo 5}. N(G) is called the neutrosophic group genera

Clearly N(G) is not a group, for I 2 = I and I is but only an indeterminate, but N(G) is d

neutrosophic group.

Thus based on this we have the following theorem:

T HEOREM 1.2.1: Let (G, *) be a group, N(G) = the neutrosophic group.

1. N(G) in general is not a group.2. N(G) always contains a group.

Proof: To prove N(G) in general is not a group ii l id ¢Z \ {0} I²



D EFINITION 1.2.2: Let N(G) = ¢ G I ² be a neutrosophi

generated by G and I. A proper subset P(G) is said to neutrosophic subgroup if P(G) is a neutrosophic group i.e. Pmust contain a (sub) group of G.

Example 1.2.3 : Let N(Z 2) = ¢Z2 I² be a neutrosophiunder addition. N(Z 2) = {0, 1, I, 1 + I}. Now we see {0,

group under + in fact a neutrosophic group {0, 1 + I} is a grunder ‘+’ but we call {0, I} or {0, 1 + I} only as psneutrosophic groups for they do not have a proper subset whis a group. So {0, I} and {0, 1 + I} will be only called as pseneutrosophic groups (subgroups).

We can thus define a pseudo neutrosophic group neutrosophic group, which does not contain a proper suwhich is a group. Pseudo neutrosophic subgroups can be foas a substructure of neutrosophic groups. Thus a psneutrosophic group though has a group structure is nneutrosophic group and a neutrosophic group cannot b

pseudo neutrosophic group. Both the concepts are different.

Now we see a neutrosophic group can have substrucwhich are pseudo neutrosophic groups which is evident fthe following example:

Example 1.2.4: Let N(Z 4) = ¢Z4 I² be a neutrosophiunder addition modulo 4. ¢Z4 I² = {0, 1, 2, 3, I, 1 + I, 2+ 2I, 1 + 3I, 2 + I, 2 + 2I, 2 + 3I, 3 + I, 3 + 2I, 3 + 3I}. o(I²) = 4 2.

Thus neutrosophic group has both neutrosophic subgr



We define I 2 = I, I + I = 2I i.e., I +…+ I = nI

then k.I = kI, 0I = 0. We denote the neutrosophicwhich is generated by K I that is K(I) = ¢ denotes the field generated by K and I.

Example 1.2.5: Let R be the field of reals. Thfield of reals is generated by R and I denoted by

clearly R ¢R I². Example 1.2.6: Let Q be the field of rationals. Thfield of rationals is generated by Q and I denoted by

D EFINITION 1.2.4: Let K(I) be a neutrosophic fi

is a prime neutrosophic field if K(I) has no prowhich is a neutrosophic field.

Example 1.2.7: Q(I) is a prime neutrosophic fieldis not a prime neutrosophic field for Q(I) R(I

D EFINITION 1.2.5: Let K(I) be a neutrosophic fia neutrosophic subfield of P if P itself is a neutro

K(I) will also be called as the extension neutrosthe neutrosophic field P.

We can also define neutrosophic fields of prime ch

(p is a prime).

D EFINITION 1.2.6: Let Z p = {0,1, 2, …, p – 1} bof characteristic p. ¢ Z p I ² is defined to be tfi ld f h i i I f ¢Z I² i



Elements of these neutrosophic fields will also be knowneutrosophic numbers. For more about neutrosophy please r[8-15]. We see Z nI = {aI | a Zn} is a neutrosophic field

pure neutrosophic field. Likewise QI, RI and Zneutrosophic fields where p is a prime. Thus Z 5I = {0,4I} is a pure neutrosophic field.

Pure neutrosophic structures as QI or RI or Zcontain any real numbers. However 0.I = 0, so 0 belongthem.



Chapter Two

NEW CLASSES OF GROUPOIDS

This chapter introduces seven new classes of groupseven sections. Section one introduces the new clgroupoids using Z n or Z or Q or R or C. Polynoof five levels are introduced in section two. Spgroupoids are introduced in section three. Polynogroupoids are introduced in section four. Section fivi l i id S d h i l



row matrix will be known as usual or real row matrix. If

ZnI or N(Z n) or QI or ZI or RI or N(Q) or N(Z) or N(R) thecall the row matrix (x 1, …, x n) to be a neutrosophic row Likewise we define

1

m

y

y

ª º« »« »« »¬ ¼

#

if y i Zn or Q or R or Z to be a usual column matrix and ifZnI or QI or RI or N(Z n) or N(Q) or N(R) as neutrosophic matrix 1 d i d m.

A matrix M num = (m ij) with m ij in Z n or Q or R or Z

known as the real matrix and if they belong to N(Z n) oror N(Z) or N(Q) or QI or N(R) or RI will be known aneutrosophic matrix. With this understanding we proceed describe and define some new classes of groupoids.

D EFINITION 2.1.1: Let G = {(x 1 , …, xn ) | x i Z m; 1 d i

3 be a collection of 1 u n row matrices with entries frmodulo integer Z m. Define * a binary operation on G as fo(x1 , …, xn ) * (y 1 , …, yn )

= t(x 1 , …, xn ) + u(y 1 , …, yn )= (tx 1 + uy 1(mod m), tx 2 + uy 2(mod m), …, (t

(mod m))

where t, u Z n \ {0}, t z u and (t, u) = 1 for all (x 1 , x(y1 , y2 , …, yn ) G. We define (G, *, (t, u)) to be a row

groupoid using Z n.

ll ll h b l l



Consider (3 2 1) * [(1 0 3) * (0 2 2)] and [(3 2 1) *

2 2). We see[(3 2 1) * (1 0 3)] * (0 2 2) = (1 0 3) * (0 2= (2 0 2) + (= (2 2 0)

(3 2 1) * [(1 0 3) * (0 2 2)] = (3 2 1) * [(2 0

= (3 2 1) * (2= (2 0 2) + (= (0 2 2)

(2 2 0) z (0 2 2). Thus the operation * is non ass(G, *, (2, 3)) is a 1 u 3 row matrix groupoids

We see 1 u 3 row matrix groupoids using Z2) or (2, 1) or (1, 3) or (3, 1) or (2, 3) or (3, 2).Thus we have 6 distinct 1 u 3 row matrix

using Z 4.

Example 2.1.2: Let G = {(x 1, x 2, x 3, x 4, x 5, x 6)

6}. Take (t, u) = 5, 6); (G, *, (t, u)) = (G, (5, 6), *) imatrix groupoid built using Z 7. We have 22 dismatrix groupoids built using Z 7. It is left for ththe number of 1 u m matrix groupoids built using

We can in the definition 2.1.1 replace Z n bythese cases we will get infinite number of row matof infinite order. By order of a groupoid G we meaof distinct elements in G. If G has finite number ofcall G a finite groupoid and if G has infinite numbeh ll G b i fi i id W ill



Example 2.1.4: Let G = {(x 1, x2, …, x 7) / x i R; 1Take (t, u) = (0.7, – 1.52) R u R. {G, *, (t, u)} is a 1matrix groupoid built using reals R.

Clearly cardinality of G is infinite and we can consinfinite number of 1 u 7 row matrix groupoids using R.

Next we proceed on to build a new class of column mgroupoids using Z n or R or Q or Z.

D EFINITION 2.1.2: Let G = {(x 1 , x2 , …, xn )t | x i Z m; 1be the collection of all n u 1 column matrices with entri

Z m. Choose t, u Z m \ {0}; t z u, (t, u) = 1. For (x 1 , …

(y1 , …, yn )t G. Define

(x1 , …, xn )t * (y1 , …, yn )t =ª º ª º ª º ª« » « » « » « » « » « »« » « » « »¬ ¼ ¬ ¼ ¬

# # # 1 1 1

n n n

x y tx u

*

x y tx u

=ª º« »« »« »¬ ¼

#1 1

n n

tx uy (mod m )

tx uy (mod m )

=ª º« »« »« »¬ ¼

#1

n

z

z

G.

Thus (G, (t, u), *) is a groupoid. This groupoid will be knowthe n u 1 column matrix groupoid built using Z m.

We will illustrate this situation by some examples.



be the collection of all 5 u 1 column matrices w

Z12. Choose t = 2 and u = 3; (t, u) = 1. (G, (2, 3),column matrix groupoid. Select

a =

10

2

0

0

1

ª º« »« »« »« »« »« »¬ ¼

, b =

3

2

11

0

0

ª º« »« »« »« »« »« »¬ ¼

and c =

1

0

9

2

0

ª º« »« »« »« »« »« »¬ ¼

To prove (a * b) * c z a * (b * c).

Consider

(a * b) * c =

10 3 1

2 2 0

* *0 11 9

0 0 2

1 0 0

§ ·ª º ª º ª¨ ¸« » « » ¨ ¸« » « » ¨ ¸« » « » ¨ ¸« » « » ¨ ¸« » « » ¨ ¸« » « » ¬ ¼ ¬ ¼© ¹

=

8 9 1

4 6 0*0 9 9

0 0 2

2 0 0

§ ·ª º ª º ª º¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¬ ¼ ¬ ¼ ¬ ¼© ¹

=

5 1

10 0*9 9

0 2

2 0

ª º ª« » «« » «« » «« » «« » «« » «¬ ¼



=

10 3 1

2 2 0* *0 11 9

0 0 2

1 0 0

§ ·ª º ª º ª º¨ ¸

« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¨ ¸« » « » « »¬ ¼ ¬ ¼ ¬ ¼© ¹

=

10 6 3

2 4 0*0 10 3

0 0 6

1 0 0

§ ·ª º ª º ª º¨ ¸

« » « » « ¨ ¸« » « » « ¨ ¸« » « » « ¨ ¸« » « » « ¨ ¸« » « » « ¨ ¸« » « » « ¬ ¼ ¬ ¼ ¬© ¹

=

10 9 8 3

2 4 4 0

* *0 1 0 3

0 6 0 6

1 0 2 0

ª º ª º ª º ª º« » « » « » « »« » « » « » « »« » « » « » « »« » « » « » « »« » « » « » « »« » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼

=

11

4

3

6

2

ª º« »« »« »« »« »« »¬ ¼Clearly

1 11

8 4

9 3

6 64 2

ª º ª º« » « »« » « »« » « »z« » « »« » « »« » « »¬ ¼ ¬ ¼

.

Hence (G, *, (2, 3)) is a groupoid as the operation * in genernon associative.

Example 2.1.6: Let

G =1

2 i 19

x

x x Z ;1 i 3

- ½ª º° °« » d d® ¾« »° °



be column matrix groupoids of finite order groupoids built using Z or R or Q will be infinite gr

Further the number of n u 1 column matrix Zm (for a fixed n and m) will be finite where as theu 1 column matrix groupoids built using Z or Qinfinite in number.

Just we will give only examples of a c

groupoid of infinite cardinality before we proceedthe notion of m u n matrix groupoids m z 1, nn can occur).

Example 2.1.7: Let

G =

1

2i

3

4

x

xx Z;1 i 4

x

x

- ½ª º° °« »° °« » d d® ¾« »° °« »° °¬ ¼¯ ¿

be the collection of all 4 u 1 column matrices wthe set Z. Choose (t, u) = (5, –12). Clearly (G, (5, –u 1 column matrix groupoid of infinite order.

The reader is left with the task of verifying *associative in general.

Example 2.1.8: Let

1

2

3

x

x

x

- ½ª º° °« »° °« »° °« »° °



(–7/8, 5/19)) is a 8 u 1 column matrix groupoid. It is

verified that (G, *, (–7/8, 5/19)) is a groupoid of incardinality. Now we proceed onto define substructures in

groupoids.

D EFINITION 2.1.3: Let (G, *, (t, u)) be a row (column)

groupoid H G (H be a proper subset of G); we call (H,*) to be a row (column) matrix subgroupoid of G if (H, (t, uis itself a row (column) matrix groupoid of G.

We will illustrate this be examples.

Example 2.1.9: Let G = {(x 1, x 2, x 3, x 4, x 5, x 6) | x i Z6} be the collection of all 1 u 6 row matrices. Choose (t, u8) Z12 u Z12. (G, (t, u), *) is a row matrix groupoid built Z12. Take H = {(x, x, x, x, x, x) / x Z12} G; {H, *, a row matrix subgroupoid of {G, *, (5, 8)}.

Example 2.1.10: Let G = {(x 1, x2, x3, x4) | x i Q, 1 d i1 u 4 row matrices with entries from Q. Take (t, u) = (7/2, (G, (7/2, –5), *) is a 1 u 4 row matrix subgroupoid. Tak{(x, 0, y, z 1) | x, y, z 1 Q} G; {H, *, (7/2, – 5)} is a 1matrix subgroupoid of {G, *, (7/2, – 5}.

Example 2.1.11: Let

1

2

a

a

- ½ª º° °« »° °« »° °° °



H = i 171

2

3

0

00

a Z ;1 i 3a

a

a

- ½ª º° °« »° °« »° °« »° ° d d« »® ¾« »° °« »° °« »° °« »° °¬ ¼

¯ ¿{H, *, (9, 8)} is a 6 u 1 column matrix subgroupo

Example 2.1.12: Let

G =

1

2

3 i

4

5

a

aa a R;1 i 5

a

a

- ½ª º° °« »° °« »° °« » d d® ¾« »° °« »° °« »° °¬ ¼¯ ¿

be the collection of all 5 u 1 column matrices wR. (G, (– 2 , 17 ), *) is a 5 u 1 column matriusing R. Take

H =

a

aa R a

a

0

- ½ª º° °« »° °« »° °« »® ¾« »° °« »° °« »° °¬ ¼¯ ¿

G.



= (tm ij + un ij )= P.

P is a n u m matrix in G. Thus (G, (t, u), *) is a matrix groupoid built using Z n (or R or Q or Z).

We will illustrate this situation by examples.

Example 2.1.13: Let

G = 9

a b ea, b,c,d,e,f Z

c d f

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

be the collection of all 2 u 3 matrices with entries from Z(t, u) = (5, 6); we see {G, *, (5, 6)} is a 2 u 3 mgroupoid with entries from Z 9.

Example 2.1.14: Let

G =

1 2

3 4i

5 6

7 8

a a

a aa Q;1 i 8

a a

a a

- ½ª º° °« »° °« » d d® ¾« »° °« »° °¬ ¼¯ ¿

be the collection of all 4 u 2 matrices with entries from Q(u, v) = (7, –3/2) from Q u Q. (G, (u, v), *) is a 4 ugroupoid with entries from Q.

Example 2.1.15: Let G = {All 9 u 9 upper triangular m



Choose (t, u) = (2, 1), (G, *, (2,1)} is a 2 u 2The operation carried out in G is in general non ass

Let

A =2 1

0 1§ ·¨ ¸© ¹

and

B =1 2

0 1§ ·¨ ¸© ¹

in G.

(A*B) = 22 1

0 1§ ·¨ ¸© ¹

+1 2

0 1§ ·¨ ¸©

=1 2

0 2§ ·¨ ¸© ¹

+1 2

0 1§ ·¨ ¸© ¹

=2 1

0 0§ ·¨ ¸© ¹

Take C =2 2

0 2

§ ·¨ ¸© ¹

G

(A * B) * C =2 1

0 0§ ·¨ ¸© ¹

*2 2

0 2§ ¨ ©

= 1 20 0

§ ·¨ ¸© ¹

+ 2 20 2

§ ·¨ ¸© ¹

0 1§ ·



0 10 2§ ·¨ ¸© ¹z

2 20 0§ ·¨ ¸© ¹

.

Thus the operation * defined in G in general isassociative.

As in case of row (and column) matrix groupoids wedefine the notion of matrix subgroupoids. We will only give example of a matrix subgroupoid.

Example 2.1.17: Let

G =

a b g

c d h a,b,g,c,d,h,e,f , i Z

e f i

- ½ª º° °« »® ¾« »° °« »¬ ¼¯ ¿

.

{G, *, (3, -2)} is a matrix groupoid built over Z.Let

H =

a a a

a a a a Z

a a a

- ½ª º° °« »® ¾« »° °« »¬ ¼¯ ¿

G;

{H, *, (3, –2)} is a matrix subgroupoid of {G, *, (3, –2)}.We see all matrix groupoids built using Z n (n < f

integers are finite where as all matrix groupoids built using R or Q are nZ are of infinite order



We also call a matrix subgroupoid V of G tosubgroupoid if

aV = Va(Vx) y = V(xy)y (xV) = (yx) V

for all x, y, a V.A matrix groupoid G is normal if

xG = GxG (xy) = (Gx)yy (xG) = (yx)G

for all x, y G.Now we define yet another class of matrix grou

D EFINITION 2.1.5: Let G = {row matrix or colum u n matrix with entries from Z n or Q or Z or R

a mutually exclusive sense. Now choose t, u Z n (or Z or Q or R) such

u) z 1. Then if for x, y G define x * y = tx + u*) is a matrix groupoid of type I which is differe

defined earlier.

We illustrate it by some examples.

Example 2.1.18: Let G = {all 2 u 2 upper triwith entries from Z 8}. Choose (t, u) = (2, 4); 2, 4

*, (2, 4)} is a matrix groupoid.We will just show how * is defined.

Take

A =3 5§ ·

¨ and B =2 3§ ·

¨



We see (G, (2, 4), *) is a finite groupoid. If we do permit (t, u) = 1 we see this class of matrix groupoidsmatrix groupoids of type I form a disjoint class from the magroupoids in which (t, u) = 1.

Now as in case of the other matrix groupoids of type I wcase of these groupoids also define all the properties witany modifications.

Example 2.1.19: Let

G =

1

2

3 i

4

5

x

x

x x Z;i 1,2,3,4,5

x

x

- ½ª º° °« »° °« »° °« » ® ¾« »° °« »° °« »° °¬ ¼¯ ¿

choose (t, u) = (5, 10); 5, 10 Z. {G, *, (5, 10)} is agroupoid of type I of infinite order.

Now we proceed onto define matrix groupoids of type II

D EFINITION 2.1.6: Let G = {collection of all row matrcolumn matrices or m u n matrices with entries by Z n

or R} ‘or’ is used in a mutually exclusive sense. Now choose t, u such that u = t. Then (G, *, (t, t)) is ano

new class of matrix groupoids which we choose to camatrix groupoids of type II. Clearly class of type I m

groupoids are disjoint from the class of type II group Further they are also disjoint from the usual class of m



Z = (1, 1, 3, 2, 2, 0 1), (3, 2, 0 1, 20, 18, 7) = X20, 4, 0, 7, 17, 3) G.

Z* (X * Y)= Z* [(3, 2, 0, 1, 20, 18, 7) * (1, 20, 4, 0, 7, 1= Z* (3, 16, 0, 8, 13, 18, 14) + (8, 13, 3, 0, 14= (1, 1, 3, 2, 2, 0, 1) * (11, 8, 3, 8, 6, 7,17)= (8, 8, 3, 16, 16, 0, 8) + (11, 3, 3, 3, 6, 14, 1= (19, 11, 6, 19, 1, 14, 18)

(Z*X) * Y= {(1, 1, 3, 2, 2, 0, 1) * (3, 2 0, 1, 20, 18, 7)}

(1, 20, 4, 0, 7, 17, 3)= [(8, 8, 3, 16, 16, 0, 8) + (3 16 0 8 13 18 14)

(1, 20, 4, 0, 7, 17, 3)= (11, 3, 3, 3, 8, 18, 1) + (8, 13, 11, 0, 14, 10= (19, 16, 14, 3, 1, 7, 4)

We see (19, 11, 6, 19, 1, 14, 18) z (19, 16, 14, 3is non associative.

Now we derive some properties of these matof type II.

T HEOREM 2.1.1: The matrix groupoids (G, *, (t P-groupoids. G is the collection of row (or colu

matrix) with entries from Z n.

Proof: Let A = (m ij) and B = (n ij) be row (or cmatrices with entries from Z n. Let t Zn. Tomatrix groupoid we have to prove (A * B) * A = A



We seeA * (B*A) = (A*B) * A

Thus (G, (t, t), *) is a matrix P-groupoid of type II.

C OROLLARY 2.1.1: We see if G = {(m ij ) / m ij Z or Q o*, (t, t)) is a matrix P-groupoid of type II.

Proof is left as an exercise to the reader.

T HEOREM 2.1.2: {G, *, (t, t)}; G is row (or column or mmatrix with entries from Z p , p a prime. {G, *, (t, t); 1< t not an alternative matrix groupoid.

Proof: To show {G, *, (t, t); 1 < t < n} is not an alternmatrix groupoid we have to show (x * y) * y z x (ysome x, y G. (x = (m ij) and y = (n ij). Consider

(x * y) * y = ((m ij) * (n ij)) * (n ij)= (t (m ij) + t (n ij)) * (n ij)

= t2

(m ij) + t2

(n ij) + t (n ij)= t 2 (m ij) + (t 2 + t) (n ij)

x * (y * y) = (m ij) * (t n ij + t n ij)= tm ij + t 2 nij + t 2 nij

= tm ij + 2t 2 (n ij)

Now I and II are identical if and only if t 2 { t (mothis is impossible as p is a prime. Hence I and II are nidentical for 1 < t < n.

Thus (G * (t t )) is not a matrix alternative groupo



(A * B) * B = [(m ij) * (n ij)] * (n ij)= (tm ij + tn ij) u (n ij)= (t 2 m ij + t 2 nij) + tn ij)= (tm ij + tn ij + t n ij) as t 2 { t

A * (B * B) = (m ij) * (n ij * n ij)= (m ij) * (tn ij + tn ij)= t m ij + t 2 nij + t 2 nij

= tm ij + tn ij + tn ij (as t 2 = t (m

Thus A * (B * B) = (A * B) * B. Hence (G, (alternative matrix groupoid of type II. Also all matrof type II are commutative.

Next we proceed onto define the notion of matof type III.

D EFINITION 2.1.7: Let G = {set of all column mmatrices or m u n matrices with entries from the

Z or R or Z n ) ‘or’ is in the mutually exclusive se

for any A, B G (A, B can be a row matrix or coa m u n matrix with entries for Z n or Q or Z oboth the places only in the mutually exclusively stA + uB where u or t is zero u, t Z n (or Z or R(G, *, (t, u); u = 0, or t = 0) to a matrix groupoid of

We will illustrate this by the following exampl

Example 2.1.21: Let G = {(a 1, a 2, a 3, a 4¸a5, a 6)6} be the set of all 1 u 6 row matrices with e



(A * B) * C = 3 (7, 1, 6, 3, 2, 5) + 0 (2, 1, 0, 7, 5, 3)= (5, 3, 2, 1, 6, 7).

Consider A * (B * C) = A * (3 (1 0 6 5 3 2 ) + 0 (2, 1, 0, 7, 5, 3

= (5, 3, 2, 1, 6, 7) * (3, 0, 2, 7, 1, 6)= (7, 1, 6, 3, 2, 5)

A * (B * C) z (A * B) * C evident from I and II. Thus (G, 0)) is a matrix groupoid of type III.

T HEOREM 2.1.4: (G, *, (0, t)) where G is row (column ormatrix) matrix with entries from Z n; n is not a prime. T*, (0, t)) is a matrix P-groupoid of type III if and only ift(mod n)

Proof: Consider A, B G

A * (B * B) = (m ij) * [(n ij) * (n ij)]= (m ij) * (t.n ij) (where A = (m ij) and = t 2 nij

= tn ij if and only if t 2 { t (mod n).

Consider

(A * B) * B = (0 + tB) * B= 0.t B + tB = tB= tn ij.

Th s A * (B * B) (A * B) * B if and onl if t 2 { t



Choose t, u Zn \ {0}; t 2 { t (mod n), u 2

u) = 1. To show (G, (t, u), *) is a matrix semigroup.Let A = (a ij), B = (b ij) and C = (c ij) be thr

Now(A * B) * C = ((a ij) * (b ij)) * (c ij)

= ((ta ij) + (ub ij)) * (c ij)= (t 2 aij) + (tu b ij) + (uc ij)= ta ij + (tub ij) + uc ij

as t 2 { t (mod n).Consider

A * (B * C) = (a ij) * ((b ij) * (c ij))= (a ij) * ((tb ij) * (uc ij))= (ta

ij) + (tub

ij) + (u 2c

ij) (u 2

= (ta ij) + (tub ij) + (uc ij)

I and II are the same. Hence (G, *, (t, u)) is a se

Note: If n is a prime (G, *, (t, u)) is never a semig

(mod n) and t2

{ t (mod n) can never occur.T HEOREM 2.1.6: The matrix groupoid (G, *idempotent matrix groupoid if and only if t + uis the set of all m u n matrices; 1 d m < f anentries from Z n }.

Proof: Given (G, *, (t, u)) is matrix groupoid rowm u n matrix; or in the mutually exclusive sense fr= (a ij) G



the collection of matrices with entries from Z n and2.1.7.

T HEOREM 2.1.7: Let (G, *, (t, u)) and (G, *, (u, t)) be a m groupoids with entries from Z n. P is a left ideal of (G, *, (and only if P is a right ideal of (G, *, (u, t)).

Recall a matrix groupoid (G, *, (t, u)) is simple if andif (G, *, (t, u)) has no normal subgroupoids.

It is left as an exercise for the reader to prove the followingtheorems.

T HEOREM 2.1.8: Let (G, *, (t, u)) be a matrix groupoientries from Z n. If n = t + u where both t and u are primes(G, *, (t, u)) is a simple matrix groupoid.

T HEOREM 2.1.9: Let (G, *, (t, u)) be a matrix groupoientries from Z n , n even and t + u = n with (t, u) = t. Then (t, u)) has only one matrix subgroupoid of order n / t and it normal matrix subgroupoid of (G, *, (t, u)).

From the above theorem the following conclusion is obvious

T HEOREM 2.1.10: Let (G, *, (t, u)) be a matrix groupoientries from Z n , n even with (t, u) = t and t + u = n. Then ((t, u)) is not a simple matrix groupoid.

We say as in case of usual groupoid a matrix groupoid*, (t, u)) is a Smarandache matrix groupoid if (G, *, (t, u)) hmatrix subgroupoid (H, *, (t, u)) which is a matrix semigrou



of (G, *, (t, u)) and H has a proper subset (K, *, (t,(K, *, (t, u)) is a matrix semigroup.

The following theorem can be proved by any intere

T HEOREM 2.1.11: Every matrix subgroupo groupoid (G, *, (t, u)) need not in general be a S subgroupoid of (G, *, (t, u)).

The reader is expected construct examples of the abAlmost all properties enjoyed by groupoids c

in case of matrix groupoids more so the properties.

Also these matrix groupoids are both finite andcan also study about properties like isomhomomorphism of matrix groupoids. Interesteexpected to do this regular exercise. However the contains problems for the reader.

2.2 Polynomial Groupoids

In this section we introduce for the first time, t polynomial groupoids built using Z n[x], n < for Q[x].

Some operation is defined on these sets sotogether with the operation becomes a groupoid was polynomial groupoids.

We follow the notation if a 0 + a 1x + polynomial of degree n then it is represented by th



collection of all polynomials of degree less than or equal with coefficients from Z m (m < f ; n < f ).

We define f (x) * g (x) as follows f(x) = (a 0 , a 1 , … g(x) = (b 0 , b 1 , …, b n )

f(x) * g(x) = (a 0 , a 1 , …, a n ) * (b 0 , b 1 , …, b n )= (a 0b1 , a 1b2 …, an-1bn , a n )= a 0 b1 + a 1b2 x + … + a n-1 bn xn-1 + a n x= h(x) n

m Z [x] (multiplication of a ib j

m).Thus the coefficients are reshuffled in this way. (

is defined as the polynomial groupoid of degree n with en from Z m. If

g(x) = a 0 + a 1 x + … + a n xn

h(x) = b 0 + b 1 x + … + b n xn

t(x) = t 0 + t 1 x + … + t n xn

a i , b i , t i Z m; 1 d i d n.

(g(x) * h (x)) * t(x) = (a 0 a1 … an ) * (b 0 , …, b n )) * (t 0 = (a 0 b1 a 1b2… an-1 bn , a n ) * (t 0 , t = (a 0 b1t 1 , a 2b2t 2 , …, a n-1 b t n , a n )

g (x) * h (x) * t(x)) = g(x) [(b 0 , b 1 , …, b n ) * (t 0 , t 1 , …= (a 0 , a 1 ,…, a n )* (b 0 t 1 , b 1 t 2 , …, b= (a 0 b1 t 2 , a 1 b2 t 3 , …, a n-1 bn , a n

Clearly the polynomials given by I and II are different. Tthe operation * in general is non associative.



f(x) * g(x) = (1, 4, 3, 0, 0) * (4, 0, 0, 1, 4)= (0, 0, 3, 0, 0)= 3x 2

g(x) * f(x) = (4 0 0 1 4) * (1 4 3 0 0)= (1 0 0 0 4)= 1 + 4x 4.

We see in the first place in general f(x) * g(x) Now we show ‘*’ is non associative .Choose

h(x) = 3x 4 + 2x 3 + x 2 + 4x + 1= (1, 4, 1, 2, 3).

Now(f(x) * g(x)) * (h(x)) = (0 0 3 0 0) * (1 4 1 2

= (0 0 1 0 0)= x 2.

[f(x) * g(x)] * h(x) = x 2

Consider f(x) * [g(x) * h(x)] = f(x)[(4 0 0 1 4) * (1 4

= f (x) * [1 0 0 3 4]= (1 4 3 0 0 ) (1 0 0 3= (0 0 4 0 0 )= 4x 2

We see I and II are different.It is important to mention here that one nee

one’s study to polynomials with coefficients frommodulo integers) We can define polynomial group



Z or R or Q}. (P, *) is a semigroup. Hence (G, *) polynomial Smarandache groupoid. Take p(x) = 20x4xn and r(x) = –5x n.

(p(x) *q(x) * r(x)) = [(0, …, 0, 20) * (0, 0, …, 0, (0, …, 0, -5)

= (0, 0, …, 0, 20) * (0,0, …, -5)= (0, 0, …, 20)= p(x).

p(x) * [q(x) * r(x)] = [(0, …, 0, 20) * (0,0, …, 0, (0, …, 0, -5)

= (0, 0, …, 0, 20) * (0,0, …, 0, 4)= (0, 0, …, 20)= p(x)

Thus * on P is associative. Hence (P, *) is a semigrou(G, *). Thus (G, *) is a polynomial Smarandache grouwhich is of infinite order.

Now we have mainly constructed this class of polynomto show that a solution to the open problem in [20] exists.

Before we proceed onto define classes of polyngroupoids in a very usual way similar to the one done in magroupoids, we extend the notion of polynomial groupoids to

polynomial ring which has polynomials of all degrees.Thus if

G = ii i

i 0

a x a Z;0 if- ½ d d f® ¾

¯ ¿¦

all polynomials of any degree with coefficients from Z in



Thus (G, *) is again an infinite groupoid.

Now we proceed onto define a different type of ope

D EFINITION 2.2.2: Let G = {all polynomials inwith coefficients form Z m , m < f of degree equal to n}. Define for any p(x) = p 0 + p 1 x q(x) = q 0 + q 1 x + …. + q n xn a binary operation

p(x) * q(x) = (tp 0 + uq 0 ) + (tp 1 + uq 1 ) x + …

where t, u Z m \ {0} (t, u) = 1 and t z u; p i , q i

is easily verified (G, *, (t, u)) is a groupoid. Thidefined as polynomial groupoid of type I.

We will illustrate this situation by some examp

Example 2.2.3: Let

G =5

ii i 8

i 0

a x a Z ;0 i 5- d d®¯¦ x an indete

(G, *, (3, 2)) is a polynomial groupoid of finite orde

Example 2.2.4: Let

G =

ni

i ii 0

a x a Z;n ,0 i-

f d d® ¯ ¦all polynomial of degree less than or equal to n wit



Example 2.2.5: Let

G =3

ii i 3

i 0

a x a Z ;0 i 3- ½ d d® ¾¯ ¿¦

all polynomials in the variable x of degree less than or equ3 with coefficients from Z 3.

We see we have only two polynomial groupoids of tyconstructed in this way. They are {G, *, (2, 1)} and {G, *2)}. Here it is important to note that we can have innumber of polynomial groupoids built using Z 3. This isvarying the degree of the polynomials from 1 to n; nHowever for each degree fixed as m; m d f we can htwo distinct polynomial groupoids of type I built using Z

We consider the following examples to give subgroupoidtype I.

Example 2.2.6: Let

G =2

ii i 10

i 0

a x a Z ;0 i 3- ½ d d® ¾¯ ¿¦

be all polynomials in the variable x with coefficients from Zdegree less than or equal to two. (G, *, (1, 5)) is a polyno

groupoid of type I built using Z 10.Consider

P =2

ii

i 0

a x a 0 or 5- ½® ¾¯ ¿¦ ;



be all polynomials in the variable x with coefficieall possible degrees including infinity. Defioperation * on G as follows.

If

p(x) = ¦n

ii

i 0

p x and q(x) = ¦n

ii 0

q

then

p(x) * q(x) = ¦n

ii i

i 0( tp uq )x

‘+’ modulo n where (t,u) Z n \{0} but (t,u) z 1is defined as the polynomial groupoid of type II bu

Note: Zn can be replaced by Q or Z or R and stgroupoid will continue to be of type II groupoid.


Example 2.2.7: Let

G =

12i

i i 18i 0

a x a Z ;0 i 12-

d d® ¯ ¦all polynomials in the variable x with coefficientsdegree less than or equal to 12. Define * on G as fo

p(x) * q(x) = 12i i

i 0

(tp uq )(mod18¦

where (t u) = (4 6) ; 4 6 Z18 (G * (4 6))



all polynomials of degree less than or equal to n coefficients from Z m }. Define the operation * on G as fo

for any polynomial

p(x) = ¦n

ii

i 0

p x

and

q(x) =

¦

ni

ii 0

q x .

Define

p(x) * q(x) = ¦n

i ii 0

( tp uq ) (mod m) x i

where t, u Z n \ {0} and t = u.

It is easily verified (G, *, (t, t)) is a polynomial groupoidthis is defined as a polynomial groupoid of type II.

We will illustrate this by a few examples.

Example 2.2.8: Let

G =7

ii i 24

i 0

a x a Z ;0 i 7- ½ d d® ¾¯ ¿¦

all polynomials in the variable x with coefficients from Zdegree less than or equal to 7.

Define * on G by

p(x) * q(x) =7

ii

i 0

p x§ ·¨ ¸© ¹¦ *

7i

ii 0

q x§ ·¨ ¸© ¹¦



(p(x) * q(x)) * r(x)= {(5x 3 + 7x + 3) * (8x 5 + 4x 2 + 2x + 1)}

(x6 + 5x 5 + 7x 2 + 13)= (20x 3 + 8x 5 + 16 + 16x 2 + 12x) * (x 6 + = (4x 6 + 4x 5 + 3x 3 + + 20x 2+20)

Consider p(x) * (q(x) * r(x))

= (5x 3 + 7x + 3) * [(8x 5 + 4x 2 + 2x + 1) *(x6 + 5x 5 + 7x 2 + 13)]

= (5x 3 + 7x + 3)* [4x 6 + 8x + 8 + 20x 2 + = [16x 6 + 16x 5 + 20 + 12x + 8x 2 + 20x 3]

We see I and II are not the same polynoperation * in general is non associative. Thus (G,

polynomial groupoid of type III built using Z 24

Example 2.2.9: Let

G =4

ii i 7

i 0

a x a Z ;0 i 4- ½ d d® ¯ ¦

all polynomials in the variable x of degree less tha4 with coefficients from Z 7.

Define * on G by a * b = 4a + 4b(mod 7) i.e. (Ga polynomial groupoid of type III. Take

a = (3x + 2) b = (4x 2 + 2x + 5)

andc = 6x 4 + 4x 3 + 5x + 1.



I and II are different. Hence (G, *, (4, 4)) is a polynogroupoid with entries from Z 7 of type III.

Now we proceed onto define type IV polynomial groupoids.

D EFINITION 2.2.5: Let G =-®¦̄

ni

ii 0

a x ; 0 d id n; a i Z

polynomials with coefficients from Z m. Define * on G ah(x) = [tg(x) + 0 h(x)]; t Z m. {G, *, (t, 0)} is a pol groupoid of type IV with entries from Z m.


Example 2.2.10: Let Z 8 = {0, 1, 2, …, 7} ring of mintegers eight. Choose

G =26

ii i 8

i 0

a x a Z ;0 i 26- ½ d d® ¾¯ ¿¦ ;

all polynomials of degree less than or equal to 26 coefficients from Z 8 in the variable x.Let t = 5 Z8. Define for any two polynomials g(x), h

G. g(x) * h(x) = tg(x) + 0 h(x). {G, *, (5, 0)} is a polynogroupoid of type IV with entries from Z 8.

Example 2.2.11: LetG =

5i

i i 23i 0

a x a Z ;0 i 5- ½ d d® ¾¯ ¿¦



When t = 1 we get (Z p[x], *, (1, 0)) to be a se p is a prime if in Z n[x]; n is not a prime and 1such that t 2 = t(mod n) then we see [Z n[xsemigroup.

We will illustrate this situation by an example.

Example 2.2.12: Let Z 6[x] be a polynomial ring(3, 0)] be the polynomial groupoid of type IV. Taksee p(x) * q(x) = 3p(x).

Now(p(x) * q(x)) * r(x) = (3p(x) + 0) * r(x)

= 3p(x) * r (x)= 3p(x) ( ' 32 = 3 (mo

p(x) * (q(x) * r(x)) = p(x) * [3q(x)]= 3p(x) + 0= 3p(x).

Thus * on Z 6[x] is associative when t = 3; t ZMotivated by this example we see we have fo

polynomial groupoids {Z n[x], *, (t, 0)} for tof them to be polynomial semigroups, for if t = 1 0)} is a polynomial semigroup. In view of this wSmarandache special class of groupoids.

D EFINITION 2.2.6: Let G(S) = {Class of grusing the same set S}. If G(S) has atleast one semigcall G(S) to be a Smarandache Special Class of gro

groupoids).



Proof: Since 1 Zn we see {Z n[x], *, (1, 0)} G(Z polynomial semigroup, hence G(Z[x]) is a SSC-groupoid.

Example 2.2.14: Let G(Z 12[x]) be the class of groupoidusing Z 12[x]. Take H 1 = {Z 12[x], *, (4, 0), 4 Z12}, H 2

*, (9, 0)} and H 3 = {Z 12[x], *, (1, 0)} in G(Z 12[x]). HH3 are polynomial semigroups. Hence G(Z 12[x]) isgroupoid.

Example 2.2.15: Let G(Z 30[x]) be the class of polygroupoids built using Z 30[x]. Take H 1 = {Z 30[x], *, (15= {Z 30[x], *, (1, 0)}, H 3 = {Z 30[x], *, (10, 0)} and H 4

*, (6, 0)} are polynomial semigroups. So G (Z 30[x]) igroupoid.

Now we will give classes of SSC-groupoid.

T HEOREM 2.2.2: The class of groupoids Z(n) = {Z n , *,not a prime; is a SSC-groupoid.

Proof: We see Z(n) has semigroups. For if (t, u) = 1 with t,Zn \ {0} with t 2 = t (mod n) and u 2 = u (mod n) then (Zu)} is a semigroup. Hence Z(n) is a SSC-groupoid.


Example 2.2.16: Take {Z 12, *, (4, 9)} Z(12). (Z 12, *,a semigroup as 4 2 = 4 (mod 12) and 9 2 = 9 (mod 12).



D EFINITION 2.2.8: Let G* = {Q[x], *, (m, n) / m= d}. This groupoid is known as the polynomial grational coefficients (d z 1. If d = 1 G = G*).

Example 2.2.17: Let G = {Q[x], *, (5, 7)} begroupoid with rational coefficients. Let

p(x) = x + 1,

q(x) = 3x2

+72 x +

53

and r (x) = 5x 3 – 32

x + 9 Q[x

p(x) * (q(x) * r (x))

= p(x) * [15 x 2 + 35x2

+ 253

+ 35x 3 – 212

= p(x) * [35x 3 + 15x 2 + 7x +25

633

§ ·¨ ¸© ¹

]

= 5(x + 1) + 35 u 7x3 + 105x 2 + 49x +§

¨ ©

Consider (p(x) * q(x)) * r(x)

= (5 (x+1) + 21x 2 +49x 35

2 3) * (5x

= 25x + 25 + 105x 2 +49 5

x2u

+35 5

3u

+

= 35x 3 + 105x 2 + (25 +49 5 21u

)x + (2



Example 2.2.18: Let G* = {Q[x], *, (6, 3) = 3} be a polyngroupoid. Take

p(x) = x 7 + 1,q(x) = 8x 4

andr(x) = 3x 2 + 5x + 1.

(p(x) * q(x)) * r (x) = [6 (x 7 + 1) + 3 u 8x4] * (3x 2 += (6x 7 + 6 + 24x 4] * (3x 2 + 5x + 1= (36x 7 + 36 + 14 4x 4) + 9x 2 + 15= 36x 7 + 144x 4 + 9x 2 + 15x + 39

p(x) * (q(x) * q(x)) = p(x) * [48 x 4 + 9x 2 + 15x + 3]= 6x 7 + 6 + 144x 4 + 27x 2 + 45x += 6x 7 + 144x 4 + 27x 2 + 45x + 15

I and II are not equal. Hence G* is a groupoid and is nsemigroup.

D EFINITION 2.2.9: Let G** = {Q[x], *, (t, 0) / t G** is a polynomial groupoid with polynomials from Q[x].


Example 2.2.19: Let G** = {Q[x], *, (5, 0)} be a polyngroupoid. Take p(x) = 8x 3, q(x) = 5x 2 + 1 and r(x) = x 5

(p(x) * q(x)) * r (x) = (40x 3 + 0) * r (x)= 200x 3

p(x) * (q(x) * r (x)) = 8x 3 [25x 2 + 5 + 0]



2.3 Special Interval Groupoids

In this section we introduce yet another spegroupoids called interval groupoids. Here we coninteresting properties about them.

Notations: Let

IZ = {[a, b] | a, b Z+ {0}, a dIQ = {[a, b] | a, b Q+ {0}, a

IR = {[a, b] | a, b R + {0}, a

Clearly IZ IQ IR . Consider InZ = {

the set of intervals in Z n.

Now using these four classes of intervals we bugroupoids.

D EFINITION 2.3.1: Let G * ( I n Z ) = {[0, r], *, (m

Z n with (m, p) = 1, m and p primes} where [0, r] mr + ps (mod n)]. G * ( I

n Z ) is a class of groupmodulo integer interval groupoids of level one.

Clearly G * ( I n Z ) has only finite number of e

We will illustrate this by some examples.

Example 2.3.1: Let G * ( I12Z ) = {[0, s], *, (5, 7

be a modulo integer interval groupoid of level on



D EFINITION 2.3.2: G(S) = {[0, a], *, (p, q) = 1; a, p, q{0}} (or C + , Q + or R + ) p and q primes} where * is defineda] * [0, b] = [0, pa + qb] G(S) and S = I C or ZG(S) is a class of interval groupoids of level one.

As p, q Z + \ {0}, vary over Z + \ {0} we get infiniteof interval groupoids. S can be I R I C or I Q .


Example 2.3.3: Let G*( IZ ) = {[0, a], *, (5, 19); a, 5, 19 be an interval integer groupoid of level one. It is easily veri* in general is non associative. For take [0, 1], [0, 3] and [0in G*( IZ ).

[0 1] * ([0 3]* [0 2]) = [0 1] * ([0, 15] + [0, 38])= [0 1] * [0, 53]= [0, 5] + [0, 53 u 19]= [0, 5 + 53 u 19]

([0, 1] * [0 3]) * [0 2] = ([0, 5] + [0, 57]) * [0, 2]= [0, 62] * [0, 2]= [0, 310] + [0, 38]= [0, 348]

Clearly I and II are not equal; hence G * ( IZ ) is interval groupoid of level one.

Example 2 3 4: Let G * ( Q ) = {[0 a] * (2 3); a 2 3



Example 2.3.7: Let G*( IR ) = {[0, a], *, (29¸5R +} be a real interval groupoid of level one.

Now we proceed onto define the notion of real ointeger or rational interval subgroupoid of level one

D EFINITION 2.3.3: Let T G * (S) be a realinteger or modulo integer or rational interval grouone.

Let P T (P a proper subset of T). If P itsecomplex or integer or modulo integer or rati

groupoid of level one then we call P to be a real ointeger or modulo integer or rational interval subgof level one.


Example 2.3.8: Let {[0, a], *, (3, 2); a, 3, 2 be a modulo integer groupoid of order 12 of level o

{[0, 2], [0, 0], [0, 4], [0, 6], [0, 8] [0, 10], *, (3, 2)}(3, 2), a, 3, 2, Z12} G * ( I

12Z ); P is a modulosubgroupoid of order 6.

Example 2.3.9: Let T = {[0, a], *, (5, 3) | 5, 3,modulo integer interval groupoid of level one.

The reader is requested to find modulo insubgroupoids of level one.



Remark: It is important and interesting to note that P is real interval groupoid but only rational interval groupoid so

much are we justified in calling it as a real interval subgrouBut by the rule of convention we call so.

Example 2.3.12: Let T = {[0, x], *, (7, 3) / x, 7, 3 Q be a rational interval groupoid of level one. Take P = {[0, a(7, 3) / a, 7, 3 Z+ {0}} T; P is a rational subgroupoid of T of level one.

Example 2.3.13: Let T = {[0, a], *, (17, 2) / a, 17, 2{0}} be a complex interval groupoid. Let S = {[0, a], *, (17,a, 17, 2 Z+ {0}} T; S is a complex interval subgroof T of level one.

Now we proceed onto define interval groupoid of level t

D EFINITION 2.3.4: Let G** ( I n Z ) = {[0 a], *, (p, q) =

and q are in Z n;} where [0, a] * [0, b] = [0, pa + qb (mod

G** ( I n Z ) is defined as the class of modulo integer in

groupoid of level two.

Example 2.3.14: Let T = {[0, a], *, (5, 8), a, 5, 8 Z( I

9Z ) be the modulo integer interval groupoid of level two.

Example 2.3.15: Let P = {[0, a], *, (9, 10), a, 9, 10G** ( I

11Z ) be the modulo integer interval groupoid of level Like wise we can define class of integer interval grou



Example 2.3.16: Let T = {[0, a], *, (8, 15), 15, 8, be an integer interval groupoid of level two.

Example 2.3.17: Let S = {[0, a], *, (19, 16), athe integer interval groupoid of level two.

Example 2.3.18: Let P = {[0, a], *, (27, 64), a, {0}} be the rational interval groupoid of level two.

Example 2.3.19: Let T = {[0, x], *, (27, 43), x, {0}} be the real interval groupoid of level two.

Example 2.3.20: Let W = {[0, y], *, (43, 7); y,{0}} be the complex interval groupoid of level two

It is both interesting and important to note the follo

The class of integer interval groupoids of level twoof rational interval groupoids of level two

interval groupoids of level two The class of cgroupoids of level two; that is G** ( IZ )

G**( IR ) G** ( IC ).Same type of containment is true in cas

groupoids of level one. Now we will call a level one interval groupoid

if it has no proper interval subgroupoids of level onWe see G*( I

nZ ); when n is a prime is a integer interval groupoid of level one.



D EFINITION 2.3.5: Let T = {[0, a], *, (p, q), a, p, qmodulo integer interval groupoid of level two. If P that P = {[0, a], *, (p, q)} T is a proper subset of Titself is a modulo integer interval groupoid of level two; thecall P to be a modulo integer interval subgroupoid of levelof T. When T has no interval subgroupoid of level two; thecall T to be a simple modulo integer interval groupoid of two.

We will illustrate both the situation by some examples.

Example 2.3.22: Let T = {[0, a], *, (3, 8), a, 3, 8 Q be a rational interval groupoid of level two. Take P = {[0, a(3, 8), 3, 8, a Z+ {0}} T; P is a rational subgroupoid of level two.

Example 2.3.23: Let T = [0, a], *, (9, 8), 9, 8, a R +

a real interval groupoid of level two. Take P = {[0, a], *, (9a, 9, 8 Q+ {0}} T, P is a real interval subgrouplevel two.

Example 2.3.24: Let T = {[0, a]; *, (15, 8); 15, 8, amodulo integer interval groupoid of level two. Take P = {[0[0, 10], [0, 20], *, (15, 8), 15, 8 Z30} T. P is a

integer interval subgroupoid of level two of T.

Example 2.3.25: Let S = {[0, a], *, (9, 8); a, 9, 8modulo integer interval groupoid of level two.

N d d fi i l



Now we proceed onto define interval groupthree.

D EFINITION 2.3.6: Let G**( I n Z ) = {[0, a]; *,

where * is such that [0, a] * [0, b] = [0, pa+qb (define G***( I

n Z ) to be the class of modulo groupoid of level three.

If we replace I n Z by I Z we call G***( Z

of integer interval groupoid of level three. If I

n Z in the definition is replaced by I Q wto be the class of rational interval groupoids of l

I n Z in the definition is replaced by I R we call

the class of real interval groupoids of level thdefinition I n Z is replaced by I C we call G**

class of complex interval groupoid of level three.

We will illustrate each of the situation by some exa

Example 2.3.26: Let T = {[0, a], *, (8, 24); a, 8, modulo integer interval groupoid of level three.

Example 2.3.27: Let T = {[0, a], *, (9, 3); a, 9,modulo integer interval groupoid of level three.

Example 2.3.28: Let P = {[0, a], *, (27, 30); a, {0}} be the integer interval groupoid of level three.

Example 2 3 29: Let P = {[0 a] * (11 66); a

E l 2 3 32 L t E {[0 b] * (7 497) b 7 497



Example 2.3.32: Let E = {[0, b], *, (7, 497); b, 7, 497{0}} be the complex interval groupoid of level three.

Now we proceed onto define the notion of interval subgrouof level three.

D EFINITION 2.3.7: Let T = {[0, a], *, (p, q) = d z 1 p, Z n } be a modulo integer interval groupoid of level three. L T; if P is a modulo integer interval groupoid of level tand P is a proper subset of T we call P to be the modulo inteinterval subgroupoid of T of level three.

If T has no proper modulo integer interval subgrothen we call T to be a simple modulo integer interval groupo

Analogous definitions hold good in case of integer or rearational or complex interval groupoid of level three.

We will illustrate this situation by some simple example

Example 2.3.33: Let T = {[0, a], *, [15, 10], a, 10, 15,a modulo integer interval groupoid of level three.

Let P = {[0, a], [0, 5], [0, 10], [0, 15], [0, 20], [0, 2530], [0, 35], [0, 40], [0, 45], [0, 50], [0, 55], *, (15, 10)}the modulo integer interval subgroupoid of T of level three.

Example 2.3.34: Let T = {[0, a], *, (8, 24), a, 8, 24 Z be the integer interval groupoid of level three. Take P = {[0, *, 2a, 8, 24 Z+ {0}} T, P is an integer subgroupoid of T of level three.

E l 2 3 35 L T {[0 ] * (27 18) 27 18

Example 2 3 37: Let Y = {[0 x] * (28 42) x



Example 2.3.37: Let Y = {[0, x], *, (28, 42), x, {0}} be a complex interval groupoid of level three

*, (28, 42), x R + {0}} Y, is the csubgroupoid of level three.

Example 2.3.38: Let B = {[0, a], *, (8, 12), a, 8, 1 be a modulo integer interval groupoid of level simple modulo integer interval groupoid of level th

The reader is expected to answer the following probLet X = {[0, a], *, (r, s) = t z 1, a, r, s, t

be a modulo interval groupoid of level three. Is X si

Before we proceed onto describe more propertiesinterval groupoid we give the definition of level groupoids.

D EFINITION 2.3.8: Let G **** ( I n Z ) = {[0, a], *

Z n } where [0, a] * [0, b] = [0, pa + 0b (mod

(mod n)]; be a modulo integer interval groupoid dmodulo integer interval groupoid of level four.

If we replace ( InZ ) in the definition 2.3.8 by

integer interval groupoid of level four. By replacinin the definition 2.3.8 we get the rational intervallevel four. If I

nZ is replaced by IR in definitthe real interval groupoid of level four and isG****( IR ). Similarly by replacing I

nZ by

Example 2 3 40: Let C = {[0 x] * (6 0) x 6



Example 2.3.40: Let C = {[0, x], *, (6, 0), x, 6modulo integer interval groupoid of level four.

Example 2.3.41: Let D = {[0, b], *, (8, 0), b, 8integer interval groupoid of level four.

Example 2.3.42: Let X = {[0, a], *, (7, 0), a, 7

rational interval groupoid of level four.

Example 2.3.43: Let Y = {[0, b], *, (141, 0), b, 141real interval groupoid of level four.

Example 2.3.44: Let P = {[0, t], *, (15, 0), t, 15complex interval groupoid of level four.

We will now proceed onto define interval subgroupoidlevel one.

D EFINITION 2.3.9: Let S = {[0, a], *, (p, 0), a, p modular integer interval groupoid of level four. T = {[0, a(p, 0), a X Z n } S be a modulo integer interval grouplevel four; then we call T to be a modulo integer int

subgroupoid of level four. If S has no proper modulo integer interval subgrou

then we call S to be a simple modulo integer interval grouof level four.

We can analogously define these concepts in case of ol d f l l f

Example 2 3 46: Let W = {[0 a] * (5 0); a



Example 2.3.46: Let W = {[0, a], , (5, 0); a,modulo integer interval groupoid of level four. W

simple modulo integer interval groupoid of level fo

In view of this we have the following theorem.

T HEOREM 2.3.2: Let P = {[0, a], *, (t, 0), a, t be a modulo integer interval groupoid of level four

The proof of the theorem is left as an exercise for th

Example 2.3.47: Let T = {[0, a], *, (9, 0), a, 9an integer interval groupoid of level four. Take W(9, 0), a 8Z + {0}} T is an integer intervallevel four of T.

Example 2.3.48: Let W = {[0, a], *, (12, 0), aan rational interval groupoid of level four. S = {[0, a 3Z + {0}} is an integer interval subgroupoiof W.

Example 2.3.49: Let T = {[0, a], *, (19, 0), areal interval groupoid of level four. S = {[0, a], *,Q+ {0}} T; S is a real interval subgroupoid of

Example 2.3.50: Let W = {[0, a], *, (23, 0), acomplex interval groupoid of level four. P = {[0, a]R + {0}} W; P is a complex interval subgro

four.



Proof: Given T = {[0, a]; *, (t, u), a, t, u Zn} G**(

+ u { 1 (mod n). To show T is an idempotent interval grouwe have to prove [0, a]*, [0, a] = [0, a] for all [0, a]

Consider [0, a] * [0, a] = [0, ta + ua (mod n)]. If T is tan idempotent interval groupoid we need [0, a] * [0, a] = [0So that [0, ta + ua (mod n)] = [0, a]. That is ta + ua = a (mo

thus (t + u – 1)a { 0(mod n).This is possible if and only if t + u { 1 (mod n). Hclaim.

We will illustrate this by a simple example.

Example 2.3.51: Let T = {[0, a], *, [4, 5], a, 4, 5G**( I

8Z ). T is a modulo integer idempotent interval grouFor consider [0, a] * [0, a] = [0, 4a + 5a(mod 8)] = [0, a] as 9a(mod 8). This is true for all [0, a] I

8Z . Hence T is ainteger idempotent interval groupoid.

C OROLLARY 2.3.1: The above theorem is also true in cthe class of interval groupoids of level one.

We now proceed onto define normal subgroupoids and idealinterval groupoids.

D EFINITION 2.3.10: Let G*( I n Z ) or G*( I Z ) or G*

G*( I R ) or G*( I C )) be a class of interval groupoids. A in

b id V f T G*( Z ) i id t b l i

Note: An interval groupoid T G*( IZ ) is norm



Note: An interval groupoid T G ( nZ ) is norm(a) [0, x] T = T [0, x](b) T ([0, x] [0, y]) = (T [0, x]) [0, y](c) [0, y] ([0, x] T) = ([0, y] [0, x]) T

for all [0, x], [0, y] T G*( InZ ).

This same definition holds good for interval gr

using IZ , IQ , IR or IC . Further the same definfor all the four levels of interval groupoids.

Thus from here onwards by interval groupomean any interval groupoid built using I

nZ or

IC and from the context it will be easily underst

level they belong to and built using which set.

Now we proceed onto define ideal of interval g

D EFINITION 2.3.11: Let T be any interval groempty subset of T. P is said to be a left ideal of T if

(1) P is an interval subgroupid of T (2) For all x T and a P, xa P. P is called a right ideal if P is an interval subg

for all x T and a P ax P. If P is both leftof T then we call P to be the interval ideal of T orof T. As in case of usual groupoids we call an inter

to be simple if it has nontrivial interval subgrinterval groupoids of level one, two, three and fhave {0} as an ideal.

T HEOREM 2.3.5: Let T G*( I n Z ) where T = {[0, a], *, (



et G ( n ) w e e {[0, a], , (t, u Z n with t and u primes and t + u = n} be an int

groupoid then T is simple.

The proof uses simple number theoretic properties and is lefthe reader to prove.

Note: Even if n is a prime and t + u = p, u and t primes thenT = {[0, a], *, (t, u); t + u = p; t, a, u Z p, p a priinterval groupoid is simple.

T HEOREM 2.3.6: The interval groupoid T = {[0 a], *, (0, t Z n } of level four is a interval P-groupoid and altern

interval groupoid if and only if t 2 { t (mod n).

The proof is left as an exercise for the reader. None of the above results hold good in case of in

groupoids built using IZ , IQ , IR or IC . These prope

only valid for the interval groupoids built using InZ .

2.4 New Classes of Polynomial Interval Groupoids

In this section we introduce the new notion of polynointerval groupoids. We will first briefly describe the essenotations. I

nZ [x] = {collection of all polynomials in the va

x with coefficients from the interval set InZ }

f- ½¦

polynomials in the variable x with interval coeff



form [0, a] from the interval set IZ } =

ii i

i 0

[0,a ]x a Z {0}f- ½ ® ¾

¯ ¿¦ .

This collection will be known as the po

interval polynomials or integer interval polynomi{collection of all polynomials in the variable x withfrom the intervals in IQ of the form [0, a i];

aiI

Q {0}} = i

i ii 0

[0,a ]x a Qf-

® ¯ ¦Thus

IR [x] = ii i

i 0

[0,a ]x a R {f- ®

¯ ¦

and

IC [x] = ii i

i 0

[0,a ]x a C {0f- ®

¯ ¦

are defined as real interval polynomials and com polynomials respectively.

Now using these 5 types of interval polynodefine 4 levels of polynomial interval groupoid

polynomial groupoids.

q(x) =f

¦ j[0 b ] x



q(x) = ¦ j j 0

[0,b ] x

belong to T;

p(x) * q(x) =f§ ·

¨ ¸© ¹¦ ii

i 0

[0,a ] x *f§

¨ ¸© ¦ j

j j 0

[0,b ] x

=f

¦ i ji j

k 0

[ 0,ta ub (mod n )] x ;

(t, u Z n \ {0} such that both t and u are primes and (t, u) =

=f

¦ k k

k 0

[0,c ] x

where k = i + j and c k = ta i + ub j (mod n). [T, *, (defined as the modulo integer interval polynomial groupolevel one. We by varying t and u get a class of modulo ininterval polynomial groupoid of level one.

We will illustrate this by some simple examples.

Example 2.4.2: Let {T, *, (u, t) | T =i

ii 0 [0,a ]x

f

¦ ; a i

= 3 and t = 2} be the modulo integer interval polynogroupoid of level one.

Example 2.4.3: Let {T, *, (u, t) | T = ii

i 0

[0,a ]xf

¦ ; a i

= 5 and t = 7} be the modulo integer interval polynogroupoid of level one.

We can built rational interval polynomial groupoid of l

Example 2.4.4: Letf



T =ii 0

[0, a ]f-

®̄¦; *, (5, 7); a

i, 5, 7 Z+

be the integer interval polynomial groupoid of levthe integers Z + {0}.

Example 2.4.5: Let

W = ii

i 0[0,a ]x

f

-®̄¦ , *, (11, 43); a i, 11, 43

be the rational interval polynomial groupoid of leve

Example 2.4.6: Let

S =i

ii 0 [0,a ]x

f-®̄¦ , *, (47, 19); a i, 19, 47

be the real interval polynomial groupoid of levreals.

Example 2.4.7: Let

V = ii

i 0

[0,a ]xf

-®¦̄ ; *, (23, 2); 2, 23, a i

be the complex interval polynomial groupoid of levcomplex numbers.

We see each of the classes of interval polynomof level one have infinite cardinality, since pairs ofinfinite collection. We can define intervalsubgroupoids of level one built using modulo inte

i l i l i l

W ill ill hi i i b i l l



We will illustrate this situation by some simple examples.

Example 2.4.8: Let

T = ii

i 0

[0,a ]xf-

®¦̄ ; *, (5, 7), a i, 5, 7 Z10}

be a modulo integer interval polynomial groupoid of level onTake

P = ii

i 0

[0,a ]xf-

®¦̄ ; *, (5, 7), 5, 7 Z10 ; a i {0, 5} Z

P is a modulo integer interval polynomial subgroupoid of lone.

Example 2.4.9: Let

S = ii

i 0

[0,a ]xf-

®¦̄ ; *, (19, 11), a i Z+ {0}}

be a integer interval polynomial groupoid. Take

P = ii

i 0

[0,a ]xf-

®¦̄

; *, 19, 11), a i 3Z + {0}}

P is a integer interval polynomial subgroupoid of S.

Example 2.4.10: Let

T = ii

i 0

[0,a ]xf-

®¯¦ ; *, (2, 3), a i Q+ {0}}

be a rational integer polynomial groupoid.Let

if- ¦

W = i[0 a ]xf-

®¦ ; * (29 3) a i 3Z +



W ii 0

[0,a ]x®¦̄ ; , (29, 3), a i 3Z

W is a real interval polynomial subgroupoid of M.

Example 2.4.12: Let

V = ii

i 0

[0,a ]xf-

®¯¦ ;*, (3, 5), a i C+

be the complex interval polynomial groupoid.Take

T = ii

i 0

[0,a ]xf-

®¦̄ ; *, (3, 5), a i R + {

T is a complex interval polynomial subgroupoid of

Example 2.4.13: Let

P = ii

i 0

[0,a ]xf-

®¦̄ ;*, (3, 2), a i Z

be the modulo integer interval polynomial grouposimple modulo integer interval polynomial grthough Z 7 is a prime field for

S = 2ii i 7

i 0

[0,a ]x a Zf-

®¦̄ , *, (3, 2)}

is a modulo integer interval polynomial subgrou

level one.

Now we proceed onto define level two intervgroupoids using Z Z+ {0} R + {0} Q +

polynomial groupoids of level two using Z + {0} or R + {0} C + {0}



or R + {0} or C + {0}.


Example 2.4.14: Let

S = ii

i 0

[0,a ]xf-

®¦̄ ; *, (2, 9), a i, 2, 9 Z10}

be the modulo integer interval polynomial groupoid of two.

Example 2.4.15: Let

P = ii

i 0

[0,a ]xf-

®¯¦ , *, (27, 32), a i, 27, 32 Z+ {

be the integer interval polynomial groupoid of level two.

Example 2.4.16: Let

S = ii

i 0

[0,a ]xf-

®¯¦ ; a i R + {0}, *, (64, 81)}

be the real interval polynomial groupoid of level two.

Example 2.4.17: Let

W = ii

i 0

[0,a ]xf-

®¦̄ ; *, (14, 27), a i, 14, 27 Q+ {

be the rational interval polynomial groupoid of level two.

Example 2.4.18: Letf



W = { ii[0, a ] x

f

¦ , *, (27, 8), a i Z+} V



i 0¦

is the rational interval polynomial subgroupoid of level two.

Example 2.4.23: Let

S = { ii

i 0

[0, a ] xf

¦ , *, (47, 81), a i R + {0}}

be the real interval polynomial groupoid of level two.Choose

W = { 2ii

i 0

[0, a ] xf

¦ , *, (47, 81), a i R + {0}}

S is a real interval polynomial subgroupoid of S of level two

Example 2.4.24: Let

S = { ii

i 0

[0, a ]xf

¦ , *, (27, 2), a i C+ {0}}

be a complex interval polynomial groupoid of S of level two

T = { ii

i 0

[0, a ]xf

¦ , *, a i R + {0}} S,

T is a complex interval polynomial subgroupoid of S of two.

We now proceed onto describe and define level three inte polynomial groupoid.

D EFINITION 2.4.4: Let

T = { f

¦ ii[0, a ] x , *, (p, q) = d z 1, p, q, a i Z +

We can on similar lines define polynomial intervalevel three using modulo integers, rationals, reals



level three using modulo integers, rationals, reals

numbers.We will only give examples of them and theasily understand which type it belongs to by the co

Example 2.4.25: Let

S = { ii

i 0

[0, a ] xf

¦, *, (21, 6), 21, 6, a

be the modulo integer interval polynomial grouthree.

Example 2.4.26: Let

V = {i

ii 0 [0, a ] x

f

¦ , *, (24, 32), a i, 24, 32,

be the rational interval polynomial groupoid of leve

Example 2.4.27: Let

B = { ii

i 0

[0, a ] xf

¦, *, (27, 42), a i, 42, 27,

be the real interval polynomial groupoid of level th

Example 2.4.28: Let

C = { ii

i 0

[0, a ] xf

¦ , *, (2, 128), a i, 2, 128,

be the complex interval groupoid of level three.

As in case of other types of interval groupoi




T {f

i[0 ] * ( 0) 0



T = {

¦i

ii 0

[0,a ]x , *, (t, 0), 0 z t, a i

where * is defined by

( f

¦ ii

i 0

[0, a ] x ) * ( f

¦ j j

i 0

[0,b ] x )=f

¦

i j 0

[

T is a real interval polynomial groupoid of level fou As we vary t in R + we get a class of real inter groupoid of level four.

Now we can define level four interval polynomusing Z n or Z + {0} or Q + {0} or C + {0}.

We will illustrate all these situations by some exam

Example 2.4.33: Let

T = { ii

i 0

[0, a ]xf

¦ , *, (9, 0), 9, a i

be a modulo integer interval polynomial groupoid o

Example 2.4.34: Let

T = { ii

i 0

[0, a ]xf

¦ , *, (11, 0), a i, 11

be the modulo integer interval polynomial groufour.

be the real interval polynomial groupoid of level four.



Example 2.4.37: LetZ = { i

ii 0

[0, a ]xf

¦ , *, (31, 0), a i, 31 Q+ {0}

be the rational interval polynomial groupoid of level four.

Example 2.4.38: Let

S = { ii

i 0

[0, a ]xf¦ , *, (22, 0), a i, 22 C+ {0}

be the complex interval polynomial groupoid of level four.

Now having seen examples of level four interval polyn

groupoids, we can as in case of other level groupoids dinterval polynomial subgroupoids.

We only illustrate this situation by some examples.

Example 2.4.39: Let

T = { ii

i 0[0, a ]x

f

¦ , *, (12, 0), 12, a i Z24}

be the modulo integer interval polynomial groupoid of four.

Take

P = {i

ii 0 [0, a ]x

f

¦ , *, (12, 0), a i, 12{0, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22}} T; P is a modulointerval polynomial subgroupoid of T of level four.

be the modulo integer polynomial subgroupoid four.



Example 2.4.41: Let

W = { ii

i 0

[0, a ] xf

¦ , *, (9, 0), a i, 9 Z

be the integer interval polynomial groupoid of level

P = { 2ii

i 0

[0, a ] xf¦ , *, (9, 0), a i 2Z+

P is a integer interval polynomial subgroupoid ofour.

Example 2.4.42: LetS = { i

ii 0

[0, a ] xf

¦ , *, (13, 0), a i 13 Q

be the rational interval polynomial groupoid Consider

X = {i

ii 0 [0, a ] x

f

¦ , *, (13, 0), a i, 13 Z+

X is a rational interval polynomial subgroupoid ofS.

Example 2.4.43: Let

S = { iii 0

[0, a ] xf

¦ , *, (22, 0), a i , 22 R

be the real interval polynomial groupoid of level foConsider

Take

W = { 2i[0 a ] xf

* (241 0) a 241 C+ {0}}



W = { ii 0

[0, a ] x

¦, *, (241, 0), a i, 241 C {0}}

W is a complex interval polynomial subgroupoid of Z of four.

Now having defined four levels of interval polygroupoids we can define for them normal interval polynosubgroupoid, normal interval polynomial groupoid and ideainterval polynomial groupoid.

We now give some of the properties enjoyed by themsee when the interval polynomial groupoids are built uintervals of the form [0, a] from Z + {0} or Q + {0}{0} or C + {0}. We see we cannot get many nice propeOnly for modulo integer interval polynomial groupoids ware built using Z n enjoy several interesting properties.

T HEOREM 2.4.1: Let {T, *, (u, t), u, t Z n \ {0}, (uwhere

T =

f

¦i

ii 0

[0, a ] x ;

a i Z n under * is a modulo integer polynomial groupolevel two.

{T, *, (u, t); T =f

¦ ii

i 0

[0, a ] x ; a i Z n }

is a polynomial semigroup if and only if t 2 { t (mod n) u (mod n).

Y = {S =f

¦ ii

i 0

[0,a ]x , *, (u, t), a i , t,



be modulo integer interval polynomial groupoid o X is a left ideal of X if and only if P is a right ide

This proof is also left as an exercise for the reader.

Suppose we construct yet a new class of intervgroupoids using Z n say

P = { ii

i 0

[0, a ] xf

¦ ; *, (t, t), t Zn \ {0} an

be the modulo integer interval polynomial groupoi{collection of all P’s for varying t; t Zn} th

interval groupoid.The following theorem is left as an exercise for prove.

T HEOREM 2.4.3: Let

T = { f

¦i

ii 0

[0, a ] x ; *, (t, t), t, a i

be the modulo integer interval polynomial groupois a P-groupoid.

Remark 2.4.1: We have a class of n – 1 modulo i polynomial groupoids built using Z n. All these (are P-groupoids. Further as n varies in Z + we gof P-groupoids.

Proof: Given T is a modulo integer interval polyn



groupoid. To show T is an alternative groupoid we hav prove (x * y) * y = x * (y * y) for all x, y T . Now

(x * y) * y = ii

i 0

[0, a ] xf§

©̈ ¦ * j j

j 0

[0, b ] xf ·

¸¦ * j 0

[0,f§

¨ © ¦

= i ji j

j i 0

[0, (a t b t) mod p]xf

§ ·¨ ¸© ¹¦ *

j 0

[0f

¦

= 2 2 i j ji j j

j i 0

[0, (a t b t b t)(mod p)]xf

¦

x * (y * y) = ii

i 0

[0, a ]xf§ ·

¨ ¸© ¹¦ * j j

j 0

[0, b ]xf§

¨© ¦ * j

j 0

[0, bf

¦

= ii

i 0

[0, a ]xf§ ·

¨ ¸© ¹¦ [ j j j 0

[0, [tb tb ](mod p)]f

¦= 2 2 2 j i

i j j j i 0

[0, (ta t b t b )(mod p)]xf

¦

I and II can be equal if and only if t 2 = t (mod p) butis a prime and 1 < t < p, t 2 { t (mod p) is impossible.

Hence T is not an alternative interval polynomial group

Remark 2.4.2 : If f

¦ i +

T HEOREM 2.4.5: Let f



T = { f

¦ ii

i 0[0, a ] x , *, (t, t), a i Z n , n not a pr

T is an alternative modulo integer interval polynomif and only if t 2 { t (mod n). We have as in cas

some of the modulo integer interval polynomial grSmaradache groupoids.

Example 2.4.45 : Let

T = { ii

i 0

[0, a ] xf

¦ , *, (1, 5), ; a i, 1, 5

be a modulo integer interval polynomial groupoid o

TakeS = { i

ii 0

[0, a ] xf

¦ , *, (1, 5), {0, 5}

It is easily verified S is a semigroup of intervaThus T is a Smarandache modulo integer intervagroupoid of level two.

Several properties enjoyed by general gSmarandache groupoids can be derived as a matThis is left as exercise for the reader.

Next we proceed onto define the notion of ingroupoids using modulo integers, positive integ

integers, reals and complex.

2 5 Interval Matrix Groupoids



Example 2.5.10 : Let[0,1]ª º



S =

[0,12][0, 9]

[0,1]

[0, 3]

[0, 9]

« »« »« »« »« »« »« »« »¬ ¼

be a 6 u 1 modulo integer interval column matrix with efrom I

13Z .

Example 2.5.11: Let

V =

[ 1,0]

[ 7,2]

[9,12]

[ 11, 2]

[ 1,0][10,16]

[25,41]

[ 42,47]

ª º« »« »« »« » « »« »« »« »« »« »« »¬ ¼

be a 8 u 1 column integer interval matrix with entries from

Example 2.5.12: Let

Example 2.5.13: Let

[ 3, 19]ª º



S = [20,25]

[ 14, 247]« »« »« »¬ ¼

be a 3 u 1 real column interval matrix with entries

Example 2.5.14: Let

W =

[i, 2 3i]

[0, 4i]

[ 4 i,20 7i]

[ 5 7i, 0]

[21,14i 42][2i,17i]

[3 4]

[0, 5i]

[ 1, 0]

[27, 27 i]

ª º« »« »« » « » « »« »« »« »« »« »« »« »« »« »¬ ¼

be a 10 u 1 column complex interval matrix witCI.

Now we will define a usual interval matrix.

D EFINITION 2.5.4: Let V = (v ij ) where 1 d i d n u m matrix whose entries v ij are intervals from

or R I or C I . We define V to be a n u m interval m

Example 2.5.16: Let



A =

[ 3,1] [2,5] [0,7][ 7,0] [ 11,0] [1,5]

[ 3,1] [0,0] [1,1]

[2,5] [ 5,3] [37,101]

[ 7 7] [1 6] [ 7 1]

ª º« » « »« »« »« »« » ¬ ¼

be a 5 u 3 integer interval matrix with entries from Z I.

Example 2.5.17: Let

W => @

> @ > @

> @ > @

7,14 3, 11 7,03, 92 0,2 5,2

3, 41 5 / 7,1 9,91

ª ºª º ª º¬ ¼ ¬ ¼« »« »ª º ¬ ¼« »

« »ª º« »¬ ¼¬ ¼

be a 3 u 3 real interval square matrix with entries from R

Example 2.5.18: Let

[0,7] [ 3,0]

[2,4] [1,10]

[3,7] [ 3,11]

[14,19] [ 16, 247]

ª º« »« »« »« »« »

Example 2.5.19: Let



P = [2,9] [3,8] [7,9][0,7] [ 3,2] [ 7,0]ª º« » ¬ ¼

be a 2 u 3 interval matrix.

Now having defined the notion of interval mat proceed onto make them groupoids.LetP( I

nZ ) = {[0 a] / a Zn]

P( IZ {0}) = {[a, b] / b z 0, a d b a Z

P( IQ {0}) = {[a, b] / a, b z 0 Q+ {0

P(I

R {0}) = {[a, b] / a, b z 0 R + {0

P( IC {0}) = {[a, b] = [x + iy, a 1 + ib 1];x, y, a, b R +}.

We will now define operations on the interval matakes its entries from P( I

nZ ), P( IQ {0}), P(

{0}) and P( IC {0}).

D EFINITION 2.5.5: Let un

1 m Z G = {[0, a 1 ], …, [0,

i d m} denote the collection of all 1 u m interv Define a binary operation * on u

n

1 m

Z G as follows

Let A = {[0, a 1 ] , …, [0, a m ]] and B = [[0, b u

n

1 m Z G where a i , b i Z n; 1 d i d m.

Example 2.5.20: Let5

1 3ZG u = {[[0, a 1], [0, a 2], [0, a 3]]/

d ai d 3}. Choose (2, 3) = (t, u); where 2, 3 Z5. { G



u)) is a groupoid, that is it is a row interval 1 ugroupoid. Choose

A = [[0, 1], [0, 3], [0, 2]]and B = [[0, 4], [0, 1], [0, 2]]in

5

1 3ZG u .

A * B = [[0, 1], [0, 3], [0, 2]] * [[0, 4], [0, 1], [0, 2]]= [[0, 1] * [0, 4], [0, 3] * [0, 1], [0, 2] * [0, 2]]= [[0, 2+12 (mod 5)], [0, (6+3) mod 5],

[0, 4 + 6 (mod 5)]]= [[0, 4], [0, 4], [0, 0]].

It is easily verified that ‘*’ on 51 3ZG u is non associative.

Example 2.5.21: Let {3

1 4ZG u , *, (1, 2)} be a row matrix i

groupoid.

Next we can give examples of groupoids using Z + or R C+.

Example 2.5.22: LetI

1 5Z

G u = {[a 1, a2], [a 3, a4], [a 5, a6

[a9, a 10] | a i Z+ {0}, a i d ai+1 1d i d 9}. {I

1 5Z

G u , *, (3

row matrix interval groupoids using Z + {0}.

Example 2.5.23: Let 1 27Z

G u = {[[a 1, a 2], …, [a 53 , a 54]]; a


1 8R

G u = {[a 1, a2] [a 3, a4],

| { }} { 1 8 ( )}



ai+1 ;1 d i d 15 |a i R + {0}}; {I

1 8R G u , *, (7, 2)}

matrix groupoid using R + {0}.

Now we proceed onto define column matrix interva


1

I

nQ

G u =

- ª º d° « »° « » d d® « »° « »° ¬ ¼¯

#

1 2i i

3 4

i2n 1 2n

[ a ,a ]a a

[ a ,a ]1 i 2

a Q[ a ,a ]

be a column interval matrix using Q + {0}. Define * on u

I

n 1Q

G as

A * B = {[ai a i+1 ]} * {[b i , b i+1 ]}= {[ta i + ub i , ta i+1 + ub i+1 ]}

= tA + uB;t z u, (t, u) = 1, t, u Z + . { u

I

n 1Q

G , *, (t, u)} is a

matrix groupoid (column matrix interval groupoid)


Example 2.5.26: Let

1 2[a ,a ]- ª º°« »

{I

6 1Z

G u , *, (5, 19)} with binary operation *, that is for A,

I

6 1Z

G u A* B = 5A + 19B.



I

Example 2.5.27: Let (I

4 1C

G u , *, (7, 1)) where

I

4 1C

G u =

1 2i i 1

3 4

5 6

i7 8

[a ,a ]a a

[a ,a ] 1 i 8[a ,a ]

a C {0}[a ,a ]

- ½ª º d° °« »° °« » d d® ¾« »° ° « »° °¬ ¼¯ ¿

be a column interval matrix groupoid built using C +

Example 2.5.28: Let

3

3 1ZG u =

i i 11 2

3 4

5 6 i 3

a a[a ,a ]

[a ,a ] 1 i 5

[a ,a ] a Z

- ½dª º° °« » d d® ¾« »° °« »¬ ¼¯ ¿

be the collection of all matrix column groupoids with * on

A * B = 2A + B (mod 3).

D EFINITION 2.5.7: Let u

I

m nQG = {A = {[

1 2ij ija ,a ] mu n be

interval matrix; 1 2ij ija ,a Q+ {0}; d1 2

ij ija a ; 1 d i d m;

Define * on um nG by A*B = tA + uB; t u Q+ ; th

Example 2.5.29: Let

I3 2

Gu

i1 2 3 4

5 6 7 8

a[a , a ] [a ,a ]

[ ] [ ] 1

- dª º°

d®



I10ZG = 5 6 7 8

9 10 11 12 i

[a ,a ] [a ,a ] 1 a[a ,a ] [a ,a ] a

« » d ® « »° « »¬ ¼¯ be a 3 u 2 interval matrix using the groupoid Zis a 3 u 2 interval matrix groupoid using Z 10.

Example 2.5.30: Let

I

5 3Z

G u =

1 1 1 1 1 1 i1 2 3 4 5 6 j2 2 2 2 2 21 2 3 4 5 63 3 3 3 3 31 2 3 4 5 6

[a ,a ] [a ,a ] [a ,a ] a

[a ,a ] [a ,a ] [a ,a ] 1[a ,a ] [a ,a ] [a ,a ] 1

- ª º° « »

d® « »° « » d¬ ¼¯

i i j j 1a ad , j =1, 2, …, 6} be a 5 u 3 interval ma

from Z + {0} or IZ . {I

5 3Z

G u , *, (3, 19)} is

matrix groupoid using entries from Z + {0}.

Example 2.5.31: Let

I

1 1 1 1 1 1 1 11 2 3 4 5 6 7 82 2 2 2 2 2 2 22 511 12 13 14 15 16 17 18R

i i i j I j j 1

[a , a ] [a ,a ] [a ,a ] [a ,a ]

[a ,a ] [a ,a ] [a ,a ] [a ,a ]G

a R {0};a a ;1 i 2;1

u

- ª ° « ° ¬ ® ° d d d ° ̄

be a 2 u 5 interval matrix built using R + {0}

i 2 u 5 i l i id i R

be a 4 u 3 interval matrix with entries from IQ . { G

19)} is a interval matrix groupoid with entries from Q



19)} is a interval matrix groupoid with entries from IQ

If m = n then we see the interval groupoid is a square integroupoid.


Example 2.5.33: Let

I

3 3ZGu

=

i j1 1 1 1 1 1

1 2 3 4 5 6 i i2 2 2 2 2 2 j j 11 2 3 4 5 63 3 3 3 3 31 2 3 4 5 6

a Z {[a , a ] [a ,a ] [a ,a ]

a a[a ,a ] [a ,a ] [a ,a ] 1 i 3[a ,a ] [a ,a ] [a ,a ]

1 j 5

- ° ª º

d° « »® « » d d° « »¬ ¼° d d¯

be a 3 u 3 interval matrix with entries from IZ . { G

13)} is a interval groupoid matrix with entries from IZ

Example 2.5.34: Let

I8

5 5Z

G u =

1 1 1 1 1 1 1 1 11 2 3 4 5 6 7 8 92 2 2 2 2 2 2 2 21 2 3 4 5 6 7 8 9

3 3 3 3 3 3 3 3 31 2 3 4 5 6 7 8 9 1

4 4 4 4 4 4 4 4 4

[a ,a ] [a ,a ] [a ,a ] [a ,a ] [a ,a

[a ,a ] [a ,a ] [a ,a ] [a ,a ] [a ,a[a ,a ] [a ,a ] [a ,a ] [a ,a ] [a ,a

[a a ] [a a ] [a a ] [a a ] [a a

- ª ° « ° « ° « ° « ® «°




1 2R

G u = {[a 1, a2], [a 3, a 4] a

ai+1 ; 1 d i d 3, *, (31, 43)} be a row matrix int

built using IR



built using IR .Consider P = {([a 1, a2], [a 3, a4]); a i d ai+1 ,

43); a i Q+ {0}}I

1 2R

G u ; P is a row

subgroupoid of I

1 2R

G u .

Consider T = {([a 1, a 2], [a 3, a 4]); a i Z+

1, 2, 3; *, (31, 43)}I

1 2R

G u ; T is also a row

subgroupoid of I

1 2R

G u . Infact it is easy to verify T

On similar lines we can define interval matrix sub

case of column matrix interval groupoid and matrix groupoid.

We will leave this easy concept to the reasubstantiate this by some examples.

Example 2.5.41: Let

I20

3 1Z

G u =1 2

3 4

5 6

[a ,a ]

[a ,a ]

[a ,a ]

- ª º°« »®« »°« »¬ ¼¯

ai Z20; a i d ai+1 ;

be a column interval matrix using I20Z . G = {

l i t l t i id

T =1 2

3 4 i 20

5 6

[a ,a ]

[a ,a ] a {0,2,4,6,...,18 Z

[a ,a ]

- ª º°« » ®« »°« »¬ ¼¯

, *, (5, 7



5 6[a ,a ]« »¬ ¼ai d ai+1 ; 1 d i d 5} { I

20

3 1Z

G u , *, (5, 7)} = G; T is a

interval matrix subgroupoid of G.

Example 2.5.42: Consider G = {I

6 1

QG u , *, (3, 17)} where

I

6 1Q

G u =

1 2

3 4i

5 6i i 1

7 8

9 10

11 12

[a ,a ]

[a ,a ]a Q {0};

[a ,a ]a a ;

[a ,a ]1 i 11[a ,a ]

[a ,a ]

- ½ª º° °« »° °« » ° °« »° °d« »® ¾« »° °

d d« »° °« »° °« »° °¬ ¼¯ ¿

.

G is a column interval matrix groupoid built using IQ .Take

P =

> @

> @

> @

1 2

i3 4

i i 1

5 6

a ,a[0,0]

a Z {0};a ,a

a a ;[0,0]

1 i 5a ,a

[0,0]

- ½ª º° °« »° °« » ° °« »° °d« »® ¾« »° °d d« »° °

« »° °« »° °¬ ¼¯ ¿

G;

P i l i l i b id f G

L is a column interval matrix subgroupoid of G.

Example 2.5.43: Consider [a a ]- ½ª º



G =I

10 1R

G u =

1 2i i 1

3 4

i19 20

[a ,a ]a a ;

[a ,a ]1 i 19;

a R {0}[a ,a ]

½ d° °« »° °« » d d® « »° ° « »° °¬ ¼¯ ¿

#

be a column interval matrix groupoid.Let

P =

1 2i i 1

3 4

i19 20

[a ,a ]a a ;

[a ,a ]1 i 19;

a Q {0}[a ,a ]

- ½ª º d° °« »° °« » d d® ¾« »° ° « »° °¬ ¼¯ ¿

#; *, (13

P is a column interval matrix subgroupoid of G.

Example 2.5.44: LetG = 1 2

i3 4

[a ,a ]a C;i 1,2,3,

[a ,a ]

- ª º° ®« »¬ ¼°̄

are proper intervals, *, (3, 5)} be a column in

groupoid.Take

T = 1 2[a ,a ]a a C;

- ª º°®« » * (3 5)}

be a 3 u 2 interval matrix groupoid built using I36Z .

Consider Z[ ] [ ]ª º



P =i 36;1 2 3 4

7 8 i i 1

9 10

a Z ;[a , a ] [a ,a ]

[0,0] [a ,a ] a a ;

[a ,a ] [0,0] 1 i 7;

- ª º°« » d®« »°« » d d¬ ¼¯

*, (3, 11)} G}, P is a 3 u 2 interval matrix subgroupoid

Example 2.5.46: Let

G = 1 2 3 4

5 6 7 8

[a ,a ] [a ,a ]

[a ,a ] [a ,a ]

- ª º°®« »¬ ¼°̄

i i i 1a Q {0};a a ;1 i 7; d d d *, (19, 7)} be a 2 umatrix groupoid built using IQ .Take

P = 1 2 3 4

5 6 7 8

[a , a ] [a ,a ][a ,a ] [a ,a ]

- ª º°®« »¬ ¼°̄

i i i 1a Q {0};a a ;1 i 7 d d d , *, (19, 7)} G, P is interval matrix subgroupoid of G.

Example 2.5.47: Let

S =1 2 5 6

7 8 9 10 1

3 4 1

[a ,a ] [a ,a ] [0,0] [0,0

[0,0] [a ,a ] [a ,a ] [a

[a ,a ] [0,0] [0,0] [a

- ª °« ®« °« ¬ ¯



i i i 1a Z {0};a a ;1 i 13 d d d , *, (23, 2)}interval matrix subgroupoid of G.

Now we define some of the properties enjoyed by tmatrix groupoids.

D EFINITION 2.5.9: Let G be a interval matrix using Z n or I Z or I Q or I R or I C ).

subgroupoid V of G is said to be a normal in subgroupoid or interval matrix normal subgroupoid

(a) aV = Va(b) (Vx)y = V(xy)(c) y (xV) = (yx) V

for all x, y, a V.

If a interval matrix groupoid G has no innormal subgroupoid then we define G to be a simatrix groupoid; we say an interval matrix grouponormal if

(a) x G = Gx(b) G (xy) = (Gx)y(c) y (xG) = (yx)G

for all x, y G.

P =[a,b]

[c,d]

- ª º°®« »¬ ¼°̄

a, b, c, d {0, 4, 8}, *, (4, 8)}.

P is a normal interval matrix subgroupoid of G.



Inview of this we have the following theorem.

T HEOREM 2.5.1: Let G = u I n

m p Z

G = {all m u p interval

with entries from I n Z }, *, (t, u) such that t + u = n, (t, u) =and n even} be a m u p matrix interval groupoid with 1f }.

Then G has matrix interval normal subgroupoid.

The proof uses simple number theoretic techniques.

Example 2.5.49: Let G = {([a, b], [c, d], [e, f]) | a, b, c, d, Z10, *, (8, 4)} be a interval matrix groupoid. Take P = {([a[c, d], [e, f]) | a, b, c, d, e, f {0, 4, 2, 8, 6}}, *, (8, 4)}matrix interval subgroupoid of G but need not be a int

matrix normal subgroupoid of G as 4/10 and 4 + 8 z 10D EFINITION 2.5.10: Let G be a interval matrix groupoiusing I

n Z or I Z or I Q or I R . G is said to be a interval P-groupoid if (AB) A = A (BA) for all A, B G.

Example 2.5.50: Let G = {([x, y] [a, b], [c, d]) | x, y, a, bZn; *, (t, t)} be a 1 u 3 row matrix interval groupoid;

row interval matrix P-groupoid.

= ([tx 1 + ta 1, ty 1 + tb 1], [tx 2 + ta 2, tyty3 + tb 3]) * ([a 1 b1], [a 2 b2], [a 3, b3

= ([t 2x1 + t 2a1 + ta 1, t2y1, t2 b1 + tb 1],2 2 2 2



[t2x2 + t 2a2 + ta 2, t2y2 + t 2 b2 + tb 2],

[t2 x3+ t 2a3+ta 3, t2y3 + t 2 b3 + tb 3]) Consider

A* (B*A) = ([a 1, b1] [a 2, b2], [a 3, b3]) *([x 1, y1], [x 2, y2], [x 3, y3]) *([a1, b1], [a 2, b 2], [a 3, b3])

= ([a 1, b 1] [a 2, b 2], [a 3, b 3]) * ([tx[tx 2 + ta 2, ty 2+tb 2], [tx 3+ta 3, ty 3

= ([ta 1 + t 2x1 + t 2a1, tb 1 + t 2y1 + t 2

[ta2 + tx 2 + t 2a2, tb 2 + t 2y2 + t 2 b[ta3 + t 2x3 + t 2a3, tb 3 + t 2y3 + t 2 b

It is easily verified that (A*B)*A = A* (B*A)G.

Thus G is a interval matrix P-groupoid.



m p Z

G = {{all m u p

with entries from I n Z }, *, (t, t), t Z n \ {0}} b

matrix groupoid built using I

n Z . Clearly G isinterval P-groupoid.

D 2 5 11 L G b i l

Example 2.5.51: Let

[a,b][c d]

-ª º°« »°



G =[c,d]

[e,f]

[g,h]

« »®« »°« »°¬ ¼¯

a, b, c, d, e, f, g, h Z5, *, (3, 3)}

be a 4 u 1 row interval matrix groupoid built using I5Zan alternative 3 u 1 row interval matrix groupoid. Let

A =

1 1

2 2

3 3

4 4

[a ,b ]

[a ,b ]

[a ,b ][a ,b ]

ª º« »« »« »« »¬ ¼

, B =

1 1

2 2

3 3

4 4

[x , y ]

[x , y ]

[x , y ][x , y ]

ª º« »« »« »« »¬ ¼

be in G.Consider

(A * B) * B =

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

[3a 3x ,3b 3y ][3a 3x ,3b 3y ]

[3a 3x ,3b 3y ]

[3a 3x ,3b 3y ]

ª º« » « »« » « » ¬ ¼

*

1 1

2

3

4

[x , y[x , y

[x , y

[x , y

ª « « « « ¬

=

1 1 1 1 1 1

2 2 2 2 2 2

[9a 9x 3x ,9b 9y 3y ][9a 9x 3x ,9b 9y 3y ]

[9a 9x 3x 9b 9y 3y ]

ª º« » « »« »

Now consider

A ( )

1 1

2 2

[a ,b ]

[a ,b ]ª º« »« »

1

2

[6x ,6

[6x ,6ª « «



A* (B * B) = 2 2

3 3

4 4

,

[a ,b ]

[a ,b ]

« »« »« »¬ ¼

* 2

3

4

,

[6x ,6

[6x ,6

« « « ¬

=

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

[3a 18x ,3b 18y ][3a 18x ,3b 18y ]

[3a 18x ,3b 18y ]

[3a 18x ,3b 18y ]

ª º« » « »« » « » ¬ ¼

=

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

[3a 3x ,3b 3y ][3a 3x ,3b 3y ]

[3a 3x ,3b 3y ]

[3a 3x ,3b 3y ]

ª º« » « »« » « » ¬ ¼

I and II are not equal. Thus G is not a 4 ualternative groupoid.

In view of this we have the following theorem easily proved using simple number theoretic techni

T HEOREM 2.5.3: Let G = u I p

m p Z

G = {all colle

interval matrices with entries from I p Z , *, (t, t)

Example 2.5.52: Let G = I6

7 3Z

G u = {all 7 u 3 interval

with entries from I6Z , *, (3, 3)} be a 7 u 3 interva

alternative groupoid as 3 2 { 3 (mod 6).




m p Z

G = {all m u p interval ma

entries from I n Z , * (t, 0), t Z n \ {0}} be a m u p matrix

groupoid. G is a P-groupoid and alternative groupoid if only if t 2 { t (mod n).

Example 2.5.53: Let G = {3 u 8 interval matrices withfrom I

12Z , *, (4, 0)} be a 3 u 8 interval matrix groupoidclearly a interval matrix P-groupoid and interval m

alternative groupoid.

2.6 Smarandache Interval Groupoid

As in case of usual groupoids we can in case of m

interval groupoids also define the concept of Smarandgroupoids.We will illustrate this concept.

Example 2.6.1: Let G = { I10

3 8Z

G u , *, (1, 5)} be a 3 u

interval groupoid. Take P = {3 u 8 interval matrices from{0, 5}, *, (1, 5)} G is easily verified to be a 3 uinterval semigroup. Thus G is a Smarandache 3 ui l id 3 u 8 i l i S

Proof: For every m Zn is such that m * m =Thus every singleton is a semigroup. Hence the clai

We have yet another new result about Smarandmatrix groupoid using I

nZ .



T HEOREM 2.6.2: Let G = { u I n

m p Z

G ; *, (t, u) wher

n)} be a m u p interval matrix groupoid. G is ainterval Smarandache P-groupoid if and only if tand u 2 = u (mod n).

Proof: It is already proved G is a m u pSmarandache groupoid if t + u { 1 (mod n). Nowm u p matrix interval Smarandache P-groupoid if

= t (mod n) and u2

= u (mod n).Let A m u p = ([a ij, a i+1j+1 ]) and B m u p = ([b ij, d m – 1 and 1 d j d p – 1. To show G is a m uS-P-groupoid (Smarandache-P-groupoid) we have B) * A = A * (B * A) for all A, B G.Consider

(A * B) * A = [([a ij, a i+ij+1 ]) * ([b ij, b i+ij+1 ])= [(ta ij + ub ij) (mod n), (ta i+1j+1 + ub i+1j+1 ) (mo= [(t 2aij + tub ij + ua ij) (mod n), (t 2ai+1j+1 + tub i

Consider A * (B * A) = ([a ij, a i+ij+1 ]) * ([(tb ij + ua ij)

tb i+1j+1 + ua i+1j+1 ) mod n])= [(ta ij + utb ij + u 2 aij) mod n(ta i+1j+1 + utb i+1j+1 + u 2ai+1j+1


m p Z

G , *, (t, u); t, u Z n \

+ u { 1 (mod n)} be a m u p matrix interval groupoid. G

u p matrix interval Smarandache alternative groupoid (mmatrix interval S-alternative groupoid) if and only if t 2



matrix interval S alternative groupoid) if and only if t n) and u 2 { u (mod n).

The proof is simple and can be obtained by using numtheoretic techniques. For definition of different typeSmarandache groupoid using interval matrices please refer 21]. Thus using those definition in [20] we will prove rerelated with them.


3 1Z

G u , *, (3, 9)} be a 3 u

interval matrix groupoid built usingI12Z . G is a column

matrix Smarandache Moufang groupoid.It can be easily verified for all A, B, C G; (A * B

A) = (A * (B*C)) * A is true.

Likewise we give an example of a matrix interval SmarandaBol groupoid.


5 7Z

G u , *, (2, 3)} be a 5 uinterval groupoid. It is easily verified G is a Smarandachegroupoid. For A, B, C in P we have ((A*B)*C) * B =

[(B*C)*B)] where P = {All 5 u 7 interval matrices withfrom the subset {0, 2} Z4, *, (2, 3)} G. Howidentity is not true for all elements in G.


q p Z

G , *, (m, m); m

{ 1(mod n) and m 2 { m(modn)} be a q u

groupoid};1. G is a q u p matrix interval Smarandache strong



q p g2. G is q u p matrix interval Smarandache idempot3. G is a q u p matrix interval Smarandache strong 4. G is a q u p matrix interval Smarandache st

groupoid.5. G is a q u p matrix interval Smarandache stro

groupoid.

Note: q and p are positive finite integers 1 < p, q <also occur.

The proof is left as an exercise for the reader.

T HEOREM 2.6.5: Let G = { u I 2 m

p q Z

G , *, (2, 0), 2

interval matrix groupoid built using intervals from p u q interval matrix S-groupoid.

Proof: Follows from the fact when intervals {[0, 0m]]} I

2mZ are taken as entries of the p u q incollection is a p u q interval matrix semigroperation *. Hence the claim.

T HEOREM 2.6.6: Let p u q matrix interval{ u

I n

p q Z

G , *, (m, 0)} built using I n Z , n = 2m is a S

q matrix interval groupoid



T HEOREM 2.7.6: The subclass of groupoids C (( n I Z , m

/ t+u { 1 (mod n), u 2 = u (mod n), t 2 = t (mod n)) C (

(t, u)) are matrix interval Smarandache alternative groupoid



T HEOREM 2.7.7: The subclass of groupoids C ( n I Z , mu

/ t + u { 1 (mod n), u 2 = u (mod n), t 2 = t (mod n)) C s, (t, u)) are m u s matrix interval Smarandache stron

groupoids.T HEOREM 2.7.8: The subclass of groupoids C ( n

I Z , m

/ t + u { 1(mod n), u 2 = u(mod n), t 2 = t(mod n)) C ( Z(t, u)) are m u s matrix interval Smarandache strong Ma

groupoid.

The above theorems can be easily derived usingdefinition and simple number theoretic techniques.

Now we proceed onto define classes of groupoids usin{0} etc.

C(n

IZ , m u n, (p, q) = {(collection of all m umatrices with entries from IZ together with * binary op

onI

m nZ

G u such that for A, BI

m nZ

G u , A * B = pA + qB, w

q Z+ and p, q vary over Z + to give the class of m uinterval groupoids using IZ }. Clearly this is an

collection. Likewise C ( IQ , m u n, (p, q)) p, q Q+

class of m u n matrix interval groupoid of infinite cardinaliSi il l C( R u ( )) R+ d C ( C

Chapter Three



ON SOME NEW CLASSES OFNEUTROSOPHIC GROUPOIDS

In this chapter we for the first time introduce the neneutrosophic groupoids. The algebraic strneutrosophic groups, neutrosophic semigroups, rings, neutrosophic fields and neutrosophic vectorbeen introduced by the authors It is pertinent to me

Neutrosophic interval matrix groupoids are introducsection six and neutrosophic interval polynomial groupoidsection seven are described.



3.1 Neutrosophic Groupoids

We now proceed on to define a neutrosophic groupoid illustrate it by some examples.

D EFINITION 3.1.1: Let S be a neutrosophic set which empty. Let * be a closed binary operation on S such tha(b*c) z (a * b) * c for some a, b, c S. We call (S, *neutrosophic groupoid.

It is interesting to note that all neutrosophic semigroupsneutrosophic groupoids but a neutrosophic groupoid in genis not a neutrosophic semigroup. Thus the class oneutrosophic semigroups is contained in the clasneutrosophic groupoids.

We will illustrate this new structure by some examples.

Example 3.1.1: Let {Z I} = N(Z) = {a + bI | a, b Zthe binary operation subtraction ‘–’ on N(Z). {N(Z), –} neutrosophic groupoid.

It is easily verified ‘–’ operation on N(Z) is non associaThus (N(Z), –) is not a neutrosophic semigroup as ‘–’ operaon N(Z) is not associative.

and B = (b ij) are in G. A * B = (2a ij + 3b ij). It(G, *) is a neutrosophic groupoid as ‘*’ is aassociative binary operation on G.

Now having seen some examples of neutrosophic g



now proceed onto define the cardinality of thstructures.

D EFINITION 3.1.2: Let (G, *) be a neutrosophic gnumber of distinct elements in G is finite, then

finite cardinality or finite order and we denote it bG has infinite number of elements then we say cardinality and is denoted by |G| = f or o(G) =

We see neutrosophic groupoids given in exampl3.1.3 are of infinite order and the neutrosophic grin example 3.1.2 is of finite order.

D EFINITION 3.1.3: Let (S, *) be a neutrosSuppose H S be a proper subset of S and if (Hneutrosophic groupoid then we call (H, *) to be a

subgroupoid of (S, *).


Example 3.1.4: Let S = {N(Z), *} be a neutrosowhere for any x, y N(Z); x * y = 5x + 3y. Taka, b Z+}; (H, *) is a neutrosophic subgroupoid of

Example 3 1 5: Let S = {N(Z ) * where a * b

D EFINITION 3.1.4: Let (S, *) be a neutrosophic groupoid. H such that (P, *) is a groupoid but is not a neutroso groupoid then we call (P, *) to be a pseudo neutroso subgroupoid of (S, *).

ll f l f h d f



We will first give some examples of this definition.

Example 3.1.6: Let {N(Z 12), *} = G be a neutrosophic grwhere ‘*’ is such that x * y = 3x + 7y (where ‘+’ is addmodulo 12}; for all x, y N(Z 12). Take H = {Z 12, *} is a groupoid and is not a neutrosophic groupoid. So H

pseudo neutrosophic subgroupoid of G.

Example 3.1.7: Let V = {N(Z), * where * is such that a * b+ 9b for a, b N(Z)}. V is a neutrosophic groupoid.

Take W = {Z, *} V = {N(Z), *}; W is a neutrosophic subgroupoid of V.

D EFINITION 3.1.5: Let (V, *) be a neutrosophic groupiohas no neutrosophic subgroupoid; then we call V to neutrosophic simple groupoid or simple neutrosophic group

Example 3.1.8: Let V = {Z 7I, *; a * b = 2a + 3b(mod 7neutrosophic groupoid. Clearly V is a simple neutrosogroupoid.

Example 3.1.9: Let V = {Z 3I, *; where a * b = 2a + b(mo

* 0 I 2I0 0 I 2I

D EFINITION 3.1.6: Let (V, *) be a neutrosophichas no nontrivial pseudo neutrosophic subgroupdefine (V, *) to be a pseudo simple neutrosophic gr

W ill ill hi b l




Example 3.1.10: Let V = {QI, *} be a neutrosoClearly V is a pseudo simple neutrosophic groupoidnontrivial pseudo neutrosophic subgroupoids.

Note: If {0} V are call {0} to be a trivial pseudsubgroupoid.

Example 3.1.11: Let V = {Z 11I, *, where * is de b Z11I as a * b = 3a + 5b(mod 11)}, V is agroupoid which is a simple pseudo neutrosophic ghas no nontrivial pseudo neutrosophic subgroupoid

D EFINITION 3.1.7: Let (V, *) be a neutrosophic gboth a simple neutrosophic groupoid as well as pneutrosophic groupoid then we call (V, *) to be a dneutrosophic groupoid.


Example 3.1.12: Let V = {Z 7I, *} where for a, b+ 2b (mod 7). V is a doubly simple neutrosophic gr

Example 3 1 13: Let V = {ZI * where a * b = 8a

doubly simple neutrosophic groupoids as {Z pI, *} {is a nontrivial neutrosophic subgroupoid of V and {Z{N(Z p) *} is a nontrivial pseudo neutrosophic subgroupoid Hence the claim.

H i d fi d b f hi id



Having defined substructures of a neutrosophic groupoidnow proceed onto define the notion of neutrosophic grouhomomorphisms.

D EFINITION 3.1.8: Let (G, *) and (H, o) be two neutro groupoids. A map K : G o H is said to be a neutro groupoid homomorphism if the following conditions hold go

(i) K (I) = I (ii) K (a * b) = K (a) o K (b) for all a, b G.

The interested reader is expected to give examples of tIt is important to mention that I should be mapped only onand no other element can be substituted as I stands forindeterminacy.

Another factor which is to be noted in case of algebraic structures is that neutrosophic groupoids may or not have identity. An element e in a neutrosophic groupoid sthat e * x = x * e = x for all x G is called the identity. 0 G it does not in general imply 0 * x = x * 0 = x for allG. Thus with these special properties it is not always possib

define kernel of the homomorphism.However we can define as in case of other algestructures isomorphism. A homomorphism which is one to

d t i d fi d i hi

Having defined the notion of neutrosophomomorphism we can now proceed onto definesubstructure.

D EFINITION 3 1 9: Let (G *) be a neutrosophic g



D EFINITION 3.1.9: Let (G, *) be a neutrosophic g G be a proper subset of G. We say H is a neutrof G if the following conditions are satisfied;

i. (H, *) is a subgroupoid ii. For all g G and h H, g * h and h


Example 3.1.15: Let G = {0, I, 2I, …, 15I} = Zon G by a * b = 2a + 3b (mod 16I) for a, bneutrosophic groupoid. Take H = {4I, 8I, 0, 12I}neutrosophic subgroupoid of (G, *).

It is easily verified that H is not a neutrosophicneutrosophic groupoid G. It is easily verifieneutrosophic ideals.

Example 3.1.16: Let G = Z 8I = {0, I, 2I, 3I,define * on G by a * b = 2a + 4b (mod 8I) for a, ba neutrosophic groupoid. Take {0, 2I, 4I, 6I} = Ha neutrosophic subgroupoid of (G, *).

It is easily verified that (H, *) is a neutrosophic ideaThus we see some neutrosophic group

Can we have non trivial ideally simple neutrosophic groupoWe explicitly show a class of such neutrosophic groupoids.

T HEOREM 3.1.1: Let G = n2 Z I = {0, I, 2I, …, 2 n I – I}integers 2 n I, n t 2}. Define a operation * on G by a * b =



nb (mod 2 n I) for all a, b G and m + n = a prime in G an n2 Z , (G, *) is a ideally simple neutrosophic groupoid.

Proof: Given G is a neutrosophic groupoid of a special orden t 2. Now the operation on G is also such that a * b = ma (mod 2 n) for all a, b G with m + n = a prime in G n G.

Let H be any proper neutrosophic subgroupoid of G. element in H is of the form {2 sI | s = 1, 2, …, 2 n}. Let msuch that m + n = prime.

Now for any odd prime a G and b H we must b to belong to H.

Thus ma + nb H as ma + mb z 2sI for any s, s = 2n. For if m + n = prime at least one of m or n must be oform p.q where q or p is an odd neutrosophic prime such ultimately m + n = a neutrosophic prime, Z and Z < 2 n

This is not possible. Hence G = n2Z I is ideally

neutrosophic groupoid.

Note: When we say n is a neutrosophic prime; n = pI where

a prime. Likewise when we say pure neutrosophic mointegers we mean Z nI = {0, I, 2I, …, (nI – I) = (n – 1)I}Z3I = {0, I, 2I}; 2I + I = 3I = 0 (mod 3I).



We see from the tables that Z 3I (I, 2I) is not the sam(2I, I). However it is important to note that we can have two neutrosophic groupoids built over Z 3I.

Now consider the mixed neutrosophic modulo int N(Z 3) = {a + bI / a, b Z3} = {0, 1, 2, I, 2I, 1 + I, 1 + 2I, 2 + I}. We will study how many neutrosophic groupoids can



2 I}. We will study how many neutrosophic groupoids canconstructed using N(Z 3).

Example 3.2.2: Let N(Z 3) be the mixed neutrosophic imodulo Z 3.

Define * on N(Z 3) as follows:a * b = 1.a + (1 + I)b

for all a, b N(Z 23).The table for this neutrosophic groupoid is given in

following.

* 0 1 2 I 2I 1+I 2+I 1+20 0 1+ I 2+2I 2I I 1 2+I 1+21 1 2+I 2I 1+2I 1+I 2 I 2+22 2 I 1+2I 2+2I 2+I 0 1+I 2II I 1+2I 2 0 2I 1+I 2+2I 1

2I 2I 1 2+I I 0 2I+1 2 1+1+I 1+I 2+2I 0 1 1+2I 2+I 2I 2 2+I 2+I 2I 1 2 2+2I I 1+2I 0

1+2I 1+2I 2 I 1+I 1 2+2I 0 2+I2+2I 2+2I 0 1+I 2+I 2 2I 1 I

It is easily verified P = (N(Z 3), (1, 1 + I), *) iassociative structure. Thus P is a neutrosophic groupoid weusing N(Z 3) we can construct 56 number of neutros

In view of this we have the following resultneutrosophic groupoid to be pure if it does not

element x where x z 0 and x has no I present with*), x, y ZnI is a pure neutrosophic groupoid. W(x y) *); x y N(Z n) to be a neutrosophic gro



(x, y), ); x, y N(Z n) to be a neutrosophic gronot pure.

The following theorems can be easily proved by an

T HEOREM 3.2.1: Let (Z n I, (x, y), *) be any pu groupoid of order n. There exists exactly 2C n

groupoids of order n built using Z n I.

T HEOREM 3.2.2: Let (N(Z n ), (x, y), *) be a groupoid (which is not pure) built using N(Z)2 2n 1

C 2 ) number of such groupoids of order n 2

Let P = {0, 2I} be a left ideal of Z 4I(u, v) whereClearly P = {0, 2I} is a right ideal of Z 4I (v, u).

In view of this we have the following theorem which is straight forward.

T HEOREM 3.2.3: P is a left neutrosophic ideal of only if P is a right neutrosophic ideal of Z n I(v

{0}.Suppose P is a left neutrosophic ideal of (N(Z y N(Z n ) \ {0} then P is a right neutrosophic idea

T HEOREM 3.2.5: No neutrosophic groupoid (Z n I, [N(Z n ), (t, u), *)] has {0} to be an ideal.

Proof: Follows by the very operation in Z nI or (N(Z n)).

T HEOREM 3.2.6: The neutrosophic groupoid Z n I (t,



p g p (idempotent groupoid if and only if t + u { I (mod n).

Proof: Recall a neutrosophic groupoid Z nI (u, idempotent neutrosophic groupoid if and only if a * an I) for all a ZnI (u, v).

Now a * a = at + ua { (t + u) a { aI (mod n I) as t(mod nI). This is possible if and only if t + u { I (mod n


Example 3.2.4: Let Z 6I (2I, 5I) be a pure neutrgroupoid.

Take 0 * 0 = 0 (trivial). Consider I * I = 2I.I * 5I u I = 2I + 5I

= I (mod 6I).

2I * 2I = 2I * 2I + 5I u 2I= 4I + 10I= 2I (mod 6I).

So2I * 2I = 2I3I * 3I = 3I u 2I + 3I u 5I

6I 15I

T HEOREM 3.2.7: Let Z n I (x, y) be a neutrosophiis even there exists even number of neutrosoph

groupoids. If n is odd then there exists an oneutrosophic idempotent groupoids.



This can be proved using simple number theoretic tConsider the neutrosophic groupoid Z 6I (3

(3I, 4I) is a neutrosophic semigroup.

For (a * b) * c = (3Ia + 4Ib) * c

= (3Ia + 4Ib) 3I + 4I c= 3I a + 4Ic.

a * (b * c) = a * [3Ib + 4Ic]= 3aI + (3Ib + 4Ic) 4I

= 3Ia + 4Ic.So (a * b) * c = a * (b * c) for all a, b, c Z6I.

Thus Z 6I (3I, 4I) is a only neutrosophic semi(4I, 3I) is also a neutrosophic semigroup and not a groupoid as the operation * is associative. We s

4I(mod 6I) and 3I * 3I = 3I(mod 6I).

In view of this we have in neutrosophic groupoids tresult.

T HEOREM 3.2.8: Let Z n I (u, t) be a neutrosophic= t (mod nI) and u 2 = u (mod nI) then Z n I (u, t) neutrosophic semigroups.

Recall as in case of groupoids. We say a neutrosophic groupG is normal if

i. xG = G x

ii. G (xy) = (Gx)yiii. x(yG) = xy G for all x, y G.



Let G be a neutrosophic groupoid H and K be any proper neutrosophic subgroupoids of G with H K =H is conjugate with K if there exists x G such that H Kx.

Several properties enjoyed by groupoids built using Zwill be true for neutrosophic groupoids Z nI (u, v).

However for this new class of neutrosophic grou(N(Z n), (u, v), *) we do not know whether all propertiederivable. However every neutrosophic groupoid Z n

(N(Z n) (u, v), *).

Now we will define Smarandache neutrosophic groupoid.

D EFINITION 3.2.2: Let G be a neutrosophic groupoid undoperation *. If H G; H a proper subset of G is such th*) is a neutrosophic semigroup then we call G to Smarandache neutrosophic groupoid (S-neutroso

groupoid).

It is left as an exercise for the reader to prove that eneutrosophic subgroupoid of a Smarandache neutrosogroupoid need not in general be a Smarandache neutrososubgroupoid.

We can build yet a new class of neutrosophic group

D EFINITION 3.2.3: Let Z n I = {0, I, 2I, …, (n-1)I Define * a closed binary operation on Z n I as fob Z n I define a * b = ta + ub; (t, u) = I (t z u)

We see (Z n I, (t, u), *) is a neutrosophic group prime special neutrosophic groupoid.



Take for example the set Z 6I.

Example 3.2.5: (Z6I, (3I, 5I), *) is a prime specigroupoid. (Z 6I, (2I, 4I), *) is not a prime speciagroupoid. (Z 6I (I, 4I), *) is also a prime speciagroupoid (Z 6I, (2I, 3I), *) is also a prime speciagroupoid. (Z 6I, (2I, 5I), *) is a prime speciagroupoid. Thus we have using Z 6I, 18neutrosophic groupoids.

Further we see the class of prime specialgroupoids built using Z nI is contained inneutrosophic groupoids built using Z nI.

It is an interesting and important problem to studyof prime special neutrosophic groupoids built using

Now {Z nI, (t, u), *} is a non prime speciagroupoid if (t, u) = rI if t, u ZnI; (t, u) = r if tZn. We can also derive almost all properties for the

prime special neutrosophic groupoids.

Example 3.2.6: Let (Z 8I, (2I, 4I), *) be a nonneutrosophic groupoids which is not a pneutrosophic groupoid. (Z 8I, (4I, 4I), *) is alsogroupoid which is non prime special neutrosophic g

We see equal special neutrosophic groupoid is not a pspecial neutrosophic groupoid or non prime spneutrosophic groupoid.

We will illustrate this by an example.

E l 3 2 7 L (Z I (2I 2I) *) b l



Example 3.2.7: Let (Z 5I, (2I, 2I), *) be a equal neutrosophic groupoid given by the following table.

* 0 I 2I 3I 4I0 0 2I 4I I 3II 2I 4I I 3I 0

2I 4I I 3I 0 2I3I I 3I 0 2I 4I4I 3I 0 2I 4I I

Consider (2I * 3I) * 4I = 3I, 2I * (3I * 4I) = 2I (* 4I) = 2Thus (Z 5I, (2I, 2I), *) is a neutrosophic groupoid. Thu

have 3 neutrosophic groupoids called the equal spneutrosophic groupoids.

T HEOREM 3.2.10: Let (Z n I, (tI, tI), *) be a equal neutrosophic groupoid. There exists (n – 2) equal spneutrosophic groupoids.

Follows using number theoretic methods.

T HEOREM 3.2.11: The equal special neutrosophic grouare commutative neutrosophic groupoids.

T HEOREM 3.2.12: (N(Z n ), (tI, tI), *) is neutrosophic groupoid. Clearly (Z n I, (tI, tI), *)*). Thus (Z n I, (tI, tI), *) is a equal specia

subgroupoid of the equal special neutrosophic grou

T HEOREM 3.2.13: The equal special neutrosop(Z I ( ) *) l hi id



(Z p I, (t, t), *) are normal neutrosophic groupoids.

Proof: Clearly the equal special neutrosophic gro

t), *) are normal for we have a (Z pI, (t, t)) = (Z

Z pI.Also it is easily proved ((Z pI, (t, t)] x) y =

and (xy) (Z pI(t, t)) = x (y (Z pI, (t, t)), p a prime.Thus we have a new class of equal special

normal groupoids.

Recall a neutrosophic groupoid G is said to be if (xy) x = x (yx) for all x, y G.

We will first illustrate it by an example.

Example 3.2.8: Z6I (4I, 4I) is a neutrosophic P –take a, b Z6I (4I 4I);

a * (b * a) = a * (4Ib + 4aI)= 4aI + 4I (4Ib + 4aI)= 4aI + 16Ib + 16aI= 20aI + 16Ib.

(a * b) * a = (4aI + 4bI) * a= 16aI + 16bI + 4aI= 20aI + 16Ib

Proof: For every x, y ZnI (t, t) we have

x * (y * x) = x * (ty + tx)= tx + t 2y + t 2x.

(x * y) * x = (tx + ty) * xt 2 + t 2 + t



= t x + t y + tx.

Thus (x * y) * x = x * (y * x) for all x, y ZnI (t,

the claim.

We see we have infact a large class of neutrosophic P-groupwhich is evident from the following theorem.

T HEOREM 3.2.15: The neutrosophic groupoids (N(Z n )

are P- groupoids.

The proof of the above theorem is left as an exercise forreader to prove.

We say a neutrosophic groupoid G is said to balternative neutrosophic groupoid if (xy) y = x (yy) for all

G.

We first illustrate by an example the neutrosophic groupoid(5I, 5I) which is not an alternative groupoid.

Example 3.2.9: Let Z 9I (5I, 5I) be a neutrosophic groupo prove x * (y * y) z (x * y) * y for x, y Z9I .Consider x * (y * y) = x * (5Iy + 5Iy)

Thus x * (y * y) z (x * y) * y for all x, y ZZ9I (5I, 5I) is not an alternative neutrosophic group


T HEOREM 3.2.16: Z p I (t, t) are not alternativgroupoids t Z I \ {0}



groupoids t Z p I \ {0}.

Proof: Consider x * (y * y) for any x, y Z pI (

x * (y * y) = x * (ty + ty)= tx + t 2y + t 2y (1

(x * y) * y = (tx + ty) * y= t 2x + t 2y + ty (2)

Clearly (1) and (2) are not equal for all t.Hence the claim.

T HEOREM 3.2.17: All neutrosophic groupoids Z prime t Z n I \ {0} with t 2 { t(mod nI) are alterna

Proof: Given Z nI (t, t) is a neutrosophic groupoidnot a prime and t 2 = t (mod nI). (t ZnI)Consider

x * (y * y) = x* (ty + ty)= tx + t 2y + t 2y= tx + ty + ty (1)

Now(x * y) * y = (tx + ty) * y

= t 2 x + t 2y + ty

T HEOREM 3.2.18: Let (N(Z n ), (t, t), *) be a neut groupoid where t 2 = t (mod nI), n a non prime. Then (N(

t), *) is an alternative neutrosophic groupoid.Thus we have non trivial class of alternative neutroso

groupoids as well as a class of neutrosophic groupoids ware not alternative



are not alternative.

3.3 Neutrosophic Polynomial Groupoids

In this section we for the first time define the clasneutrosophic polynomial groupoids built using Z nI or ZI or N(Z) or QI or N(Q) or N(R) or RI or N(C) or CI.

ZnI[x] =n i

i i ni 0

a x a Z I,n- d f®¦̄

x a variable or an indeterminate}.

N(Z n)[x] =n

ii i n

i 0

a x a N(Z ),n- ½ d f® ¾¯ ¿¦ .

Clearly Z nI [x] N (Z n) [x].

ZI [x] = i ii 0

a x a ZI,nf

- ½ f® ¾¯ ¿¦

and N(Z n)[x] and infinite classes of neutrosophiinfinite order using QI [x] or ZI [x] or N(Q)[x] orso on.

Groupoids built using N(Z n)[x], QI[x] angeneral be known as polynomial neutrosophic grou



D EFINITION 3.3.1: Let G = {Z n I [x], *, (p, q); pand q are primes} be such that if

a(x) = a 0 + a 1 x + … + a n xn

and b(x) = b 0 + b 1 x + … + b m xm

thena(x) * b(x) = a 0 * b 0 + (a 1 * b 1 )x + … + (a n

+ (0 * b n+1) x n+1 + … + (0 * b

= (pa 0 + qb 0 )(mod nI ) + (pa 1

+ (pa 2 + qb 2 )(mod nI) x 2 + … (mod nI) x n + … + (p0 + qb m )

where a i , b i Z n I.

(If m < n or m = n one can easily define * on

{Z n I[x], *, (p, q) / p, q Z n \ {0}, p and q pneutrosophic polynomial groupoid of modulo intone.


Example 3.3.1: Let G = {Z 12I[x], *, (3, 5)} beneutrosophic groupoid of modulo integer of level o

Example 3.3.3: Let

G =9

ii i 4

i 0

a x a Z I,(1,3)- ½® ¾¯ ¿¦

be a neutrosophic polynomial groupoid of finite order.



Example 3.3.4: Let

G =4

ii i 23

i 0

a x a Z I,(5,13)- ½® ¾¯ ¿¦

be the finite neutrosophic polynomial groupoid.

Example 3.3.5: Let

G =n

ii i 16

i 0a x a N(Z );(7,11)- ½® ¾¯ ¿¦

be an infinite neutrosophic polynomial groupoid as n d

Example 3.3.6: Let

G = {n

ii

i 0

a x¦ ; a i N(Z 43), (23, 29); n d f }

is an infinite neutrosophic polynomial groupoid.

Example 3.3.7: LetG = {

ni

ii 0

a x¦ ; a i N (Z 3); (1, 2)}

Now we can define neutrosophic polynomial inrational or complex) groupoid similar to the onedefinition 3.3.1.

We will illustrate this only by examples.

Example 3.3.9: Letn

i- d f¦



G = ii i

i 0

a x n ;a ZI;(13,17 d f ®

¯ ¦

be a neutrosophic integer coefficient polynomial gro

Example 3.3.10: Let

G =n

ii i

i 0

a x n ;a 3ZI;(2,19- d f ® ¯ ¦

be the neutrosophic integer coefficient polynomial g

Example 3.3.11: Let

G =n

ii i

i 0

a x n ;a ZI;(23,53- d f ® ¯ ¦

be a neutrosophic polynomial integer groupoid.

Example 3.3.12: Let G = { 6aixi; a i N(Z

neutrosophic polynomial integer groupoid of infinit

Example 3.3.13: Letn

be an infinite neutrosophic polynomial integer groupoid.

Example 3.3.15: LetP =

19i

i ii 0

a x a ZI;(3,29)- ½® ¾¯ ¿¦



be an infinite neutrosophic polynomial integer groupoid.


G =0

,*, ;( , )ii i

i

a x a ZI p qf- ½

® ¾¯ ¿¦

be a neutrosophic polynomial integer groupoid. If P that P is a neutrosophic polynomial integer groupoid underoperations of G then we call P to be a neutrosophic int

polynomial subgroupoid of G.

It is to be noted in this definition we can replace ZI by N(ZZ

nI or N(Z

n) or N(Q) or N(R) or N(C) or QI or RI or CI.

We will illustrate however all these by some simple example

Example 3.3.16: Let

G =n

ii i 12

i 0

a x ,n ,a N(Z ),(3,7)- ½t f ® ¾¯ ¿¦

be a neutrosophic polynomial groupoid



are all infinite polynomial integer neutrosophic subgroupoidG.

Since m = 2, 3, …, f we have infinitely man

subgroupoids but G has no finite polynomial neutrososubgroupoids.

Example 3.3.19: Let



G = ii i

i 0

a x a QI,*, (29,3)f- ½

® ¾¯ ¿¦

be a neutrosophic polynomial rational groupoid of infinite orG has infinitely many polynomial neutrosophic subgroupoidinfinite order but has no neutrosophic polynomial subgrouof finite order.

TakeP = 2i

i ii 0

a x a QI,(29,3)f- ½

® ¾¯ ¿¦ G;

P is an infinite neutrosophic polynomial subgroupoid of infiorder.

Example 3.3.20: Let

G =20

ii i

i 0

a x a N(Q),*,( 17,23)- ½ ® ¾¯ ¿¦

be a neutrosophic polynomial rational neutrosophic groupoiinfinite order.Take

10 ½

be an infinite polynomial real neutrosophic groupoiLet

P = i

i ii 0

a x ,*,( 2,7),a N(Q)f- ½

® ¾¯ ¿¦P is an infinite polynomial real neutrosophic subgro

Example 3 3 22: Let



Example 3.3.22: Let

G =26

ii i

i 0

a x a N(C),*,(7,17)- ® ¯ ¦

be an infinite complex polynomial neutrosophic groTake

M = ii i

i 0

a x a RI,*,(7,17)f- ½

® ¾¯ ¿¦

M is also an infinite complex polynomial subgroup

Now having seen examples of these groupoids nowonto state that all properties and definitions likgroupoid, normal groupoid, normal subgroupoisubgroupoids and so on which can always b

neutrosophic polynomial groupoids with appropmodifications.

Only in case of homomorphism Kgroupoids G and G c; we demand K(I) = I angroupoid homomorphism. By no means I should bto any real number for the indeterminate c

compensated, it should always continue to rindeterminate. The reader is expected to give somepolynomial neutrosophic groupoid homomorphism


G =- ½

® ¾¯ ¿¦n

i

i ii 0

a x a N( Q ),*, ( p,q )

be a neutrosophic polynomial groupoid of level one. Let

^ `¦ i



P = ^ ,*, ( , )¦ iii a Q p qa x ;

p, q are primes in Q G. P is clearly a poly(sub)groupoid but is not a neutrosophic polynomial groupWe call P to be a pseudo neutrosophic polynomial subgroupof G. If G has no pseudo neutrosophic polynomial subgrouthen we call G to be a pseudo simple neutrosophic polyno

groupoid.

We will illustrate both these situations by some siexamples.

Example 3.3.23: Let

G =8

ii i

i 0a x a N(Q),*,(3,7)- ½® ¾¯ ¿¦

be a neutrosophic polynomial rational groupoid.Let

W =7

ii i

i 0

a x a 3Z,*,(3,7)- ½® ¾¯ ¿¦ G

is a pseudo neutrosophic polynomial subgroupoid of G

neutrosophic polynomial subgroupoid. Thus G simple neutrosophic polynomial groupoid.

Example 3.3.25: LetG = i

i ii 0

a x a RI,*,(3,23)f- ½

® ¯ ¦



be a neutrosophic polynomial groupoid. It is easihas no pseudo neutrosophic polynomial subgroupois a pseudo simple neutrosophic polynomial groupo


T HEOREM 3.3.1: Let

G =-®¦̄

ni

i i ni 0

a x a Z I ( ZI or RI or QI or CI

p and q primes in Z + } be a neutrosophic polynoclearly G is a pseudo simple neutrosophic groupoid

Proof: Since Z nI or ZI or QI or RI or QI are pursets we see the polynomial groupoids G built usinghave even a single non zero real coefficient. Hencea pseudo simple neutrosophic groupoid in such casG can have neutrosophic polynomial subgroupoids.

We give yet another theorem which gurantees thed hi l i l b id



Next we proceed onto define level two polynomial groupoids.

D EFINITION 3.3.4: Let G = { 6 a i xi | a i Z n ( N(Z) or RI or QI or N(Q) or N(R)), *, (p, q) suchare just numbers not necessarily prime but (p, qdefined to be a polynomial neutrosophic groupoid o



defined to be a polynomial neutrosophic groupoid o

We will illustrate by a few examples before we define other levels of neutrosophic polynomial grou

Example 3.3.29: Let

G = ii i

i 0

a x a ZIf-

®¦̄ , *, (9, 16)

be the neutrosophic polynomial groupoid of level tw

Example 3.3.30: Let

G = ii i

i 0

a x a N(Q)f-

®¦̄

, *, (24, 4

be the neutrosophic polynomial rational groupoid o

Example 3.3.31: Let

G =n

ii i 12i 0

a x a Z I-®̄¦ , (9, 8), *

Substructures as in case of level one groupoids cadefined in case of level two neutrosophic polynogroupoids also.

Further from the very context one can easily understanlevel to which the polynomial neutrosophic groupoid belto. Now we define level three polynomial neutrosogroupoids.




G = -®¦̄n

ii i

i 0

a x a N( Q )

(or QI, Z n I or N(Z n ) or RI or N(R) or N(C) or CI) / n dq); (p, q) = d z 1, p, q Z + }, G is a polynomial neutr

groupoid defined as level three neutrosophic polyn groupoid.


Example 3.3.33: Let

G =n

ii i 17

i 0

a x n ,a Z I- d f ®¦̄ , *, (3, 15)}

be a neutrosophic level three polynomial groupoid.

Example 3.3.34: Let

G = ii i

i 0

a x a ZI,f-

®¦̄ *, (7, 147)}



Substructure of these groupoids can be constructed acase of other level groupoids.

Finally we now proceed onto define level five groupoid


G =f-

®¦̄ ii i n

i 0

a x a N( Z )



(or Z n I, ZI, N(Z) or QI or N(Q) or N (R) or RI or CI or N*, (0,t) t z 0, t Z + } be a neutrosophic polynomial groWe define G to be a polynomial neutrosophic groupoid of

five.


Example 3.3.39: Let

G = ii i

i 0

a x a N(Q)f-

®¦̄ , *, (3,0)}

be a polynomial neutrosophic groupoid of level five.

Example 3.3.40: Let

G =6

ii i 12

i 0

a x a N(Z )-®¦̄ , *, (0,8)}

be a neutrosophic polynomial groupoid of level five.

Example 3 3 41: Let

be a neutrosophic polynomial groupoid of level five

Now we give results about these neutrosophic

groupoids.T HEOREM 3.3.3: The neutrosophic polynomialevel four are commutative groupoids.



Proof: Follows from the very fact if G is any

polynomial groupoid of level four then for a(x), bhave a(x) * b(x) = ta(x) + tb(x) and b(x) * a(x) = hence G is commutative groupoid. In case of buildiusing Z nI we can choose t, u ZnI.

T HEOREM 3.3.4: Let G be a neutrosophic polyno

of the formG =

-®¦̄

ni

i i pi 0

a x a Z I , p a prime, *, (tI, tI)

then G is a neutrosophic polynomial groupoid.

The reader is expected to prove the theorem.T HEOREM 3.3.5: Let

G =-®¦̄

ni

ii 0

a x ; n < f , a i Z n I,*, (tI

be a neutrosophic polynomial groupoid of level foa neutrosophic polynomial P-groupoid.

Proof is left as an exercise for the reader.

T HEOREM 3.3.7: Let

G =0

f-®¦̄ i

i nii

a Z I a x ; n is not a prime, *, (tI, tI)



is a neutrosophic alternative polynomial groupoid if and on

(tI)2

{ tI (mod n).

Proof: Clearly this is proved using the alternative identity.

T HEOREM 3.3.8: Let

G = 0

f-®̄¦

i

i nii a Z I a x , *, (0, tI}

is a neutrosophic polynomial P-groupoid and neutroso polynomial alternative groupoid if and only if (tI) 2 { tI


Example 3.3.43: Let

G =8

ii i 6

i 0

a x a Z I-®¦̄ , *, (4I, 5I)}

be a Smarandache polynomial neutrosophic groupoid.For



be a neutrosophic polylnomial groupoid. It is easily verifieis a Smarandache Bol groupoid and is not a Smarandache stBol groupoid.

Example 3.3.49: Let

G =120

ii i 6

i 0

a x a Z I-®¦̄ , 8, (4I, 3I)}

be a neutrosophic polynomial groupoid G is a Smarand



be a neutrosophic polynomial groupoid. G is a Smarandstrong neutrosophic polynomial P-groupoid.

Example 3.3.50: Let

G =27

ii i 4

i 0

a x a Z I-®¦̄ , *, (2I, 3I)}

be a neutrosophic polynomial groupoid. G is a Smarandstrong neutrosophic polynomial P-groupoid.

Example 3.3.51: Let

P = ii i 14

i 0

a x a Z If-

®¯¦ , *, (7I, 8I)}

be a neutrosophic polynomial groupoid. P is a Smaradastrong alternative neutrosophic polynomial groupoid.

Example 3.3.52: Let

G =3 i

i i 5i 0

a x a Z I-®¦̄ , *, (I, 3I)}

The proof is obvious by using simple numtechniques.

T HEOREM 3.3.10: Let

G =f-

®¦̄ ii i n

i 0

a x a Z I , *, (tI, uI) / t + u = I



be Smarandache neutrosophic polynomial groupSmaradache P-groupoid if and only if t 2 I { tI (muI (mod n).


T HEOREM 3.3.11: Let

G =0

ii i n

i

a x a Z I f-

®¯¦ , *, (tI, uI) with tI + uI

be a neutrosophic polynomial groupoid, G is a strong Moufang groupoid if and only if t 2 I { t= uI (mod n).


Several results of this type can be derived for polynomial groupoids built using Z nI.

P =

1

2

m

y

y

y

ª º« »« »« »« »¬ ¼

#

where y i N(R) or N(Z n) or Z nI or QI or N(Q) or so neutrosophic column matrix.

Now take



Mnum =

11 1m

21 2m

n1 nm

m m

m m

m m

ª º« »« »« »« »¬ ¼

"

"

# #

"

;

m ij N(Z n) or Z nI or N(R) or RI or so on; we call Mneutrosophic n um matrix. If n = m we call M num

neutrosophic n um square matrix.

With this convention we now proceed onto define several classes of neutrosophic matrix groupoids.

D EFINITION 3.4.1: Let G = {(x 1 , …, xn )/ x i Z n I, *, (p, q primes} be a neutrosophic row matrix groupoid withoperation for *; x = (x 1 , …, xn ) and y = (y 1 , …, yn ) G

x * y = (x1 , …, xn ) * (y 1 , y2 , …, yn )= (px 1 + qy 1 (mod nI), …, px n + qy n (mo

G is a neutrosophic row matrix groupoid of level one.We will illustrate this situation by some examples.

Example 3.4.3: Let P = {(x 1, x 2, x3) | x i RIneutrosophic row matrix groupoid of level one.

We see the row matrix neutrosophic groupoid given3.4.1 and 3.4.3 are of infinite order where as theexample 3.4.2 is of finite order.

D EFINITION 3.4.2: Let ª º



G =

- ª º

°« »°« »®« »°« »°¬ ¼¯

#

1

2

m

x

x

x

| x i ZI, *, (p, q); p and q are disti

G is a column matrix neutrosophic groupoid of lev* for any

x =

ª º« »« »« »« »¬ ¼

#

1

2

m

x x

x

and y =

ª º« »« »« »« »¬ ¼

#

1

2

m

y y

y

in G is defined by

x*y =

ª º« »« »« »« »¬ ¼

#

1

2

m

x

x

x

*

ª º« »« »« »« »¬ ¼

#

1

2

m

y

y

y

=

ª « « « « ¬

#

1

2

m

px qy

px qy

px qy

We will illustrate this situation by some simple exam

Example 3.4.5: Let

P = {

1

2

3

4

5

6

x

x

xx

x

x

ª º« »« »« »« »« »« »« »« »

| x i N(C), *, (19, 29)}



6

7x

« »¬ ¼

be a column matrix neutrosophic groupoid of level one.

Example 3.4.6: Let

R = {

1

2

3

4

x

xx

x

ª º« »« »« »« »¬ ¼

| x i Z7I, *, (3, 5)}

be the column matrix neutrosophic groupoid of level one.

Example 3.4.7: Let

T = {

1

2

3

4

5

x

x

x

x

x

x

ª º« »« »« »« »« »« »« »« »

| x i N(Z 3), *, (1, 2)}

D EFINITION 3.4.3: Let M mu n = {(m ij ) / m ij N( j d m, *, (p, q); p and q are distinct primes} be the

neutrosophic groupoid of level one.


Example 3.4.8: Let§



G =

11 12 13 14

21 22 23 24

31 32 33 34

a a a a

a a a aa a a a

-§ ·

°¨ ¸®̈ ¸°¨ ¸© ¹¯| a ij

1 d i d 3; 1 d j d 4; *, (13, 43)} be the 3 u 4 negroupoid of level one.

Example 3.4.9: Let

P =

11 12

21 22

31 32

41 42

51 52

61 62

71 72

a a

a a

a a

a aa a

a a

a a

-§ ·°¨ ¸°¨ ¸°¨ ¸°¨ ¸®̈ ¸°¨ ¸°¨ ¸°¨ ¸°¨ ¸© ¹¯

| a ij N(Z 5), *, (3, 2); 1 d i

be a 6 u 2 neutrosophic matrix groupoid of level ois a finite groupoid.

ai N(Q), *, (31, 43), 1 d i d 20} be the 4 u5 neumatrix groupoid of level one of infinite order.

When n = m in the definition 3.4.3 we get s

neutrosophic matrix groupoids of level one.We will illustrate this situation by some simple examples.

Example 3.4.11: Leta a a§ ·



P =

1 2 3

4 5 6

7 8 9

a a a

a a aa a a

-§ ·

°¨ ¸®̈ ¸°¨ ¸© ¹¯| a i N(R), *, (3, 17) 1 d i d

be the 3 u 3 neutrosophic matrix groupoids of level one. ClP is an infinite groupoid.

Example 3.4.12: Let

M =a b

c d

-§ ·°®̈ ¸°© ¹¯

| a, b, c, d ZI, *, (23I, 53I)}

be the 2 u 2 neutrosophic matrix groupoid of level oinfinite order.

Example 3.4.13: Let

W =

1 2 3 4

5 6 7

8 9

a a a a

0 a a a

0 0 a a

0 0 0 a

-§ ·°¨ ¸°¨ ¸®̈ ¸°¨ ¸°© ¹¯

| a i N(Z 19I), *, (17, 2

Now having seen level one groupoid we proceedlevel two neutrosophic matrix groupoids.

D EFINITION 3.4.4: Let G = {m u n matrices w Z n I or N(Z n ) or QI or N(Q) or ZI or N(Z) or so on,

that (p, q) = 1; p and q need not necessarily be pdefined to be the m u n neutrosophic matrix grotwo. If m = 1 then G is a 1 u n row neutgroupoid of level two If n = 1 we get a m



groupoid of level two. If n = 1 we get a m

neutrosophic matrix groupoid of level two. If m = n then we get a square neutrosophic maof level two. If m z n then G is a rectangulamatrix groupoid of level two.

We will illustrate the definition by some examples.

Example 3.4.15: Let

G =

1

2

3

4

5

6

x

x

x

xx

x

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°¬ ¼¯

| x i Z7I, 1 d i d 6, *, (3

be a column neutrosophic matrix groupoid of level

G is of finite order.

E l 3 4 16 L t W {( ) | N

Example 3.4.18: Let

M = 1 2 3 4 5 6

7 8 9 10 11 12

a a a a a aa a a a a a- ª º°®« »°¬ ¼¯

| a i N(Z 2

1 d i d 12, *, (24, 25)} be the rectangular neutrosophic mgroupoid of finite order.



Now having seen level two groupoids we will just givmethod by which level three, level four and level five mgroupoids are built.

In the definition 3.4.4 if we take p, q such that (p, q) = dthen we call G to be a neutrosophic matrix groupoid of lthree. If in the definition p = q = t is taken then we defineneutrosophic matrix groupoid to be a level four groupoid.

If instead of (p, q) we take one of p or q to be zero thedefine those matrix neutrosophic groupoids to be level matrix groupoids.


Example 3.4.19: G = {(x 1, x 2, x 3, x 4)| x i N (Z), *, (25the row matrix neutrosophic groupoid of level three. Clearis of infinite order.

Example 3.4.20: Let

1x- ª º°« »

Example 3.4.21: Let

G =a b

c d

-§ ·°

®̈ ¸© ¹°̄a, b, c, d N(R), *, (2

be the square matrix neutrosophic groupoid of linfinite order.



Example 3.4.22: Let P = {all 10 u 9 neutrosophentries from Z 11I, *, (9, 6)} be the 10 u 9 neugroupoid of level three of finite order.

Example 3.4.23: Let

G =

1

2

3

4

5

6

7

8

x

xx

x

x

x

x

x

- ª º

°« »°« »°« »°« »°« »®« »°« »°« »°« »°« »°« »¬ ¼¯

| x i N (Z), *, (5, 5); 1 d

be a 8 u 1 neutrosophic column matrix groupoid oinfinite order.

Example 3.4.24: Let G = {[x 1, x 2, x3]| x i N

Example 3.4.26: Let G = {8 u 8 neutrosophic matricentries from ZI, *, (19, 19)} be a square neutrosophic mgroupoid of level four of infinite order.

Now we proceed onto define neutrosophic groupoids of five.

Example 3.4.27: Let

1x- ª º



G =

1

2

3

25

x

xx

x

°« »°« »°« »®« »°« »°« »°¬ ¼¯

#

| x i N(Z 4), *, (3, 0); 1 d i d 25}

be a neutrosophic column groupoid matrix of level five of fiorder.

Example 3.4.28: Let G = {[x 1, x2, …, x 7] such that x(0, 8), 1 d i d 7} be a neutrosophic row groupoid of level finfinite order.

Example 3.4.29: Let V = {3 u 3 square matrices withfrom N(C), *, (0, 7)} be a square neutrosophic comgroupoid of infinite order of level five.

Example 3.4.30: Let P = {20 u 5 rectangular matric

entries from N(Z 8), *, (6, 0)} be a rectangular neutrogroupoid of level five of finite order.Here we wish to state from now on wards we will not

neutrosophic matrix subgroupoid. If G has no matrix subgroupoids then we define G to neutrosophic matrix groupoid.

We will first illustrate this situation by some simple

Example 3.4.31: Let G = {(x 1, x 2, x 3), *, (3, 5),3} be a neutrosophic row matrix groupoid of levelorder.



Choose P = {(x, x, x) | x Z7I, *, (3, neutrosophic row matrix subgroupoid of G of level

Example 3.4.32: Let G = {set of all 3 u 3 neutrwith entries from N(Z), *, (7, 8)} be a neutrosgroupoid of level two of infinite order. Choose P

neutrosophic matrices with entries from 3ZI, *, (7, a neutrosophic matrix subgroupoid of level two of i

Example 3.4.33: Let G = {all 2 u 9 neutrosophentries from N(C), *, (9, 13)} be a neutrosophic maof level two.

Take P = {all 2 u 9 neutrosophic matrices w N(R); *, (9, 13)} G; P is a neutrosophic matrof G of infinite order and of level two.

Example 3.4.34: Let G = {all 7 u 2 neutrosophentries from RI, *, (2, 15)} be the neutrosophic ma

of level two. Take W = {all 7 u 2 neutrosophentries from 3ZI, *, (2, 15)} G, W is a neu

Example 3.4.36: Let G = {all 1 u 5 row matrices withfrom Z 30I, *, (6, 15)} be a neutrosophic row matrix grouTake W = {(a, a, a, a, a)| a Z30I, *, (6, 15)} G

neutrosophic row matrix subgroupoid of G of level three.

Example 3.4.37: Let M = {all 10 u 8 neutrosophic from QI, *, (17, 34)} be a neutrosophic matrix groupoid of lthree. Take P = {all 10 u 8 neutrosophic matrices with from 13ZI, *, (17, 34)} M; P is a neutrosophic



, , ( , ) ; p

subgroupoid of M of level three.

Example 3.4.38: Let

G =

a a a

a a a

a a a

-§ ·°¨ ¸®¨ ¸°¨ ¸© ¹¯

| a Z7I, *, (3, 6)}

be a neutrosophic matrix groupoid of level three; clearly Gno proper subgroupoids. Thus G is a simple neutrosophic magroupoid.

Example 3.4.39: Let G = {(a a a a) where a Z17I, (16 be a neutrosophic matrix groupoid, clearly G is a simple mneutrosophic groupoid of level three.

Example 3.4.40: Let

a a a- ª º°

Example 3.4.41: Let

G =

a

a

aa

a

a

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°« »

| a QI, *, (9, 48)}



a« »¬ ¼¯

be a neutrosophic matrix groupoid of level infinitely many neutrosophic matrix subgroupoids.

Example 3.4.42: Let

G =

aa

a

a

a

a

a

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°« »¬ ¼¯

| a Z43I, *, (24, 40)}

be a neutrosophic matrix groupoid of level three. Gneutrosophic matrix groupoids.

Example 3.4.43: Let G = {all 5 u 5 neutrosoph

Example 3.4.45: Let G = {(a, a, a, a) | a Z6I, *, (4, neutrosophic matrix groupoid of level four. Let V = {(a, a, aa {0, 2I, 4I}, *, (4, 4)} G, V is a neutrosophicsubgroupoid of G.

Example 3.4.46: Let G =a a a a

a a a a

-§ ·°®̈ ¸°© ¹¯

| a Z23I,

be a neutrosophic matrix groupoid G is a simple neutroso



be a neutrosophic matrix groupoid G is a simple neutrosomatrix groupoid.

T HEOREM 3.4.1: Let

G =

-§ ·°¨ ¸°¨ ¸®̈ ¸°¨ ¸°© ¹¯

# #

a ... a

a ... a

a ... a

| a Z p I, p a prime, *, (t, t); 0 d

be any neutrosophic matrix groupoid. G is a sneutrosophic matrix groupoid of level four.


T HEOREM 3.4.2: Let

G =

-§ ·°¨ ¸°¨ ¸®̈ ¸°¨ ¸°̄

# # #

a a ... a

a a ... a|

Example 3.4.47: Let G = {all 7 u 7 square matrfrom Z 12I, *, (3, 0)} is a neutrosophic matrix grofive. Let P = {All 7 u 7 square matrices with entr

6I, 9I}, *, (3, 0)} G, P is a neutrosophic matrof level five.

Example 3.4.48: Leta

b

- ª º°« »°« »



G =

b

c

d

e

f

« »°« »°« »®« »°« »°« »°« »°¬ ¼¯

| a, b, c, d, e, f ZI, *, (0

be a neutrosophic matrix groupoid of level five.Take

P =

a

a

a

a

a

a

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°¬ ¼¯

| a 5ZI, *, (0, 8)}

P is a neutrosophic matrix subgroupoid of level five

Example 3.4.49: Let G = {(a, a, a, a, a, a, a, a, a(0 24)} be a neutrosophic matrix groupoid of leve

Example 3.4.51: Let

G =

a a a a

a a a a

a a a a

-§ ·°¨ ¸®̈ ¸°¨ ¸© ¹¯

| a Z11I, *, (90, 0)}

be a neutrosophic matrix groupoid of level five. G is a simneutrosophic matrix groupoid.

In view of this we have the following theorem the pro



which is left as an exercise for the reader.T HEOREM 3.4.3: Let

G =

-§ ·°¨ ¸°¨ ¸®̈ ¸°¨ ¸°© ¹¯

# # #

a a ... a

a a ... a

a a ... a

such that a Z p I, p a prime, *, (0, t) 0 d t < neutrosophic matrix groupoid of level five. G is a sneutrosophic matrix groupoid of level five.

This proof is also left as an exercise for the reader.

Now we give yet another theorem which states as follows.

T HEOREM 3.4.4: Let G = {m u n matrices with entries fr*, (t, 0)} (the entries can be from N(Z n ) or ZI or N(z) or

N(R) or N(C) or QI or RI or CI). G is not a simple neutroso



A *x y

z w§ ·¨ ¸© ¹

*a b

c d§ ·¨ ¸© ¹

= A * 5x 5a(mod 6) 5y 5b(mod5z 5c(mod 6) 5w 5d(mod

§ ¨ ©

=5a 7x 7a(mod6) 5b 7y 7b

5c 7z 7c(mod 6) 5d 7w 7

§ ¨ ©



=x y

z w§ ·¨ ¸© ¹

I and II are equal tox y

z w§ ·¨ ¸© ¹. Thus G is a neut

P- groupoid.

T HEOREM 3.4.7: Let G = {m u n neutrosophentries from Z n I; n not a prime, *, (t, t); 1

neutrosophic matrix groupoid. G is an alternative matrix groupoid if and only if t 2 { t (mod n).

Proof: To prove the theorem it is sufficient if we (a ij) and B = (b ij) in G

(A * B) * B = A * (B * B).

Consider (A * B) * B = (tb ij + tb ij)*B

2



It is easily verified that G is a Smarandache Moufand is not a Smarandache strong Moufang groupoid

Example 3.4.59: Let G = {8 u 9 neutrosophentries from Z 12I, *, (3, 4)} be a neutrosophic mG is a Smarandache strong neutrosophic Bol groupo

Example 3.4.60: Let G = {all 18 u 1 neutrosophentries from Z 4I, *, (2, 3)} be a neutrosophic groupoid.



g pIt is easily verified G is a Smarandache

neutrosophic groupoid and is not a Smarandachneutrosophic matrix groupoid.

Example 3.4.61: Let G = {all 10 u6 neutrosophentries from Z nI, *, (0, 2); n = 2m} G is aneutrosophic matrix groupoid of level five.

Several results true in case of general groupoids bcan be derived for these neutrosophic matrix grusing Z nI.

3.5 Neutrosophic Interval Groupoids

In this section we just introduce a new class of groneutrosophic interval groupoids.

We will give the basic notations.o(Z nI) = {[0, a i] | a i ZnI}

o(N(Z )) = {[0 a + bI] | a b Z

o{N(C +)) = {a + bI / a, b C + {0}}

are special neutrosophic intervals. We only take positive vaand always the least value is zero.

D EFINITION 3.5.1: Let o(Z n I) be the collection of i Define * for any two intervals [0, a], [0, b] in o(Z n I) a[0, b] = [0, ta + pb(mod n)] where t, p Z n \ {0} such t

p are primes with (t, p) = 1. We see G = {o(Z n , I), *, (t,interval neutrosophic groupoid of level one.




Example 3.5.1: Let G = {[0, a] | a Z pI, *, (7, neutrosophic interval groupoid of level one.

In the definition 3.5.1 we can replace o(Z nI) by o(No(Z +I) or o(N(Z +)) or o(R +I) or o(N(R +)) or o(N(Q +I)) Still we get only neutrosophic interval groupoids.

We will illustrate all this situation only by some examples.

Example 3.5.2: Let G = {[0, a] | a Z+I, *, (19, 17)neutrosophic interval groupoid of level one.

Example 3.5.3: Let G = [0, a + bI] | a, b Q+ {0}, * be the neutrosophic interval groupoid of level one.

Example 3.5.4: Let P = {[0, a + bI] | a, b R + {011)} be a neutrosophic interval groupoid of level one



Example 3.5.14: Let T = {[0, a + bI] | a, b R + {0 be a neutrosophic interval groupoid of level four.

Example 3.5.15: Let W = {[0, a] | a C+I; (9, 9)neutrosophic interval groupoid of level four.

Example 3.5.16: Let P = {[0, a + bI] | a, b C+

20)} be the neutrosophic interval groupoid of level four.

Now if in the definition 3.5.1 we take u = 0 or t = 0 then we



neutrosophic interval groupoid of level five.

We will illustrate this by examples.

Example 3.5.17: Let S = {[0, a] | a Z9I; *, (0, 8)neutrosophic interval groupoid of level five.

Example 3.5.18: Let S = {[0, a + bI]| a, b N(Z 28), *, (the neutrosophic interval groupoid of level five.

We will not distinguish between the levels of neutroso

interval groupoids or the neutrosophic sets which are being uto build these groupoids as it clear by the structure.

D EFINITION 3.5.2: Let G = {[0, a] | [0, a] o(Z n I), *, a neutrosophic interval groupoid.

Let P o(Z n I) such that (P, * (p, q)) is a neutro

interval groupoid. We call P to be a neutrosophic inte subgroupoid of G.

Example 3.5.20: Let T = {[0, a + bI] | a + bI be a neutrosophic interval groupoid. P = {[0, bI] | (3, 8)} T; is a neutrosophic interval subgroupoid

Example 3.5.21: Let G = {[0, a + bI] | a, bI0)} be a neutrosophic interval groupoid. Choose M

Z+I, *, (20, 0)} G, M is a neutrosophic intervof G.

l {[ ] |+

{



Example 3.5.22: Let G = {[0, a] | a R I {0a neutrosophic interval groupoid. Let V = {[0, a]{0}, *, (18, 24)} G; V is a neutrosophic intervof G.

Now it may so happen G be a neutrosophic interva

may be a proper subset of G which is just an interand not a neutrosophic interval groupoid. We ca pseudo neutrosophic interval subgroupoid of G.

If G has no pseudo neutrosophic interval subcall G to be a pseudo simple neutrosophic interval g


Example 3.5.23: Let G = {[0, a + bI], a, b be a neutrosophic interval groupoid. Take W = {[0*, (3, 4)} G is a real interval groupoid; W is caneutrosophic interval subgroupoid of G.

Example 3.5.24: Let G = {[0, a] | a Z+I {a neutrosophic interval groupoid we see G has

The proof is left as an exercise to the reader.

T HEOREM 3.5.2: Let G = {[0, a + bI] / a, b N(Q+ )or N(R + ) or N(Z + ) or N(C + ), *, (t, u)} be a neutrosophic i

groupoid. G has pseudo neutrosophic interval subgroupthat G is not a simple pseudo neutrosophic interval groupoid

This proof is also left as an exercise to the reader.

W ll hi i l id G b d bl i



We call a neutrosophic interval groupoid G to be doubly simneutrosophic interval groupoid if G has no neutrosophic intesubgroupoid as well as G has no pseudo neutrosophic intesubgroupoid.

Example 3.5.25: Let G = {[0, a] | a Z5I, *, (3, neutrosophic interval groupoid. Clearly G has no neutrosointerval subgroupoids. Further G has no pseudo neutrosointerval subgroupoid. Thus G is a doubly simple neutrosointerval groupoid.

We will illustrate this situation by a theorem which guaranthe existence of a large class of doubly simple neutrosointerval groupoids.

T HEOREM 3.5.3: Let G = {[0, a] | a Z p I, p a prime, *,< t, u d p – 1} be a neutrosophic interval groupoid Gdoubly simple neutrosophic interval groupoid.

The proof is simple and is left as an exercise to the reader

We will however give some examples of these a few interesting theorems.

Example 3.5.26: Let G = {[0, a] | a Z4Ineutrosophic interval groupoid.

This groupoid has left ideals given by P = {[02I}, *, (2, 3)} G and T = {[0, a] | a {I, 3IClearly P and T are not neutrosophic interval right i

However P and T right ideals of G c= {[0, a2)}.




T HEOREM 3.5.4: Let G = {[0, a] | a Z n I, *{[0, a] | a Z n I, *, (u, t)} be neutrosophic interva

is a left neutrosophic interval ideal of G if and oright neutrosophic interval ideal of G.

The proof is got by simple number theoretic compthe simple translation of rows to columns. Texpected to prove the same.

Example 3.5.27: Let G = {[0, a] | a Z10Ineutrosophic interval groupoid G has no left or righ

T HEOREM 3.5.5: Let G = {[0, a] | a Z p I; p a p, *} be a neutrosophic interval groupoid. G

groupoid.

P f i l f i f h d

T HEOREM 3.5.7: Let G = {[0, a] | a Z p I, p a prime,1d t < p} is a neutrosophic groupoid. G is not an alternainterval groupoid.

Proof: We have to prove for any x, y, G;(x * y) * y z x * (y * y).

Take x = [0, a] and y = [0, b]. Consider

(x * y) * y = {[0, a] * [0, b]) * [0, b]= [0, ta + tb] * [0, b]

[0 t 2 + t 2 b + tb ( d )]



= [0, t 2a + t 2 b + tb (mod p)] Consider

x * (y * y) = [0, a] * ([0, b] * [0, b])= [0, a] * ([0, tb + tb])= [0, at + t 2 b + t 2 b (mod p)]

Clearly I and II are different. Hence G is not an alternneutrosophic interval groupoid.

T HEOREM 3.5.8: Let G = {[0, a] | a Z n I, (t, t); 1 < not a prime, *} be a neutrosophic interval groupoid. G i

alternative neutrosophic interval groupoid if and only if t(mod n).

The proof is left to the reader.

Now we will illustrate these situations by some

examples.

E l 3 5 28 L t G {[0 ] | Z I * (11 1

T HEOREM 3.5.9: Let G = {[0, a] | a Z n neutrosophic interval groupoid. G is a P-galternative interval groupoid if and only if t

The proof is left as a simple exercise.


Example 3.5.30: Let G = {[0, a] | a Z40,neutrosophic interval groupoid G is an alternagroupoid and G is also a neutrosophic interval P gro



groupoid and G is also a neutrosophic interval P-gro

Example 3.5.31: Let G = {[0, a] | a Z29,interval neutrosophic groupoid. G is not a P-group1 < t < 29.

Example 3.5.32: Let G = {[0, a] | a Z10Ineutrosophic interval groupoid. G is a Smaranneutrosophic interval Moufang groupoid.

Example 3.5.33: Let G = {[0, a] | a Z12Ineutrosophic interval groupoid; G is a Smarandaneutrosophic interval groupoid which is not a strong Moufang neutrosophic interval groupoid.

Example 3.5.34: Let G = {[0, a], *, (3, 4), aneutrosophic interval groupoid. G is a Smaradach

neutrosophic interval groupoid.

E l 3 5 35 L t G {[0 ] | Z I

Example 3.5.37: Let G = {[0, a], *, a Z12I, (5, 1neutrosophic interval groupoid. G is only a Smarandneutrosophic interval P-groupoid and is not a Smarandstrong neutrosophic interval P-groupoid

Example 3.5.38: Let G = {[0, a] | a Z14I, *, (97, neutrosophic interval groupoid. G is a Smarandneutrosophic strong interval alternative groupoid.

Several properties enjoyed by interval groupoids can alsderived for neutrosophic interval groupoids with approp



derived for neutrosophic interval groupoids with appropmodifications.

When we define homomorphism of neutrosophic intgroupoids it is essential that I is mapped only on to I for duhomomorphisms it is impossible for the indeterminate I tchanged to real, this indeterminate must remain aindeterminate only.

3.6 Neutrosophic Interval Matrix Groupoids

In this section we will define for the first time the new notioneutrosophic interval matrix groupoids and describe a few o

properties associated with them. We will be using onlynotations given in section 3.5.

We will first give some essential notations.Let (a 1, a2, …, a n) = X be such that a i o(Z nI) or

o(Z +I) o(N(Q + {0}} o(Z +I) o(N(Z +I)) and o(R +I) so

where x i’s are neutrosophic intervals from o(Zo(Q + I), o(N(Q + {0})) and so on. Y will in genas the neutrosophic interval column matrix.

Let

M =11 1m

n1 nm

a ... a

a ... a

ª º« »« »« »¬ ¼

# #

be the neutrosophic interval matrix. The entries ino(Z nI) o(N(Z n)) o(Q +I) and so on



o(Z nI), o(N(Z n)), o(Q I) and so on.If m = n then we call M to be a neutrosophic in

matrix.

Now we will define groupoids using these interval matrices. We can have five levels of ingroupoids we will define them without mentioning from the very context one can easily understand tothe groupoid belongs to. We will now make a formand give examples of them.

D EFINITION 3.6.1: Let G = {(a 1 , a 2 , …, a n ) | a i

or o(N(Z n )), o(Z + I), o(N(Z + 0)) and so on, *, or Z + {0}} where for X = (a 1 , …, a n ) and Y =

X * Y = ([0, x 1 ], …, [0, x n ]) * ([0, y 1 ], …, [0, y n

qy1 ], [0, px 2 + qy 2 ], …, [0, px n + qy n ]) G.

It is easily verified G is a groupoid and G isrow neutrosophic interval matrix groupoid.

We will illustrate this by some examples

Example 3.6.3: Let G = {([0, a 1], [0, a 2], …, [0, a 1

N(Z +), i = 1, 2, …, 14; *, (9, 17)} be a neutrosophic row intmatrix groupoid of level two.

Example 3.6.4: Let G = {([0, a 1], [0, a 2], [0, a 3]) | a i

(5, 19), *} be a neutrosophic row interval matrix groupoilevel one.

Example 3.6.5 : Let G = {([0, a 1], [0, a 2], …, [0, a 140] N(R +), (4, 28), *} be a neutrosophic row interval mgroupoid of level three.



groupoid of level three.

Example 3.6.6: Let G = {([0, a 1], [0, a 2], …, [0, a 48]) |(3, 3), *} be a neutrosophic row interval matrix groupoilevel four.

Example 3.6.7: Let G = {([0, a 1], [0, a 2], …, [0, a 18]) | 1 d i d 18, *, (24, 8)} be a neutrosophic row interval mgroupoid.


G =- ª º°« »°« »®« »°« »°¬ ¼¯

#

1

2

n

x

|

xi = [0, a i ], a i o(Z n I) or o(N(Z n )) or o(Z + I) or o(Q +

on. *, (p, q); p, q Z n or Z + {0}} be such that for any

x * y =

ª º« »« »« »

« »¬ ¼

#

1

2

n

*

ª º« »« »« »

« »¬ ¼

#

1

2

n

y

y

y

=

ª « « «

« ¬

#

1

2

n

0, pa q

0, pa q

0, pa q

It is easily verified G is a neutrosophic column mgroupoid.


Example 3.6.8: Let



Example 3.6.8: Let

G =

1

2

3

4

5

x

x

x

x

x

- ª º°« »°« »°« »®« »°« »°« »°¬ ¼¯

| x i = [0, a i], a i Z8I; 1 d i d

be a neutrosophic interval column matrix groupoid

Example 3.6.9: Let

G =

1

2

3

4

5

6

x

x

x

x

xx

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°

| x i = [0, a i]; a i N(Q +), *, (8, 9

be a neutrosophic column interval matrix groupoid of level tw

Example 3.6.11: Let

G =

1

2

3

4

x

x

x

x

- ª º°« »°« »°« »°« »®« »°« »° #

| x i = [0, t i]; t i N(Z 28); 1 d i d 27; *, (13



27x

« »« »°« »°¬ ¼¯

be a neutrosophic column matrix interval groupoid of level f

Example 3.6.12: Let

G =

1

2

3

4

x

x

x

x

- ª º°« »°« »®« »°« »°¬ ¼¯

| x i = [0, n i], n i Z5I, *, (3, 0)}

be a neutrosophic column interval matrix groupoid of level fClearly G is a groupoid of finite order.

Example 3.6.13: LetG = 1x- ª º°

® | x = [0 a ] Z+I * (8 26) 1 d i d

G is a neutrosophic n u m interval matrix g= m we call G to be a neutrosophic interval s

groupoid.

We will illustrate these situation by some examples

Example 3.6.14: Let

G = 1 2

3 4

x x

x x

-§ ·°®̈ ¸°© ¹¯

| x i = [0, a i]; a i Z81I, *



© be a neutrosophic interval square matrix grouorder.

Example 3.6.15: Let

G = 1 2 3 4

5 6 7 8

y y y yy y y y

-§ ·°®̈ ¸°© ¹¯| y i [0, a i]; a i Z

be a neutrosophic interval rectangular matrix group

Example 3.6.16: Let

G =

1 2 3

4 5 6

7 8 9

10 11 12

y y y

y y y

y y y

y y y

- ª º°« »°« »®« »°« »°¬ ¼¯

| y i = [0, m i] m i N(

be a neutrosophic interval groupoid of level five

be a neutrosophic interval matrix groupoid of level three.

Example 3.6.18: Let

G =

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

a a a a

a a a a

a a a a

a a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

|

ai = [0, d i]; d i R +

I; 1 d i d 16, *, (0, 26)} be a neutrointerval matrix groupoid of infinite order.



interval matrix groupoid of infinite order.

Example 3.6.19: Let

G =

1 2

3 4

49 50

a a

a a

a a

- ª º°« »°« »®« »°« »°¬ ¼¯

# # | a i = [0, n i]; n i Z4I, (2, 2), *}

be a neutrosophic interval matrix groupoid of finite order.

Now having defined neutrosophic interval matrix groupoids

now proceed onto define substructures in them and illustrate it by examples.

D EFINITION 3.6.4: Let G = {M = (m ij ) | mij = [0, a N(Z + ) or N(Z n ) or Z n I or N(R + ) and so on 1 d i d m; 1(p, q)} be a neutrosophic interval matrix groupoid. Let P

P is proper subset of G such that P itself is a neutrosointerval matrix groupoid We define P to be a neutroso

be a neutrosophic interval matrix groupoid of level Take

P =

a a

a a

b b

- ª º°« »®« »°« »¬ ¼¯

| a = [0, m], b = [0, n]; m, n Z9

P is a neutrosophic interval matrix subgroupoid othree.

Example 3.6.21: Let G = {all 9 u 9 inter+



matrices with entries from o(Z +I), *, (11, 11)} bmatrix interval groupoid of level four.

Take W = {all 9 u 9 upper triangular intervmatrices, with entries from o(Z +I), *, (11, 11neutrosophic matrix interval subgroupoid of G of le

Example 3.6.22: Let

G =

1 2

3 4

19 20

a a

a a

a a

- ª º°« »°« »®« »°« »°¬ ¼¯

# #|

ai = [0, n i], n i N(R + {0})); 1 d i d 20; neutrosophic matrix interval groupoid of level five.

Take

1 1a aa a

- ª º°« »°

We will give some examples, a few important propeabout these structures.

Example 3.6.23: Let

G = 1 2

3 4

a aa a

- ª º°®« »°¬ ¼¯| a i = [0, m i]; m i Z4I, 1 d i d 4, *, (

be a neutrosophic interval matrix groupoid.Take

P =1 2

3 4

a a

a a

- ª º°®« »°¬ ¼¯ | a i = [0, m i]; m i {0, 2I}, *, (2, 3)}



T is also a neutrosophic interval matrix left ideal of G.

T HEOREM 3.6.1: Let G = {m u p neutrosophic interval m from o(Z n I), *, (t, u)} be a neutrosophic matrix i

groupoid. G is an idempotent neutrosophic interval m groupoid or neutrosophic interval matrix idempotent grouif and only if t + u { 1(mod n).

Proof: Let M = (m ij) G with m ij = [0, a ij], a ij ZnI; 11 d j d p. To show M * M = M.

Consider M * M = (m ij) * (m ij)

= [0, a ij] *, [0, a ij]= [0, ta ij + ua ij (mod n)]= [0, a ij (t + u) (mod n)]= [0, a ij]

= M

Proof: Simple number theoretic computations wresult.

The reader is expected to prove.

Example 3.6.24: Let G = {set of all 3 u 1neutrosophic matrices with entries from o(Z 10

neutrosophic column interval matrix groupoid. G hright ideals.

T HEOREM 3.6.3: Let G = {collection of all 1interval row matrices with entries from o(Z



interval row matrices with entries from o(Z n neutrosophic interval row matrix groupoid. If n = tand u are primes then G has no left or right ideals.

The reader can prove this theorem using some sitheoretic techniques.

C OROLLARY 3.6.1: If in the above theorem n and if (t, u) = 1 with t + u = p then also G has noideals.

Example 3.6.25: Let

G =a a a

a a a

- ª º°®« »°¬ ¼¯

| a o(Z 5I), *, (2

be a neutrosophic matrix interval groupoidsubgroupoids.

Example 3.6.26: Let

be a neutrosophic interval matrix groupoid. G hsubgroupoids.

Example 3.6.27: Let

G =a a

a a

- ª º°®« »°¬ ¼¯

| a o (Z 6I), *, (2, 2)}

be a neutrosophic interval matrix groupoid. G has subgroupo

a a- ª º°



P =a a

a a

®« »°¬ ¼¯| a o(0, 2I, 4I), *, (2, 2)} G

is a neutrosophic interval matrix subgroupoid of G.

T HEOREM 3.6.4: Let

G =- ª º°®« »°¬ ¼¯

a a

a a| a o(Z p I); p a prime , *, (t, t), t < p

be the neutrosophic interval matrix groupoid of level four. a normal groupoid.


T HEOREM 3.6.5: Let



Example 3.6.28: Let G = {(a, a, a, a, a, a, a, a) | a o(91, 5)} be a neutrosophic matrix interval groupoid. G Smarandache neutrosophic matrix interval groupoid.

Example 3.6.29: Let

G =a a

a a

- ª º°®« »°¬ ¼¯

| a o (Z 6I), *, (4, 5)}

be a neutrosophic matrix interval groupoid.

a a- ª º°



P =a a

a a

®« »°¬ ¼¯| a {[0, 0], [0, 2I], [0, 4I]}, *, (4, 5)}

is a neutrosophic interval matrix subgroupoid of G but is nSmarandache subgroupoid of G.

Example 3.6.30: Let

G =

a

a

a

a

- ª º°« »°« »®« »°« »°¬ ¼¯

| a o (Z 6I), *, (4, 5)} be

a neutrosophic matrix interval groupoid.

A =

a

a

aa

- ª º°« »°« »®« »°« »° ¼¯

/ a {[0, 0], [0, I], [0, 3I], [0, 5I]}, *, (4, 5)}

is an ideal which is not a Smarandache ideaSmarandache groupoid of G.

Example 3.6.32: Let

G =

a a a

a a a

a a a

a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

/ a o (Z 8I), *, (2

be the neutrosophic matrix interval groupoid



be the neutrosophic matrix interval groupoid.

Let

A =

a a a

a a aa a a

a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

| a {0, [0, 2I], [0, 4I], [0, 6I]},

A is clearly a Smarandache neutrosophic matrix int

groupoid of G.

Example 3.6.33: Let G = {(a, a, a, …, a)| a be a 1 u 26 neutrosophic row matrix interval groSmarandache strong Moufang groupoid.

Example 3.6.34: Let

Example 3.6.35: Let

G =

a a a

a a a

a a a

a a a

a a a

a a a

a a aa a a

- ª º°« »°« »°« »°« »°« »°« »®« »°« »°« »°« »°« »°

| a o(Z 12I), *, (3, 4)}



a a a

a a a

« »°« »°¬ ¼¯

be a neutrosophic matrix interval groupoid. G is a Smaranda

strong Bol groupoid.

Example 3.6.36: Let

G =

a a a

a a a

a a a

a a a

a a a

a a a

a a aa a a

- ª º°« »°« »°« »°« »°« »°« »°« »°« »®« »°« »°« »

| a o (Z 6I), *, (4, 3)}

Example 3.6.37: Let

G =

a a a a

a a a a

a a a a

a a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

| a o(Z 4I), *,

be a neutrosophic matrix interval groupoid. Smarandache Bol groupoid but G is not a SmaranBol groupoid.



Example 3.6.38: Let

G =a a a a a

a a a a a

- ª º°®« »°¬ ¼¯

| a o(Z 4I), *

be a neutrosophic matrix interval groupoid. G is a strong P-groupoid.

Example 3.6.39: Let

G =

a a a a

a a a a

a a a a

a a a a

a a a aa a a a

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°¬ ¼¯

| a o (Z 6I), *,

be a neutrosophic matrix interval groupoid. G is a SmarandaP-groupoid and is not a Smarandache strong P-groupoid.

Example 3.6.41: Let

G =

a aa a

a a

a a

a a

- ª º°« »°« »°« »®« »°« »°« »°¬ ¼

¯

| a o (Z 14I), *, (7, 8)}

be a neutrosophic matrix interval groupoid G is a Smaranda



be a neutrosophic matrix interval groupoid. G is a Smarandastrong alternative groupoid.

Example 3.6.42: Let

G =a a a a aa a a a a

a a a a a

- ª º°« »®« »°« »¬ ¼¯| a o(Z 12I), *, (1, 6)}

be a neutrosophic matrix interval groupoid. G is o

Smarandache strong alternative groupoid.

Example 3.6.43: Let

G =a a

a a

- ª º°®« »°¬ ¼¯

| a o(Z 9I), *, (4, 4)}

be a neutrosophic matrix interval groupoid. Clearly G is n

be a neutrosophic interval matrix groupoid. CleSmarandache groupoid.

Several other results not mentioned in this section

by general groupoids are true in case ofneutrosophic interval matrix groupoids withmodifications.

3.7 Neutrosophic Interval Polynomial Groupoids

In this section we will be defining and dish l l l d



neutrosophic interval polynomial groupoids or polynomial interval coefficients groupoids.

The notions given in section 3.5 will be section. Here also we can define five levels of

interval polynomial groupoids using o(Z nI), o(N(Z +)), o(Q +I), o(N(Q +)) and so on.

From the context one can easily understand tthe neutrosophic interval groupoid belongs to.


G = f-®¦̄ ii

i 0

[0,a ] x / a i Z n I, *, (p, q). p

where * for any two interval neutrosophic polynom

x =f

¦ ii

i 0

[ 0,a ] x

and f

¦ i

=f

¦ ii i

i 0

([0, pa qb (mod n )] x .

G is easily verified to be a groupoid which we call aneutrosophic polynomial interval groupoid.

If p and q are two distinct primes we call G to be a leve groupoid.

If p and q are distinct such that (p, q) = 1 we cal groupoid G to be a level two groupoid. If p and q are such (p, q) = d z 1 we call G to be a level five groupoid.

In definition 3.7.1 we can take instead of Z nI. N(Zor N(Z +) or N(R +) or N(Q +) or Q +7 I or so on.



( ) ( ) (Q ) Q

We will give some illustrations before we proceed on to psome results.

Example 3.7.1: Let

G =25

ii

i 0

[0,a ]x-®¦̄ | a i Q+I {0}, *, (13, 41)}

be a neutrosophic polynomial interval groupoid of level one.

Example 3.7.2: Let

V =45

ii

i 0

[0,a ]x-®

¯

¦ | a i N(R + {0}), *, (12, 25)}

Example 3.7.4: Let

W = ii

i 0

[0,a ]xf-

®¦̄ | a i C+I {0}, *,

be a neutrosophic polynomial interval groupoid of l

Example 3.7.5: Let

P = ii

i 0[0,a ]x

f

-®̄¦ | a i Z+I {0}, *, (



be a neutrosophic polynomial interval groupoid of l

Now we have seen different levels of neutrosophiinterval groupoids. Now we will define thsubgroupoids.


G =

f-®̄¦

ii

i 0 [0,a ] x / a i Z n I or Z +

I or so o

be a neutrosophic polynomial interval groupoid.Suppose P G be a proper subset of G

operations of G is a neutrosophic polynomial interv

then we define P to be a neutrosophic polyno subgroupoid of G.

Take

P =5

ii

i 0

[0,a ]x¦ | a i Z40I, *, (23, 17)} G

is a neutrosophic polynomial interval subgroupoid of G.

Suppose

S = ii

i 0

[0,a ]xf-

®¦̄ |

ai {0, 2I, 4I, 6I, 8I, 10I, 1I, …, 36I, 38I}, *, (23, 17)}i l t hi l i l i t l bg id



is also a neutrosophic polynomial interval subgroupoid oWe see P is a finite order where as S is of infinite order.

Example 3.7.7: Let

G =25

ii

i 0

[0,a ]x-®¦̄ | a i Q+I {0}, *, (24, 35)}

be a neutrosophic polynomial interval groupoid of level two.

Take

W =25

ii

i 0

[0,a ]x-®¦̄ | a i Z+I {0}, *, (24, 35)}

is a neutrosophic polynomial interval subgroupoid of G of ltwo

Example 3.7.8: Let

G = ii

i 0

[0,a ]xf-

®¦̄ | a i N (R + {0}), *,

be a neutrosophic polynomial interval groupoid.Take

P =25

ii

i 0

[0,a ]x-®¦̄ | a i 15; Z +I {0}, *, (3

is a neutrosophic polynomial interval subgroupoid th



three.

Example 3.7.9: Let

G =40 i

ii 0

[0,a ]x-®¦̄ | a i Z+I {0}, *, (

be a neutrosophic polynomial interval groupoid four.

Consider

W =20

ii

i 0

[0,a ]x-®¦̄ | a i 20 Z +I {0}, *, (1

is a neutrosophic polynomial interval subgroupoid four.

P = ii

i 0

[0,a ]xf-

®¦̄ | a i C+I {0}, *, (26, 0)}

is a neutrosophic polynomial interval subgroupoid of G.

We can as in case of other groupoids define the notioleft ideal, right ideal, ideal, normal groupoid, nosubgroupoid, Smarandache groupoids and Smarandgroupoids satisfying special identities.



Chapter Four

APPLICATION OF THESE



APPLICATION OF THESENEW CLASSES OF GROUPOIDSNEUTROSOPHIC GROUPOIDS

In this chapter we give some of the application

classes of groupoids. In the first place thesgroupoids can be used in theory of cryptog

neutrosophic matrix interval groupoids when the field of sdemands a non-associate structure with closure operationsinterval solutions.

Smarandache groupoids can be used in computin

information science.Smarandache groupoids can be used in biology to descertain aspects in the crossing of organisms in genetics anconsiderations of metabolisms.



Chapter Five

SUGGESTED PROBLEMS



SUGGESTED PROBLEMS

Here we suggest over 200 problems. Solving these the reader to understand the concepts described in t

1. Find subgroupoids of {Z +, *, (7, 9)}.

2 Fi d b id f {Q + * (3/7 2/11)}



7. Let W =1 2 3

4 5 6 i 11

7 8 9

a a a

a a a a Z ;

a a a

- ª º°« »®« »°« »¬ ¼¯

1 d i d 9, *

3 matrix groupoid. Find subgroupoids of W. cardinality of W?

8. Let R = 13

a aa Z ;

a a

- ª º°®« »¬ ¼°̄

* (7, 5)} be a 2 u 2

Find the number of elements in R. Is R simple?claim.



9. Let T = 24

a a a a

0 a a aa Z ;

0 0 a a0 0 0 a

- ª º°« »°« »®« »°« »°¬ ¼¯

*, (2, 19)} b

groupoid.a. Find the cardinality of T.

b. Is T simple?

c. Can T have normal subgroupoids?d. Is T a normal groupoid?e. Can T be a Bol groupoid?f. Is T a Moufang groupoid?g. Can T be an alternative groupoid? Justify you

10. Obtain some interesting results about matrix grusing Q + {0} using (p, q) = (3, 5).

14. Give an example of a matrix groupoid which is simple.

15. Give an example of a matrix groupoid which is normal.

16. Obtain conditions for a matrix groupoid to be a Smarandmatrix groupoid.

17. Give an example of a matrix groupoid which is not a S-magroupoid.

18. Does there exist a Bol matrix groupoid constructed usin{0}?



19. Can there be a matrix groupoid which is Moufang construusing Q + {0}?

20. Can any matrix groupoid constructed using R + {0or left alternative? Justify your claim.

21. Find subgroupoids of the matrix gro

G =

1 2 3

4 5 6

7 8 9

10 11 12

a a a

a a a

a a a

a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

where a i Z19; 1 d i d 12, *, (3

a. Is G a Bol matrix groupoid? b. Can G be right alternative?c. Does G have normal subgroupoids?

22. Let H =

1 1

2 2

3 3i i 17

4 4

5 5

6 6

a 0 b 0

0 a 0 b

0 0 a ba ,b Z ,

a 0 0 ba b 0 0

0 a b 0

- ª º°« »°« »°« »°« »®« »°« »°« »°« »°¬ ¼¯

1 d

be a matrix groupoid.a. Find atleast 2 proper subgroupoids of H.

b. What is the cardinality of H?c. Can H satisfy any one of the special identitie



23. Let P =1 2 3

4 5 i 240

6

a a a

0 a a a Z

0 0 a

- ª º°« »®« »°« »¬ ¼¯

; 1 d i d

matrix groupoid find;a. Atleast 2 subgroupoids of P.

b. Is P simple?c. Can P be a S-groupoid?d. Is P a P-groupoid?e. Can P be a strong Bol groupoid?f. Can P be a strong Moufang groupoid? Justify

Replace (2, 3) by (20, 30) and answer all the(a) to (f).

24. Let V = {(a 1, a2, …, a 9)| a i Z24; 1 d i dmatrix groupoid.

d

25. Prove G = {(x 1, …, x 20)| x i Z20; 1 d i d 20; *, (16, 5)idempotent matrix groupoid. Is G a Smarandache P-groupJustify your claim.

26. Prove T =

1 2

3 4

5 6 i 12

7 8

9 10

a aa a

a a a Z

a a

a a

- ª º°« »°« »°« »®« »°« »°« »°¬ ¼¯

; 1 d i d 10, *, (4,

matrix S-P-groupoid.

ª º



27. Consider T =

1 2 3

4 5 6i 5

7 8 9

10 11 12

a a a

a a aa Z ,

a a a

a a a

- ª º°« »°« »®« »°« »°¬ ¼¯

* (3, 2)} be

groupoid. How many groupoids are in the class of 4matrix groupoids C(Z 5, (4 u 3)) constructed using Z 5

a. How many in that class are S-matrix P-groupoids? b. Can this class of groupoids contain matrix S-strong

groupoids? Justify your claim.

28. Prove the 1 u 12 row matrix groupoid V = {{(a 1, a2

ai Z+ {0}}, *, (11, 15)} is of infinite order.a. Is V a S-matrix groupoid?

b. Find two subgroupoids of V.c. Can V have normal subgroupoids? Justify your claim.

30. Let T = { 3 822Z u , *, (1, 11)} be a 3 u 8 matr

using Z 22. Is T a S-groupoid? Justify your answe

31. Let W = { 7 246Z u , *, (1, 23)} be a 7 u 2 matr

using Z 46.a. Is W a S-idempotent matrix groupoid? Prove

b. Can W be a S-strong Bol matrix groupoid? Juc. Prove W is to a S-P-matrix groupoid.

32. Let M = { 9 3

25Z u , *, (p, q), p, q, Z

25} be

groupoid.a. Does there exists for a suitable p, q Z2

i S P t i id



i. S-P-matrix groupoid.ii. S-idempotent matrix groupoid.iii. S-alternative matrix groupoid.

iv. S-strong Bol matrix groupoid.

33. Let G = {C +, *, 10 u 8, (9, 19)} be the set of almatrix groupoid. Obtain some interesting propert

34. Let P =

10

i ii 0 a x

-®̄¦ | a i Z17, *, (3, 11)} b

groupoid built using Z 17. Find subgroupoids groupoid? Prove or disprove!

35. Obtain some interesting properties aboutgroupoids built using Z

nor Q + {0} or R

{0}.

39. Does their exist a S-strong Moufang polynomial grou built using Z + {0}? Justify your claim!

40. Construct an example of a polynomial groupoid which is n

S-polynomial groupoid.

41. Obtain a necessary and sufficient condition for polynomial groupoid to be a S-alternative polygroupoid built using Z n.

42. Can an alternative polynomial groupoid G =n

i 0

a-®¦̄Z+ {0}, *, p, q Z+ (p, q)} be built? Justify your



{ }, , p, q (p, q)} y yFind atleast 3 polynomial subgroupoids of G.

43. When is T =24

iii 0

a x-®̄¦ | a i Z28, *, (p, q), p, q Z28};

a. S-strong polynomial Bol groupoid (obtain condition and q)?

b. S-strong alternative polynomial groupoid?c. S-strong P-groupoid?

d. S-groupoid?e. S-idempotent groupoid?Obtain conditions on p and q.

44. Can V =15

ii

i 0

a x-®

¯¦ | a i Z19; *, (p, q), p, q, Z19} eve

a. S-idempotent groupoid?

47. Can the polynomial groupoid J =29

ii

i 0

a x-®¦̄

3)} have identity element? Justify your claim.a. Is J a S-groupoid?

b. Is J a S-strong Bol polynomial groupoidclaim.

48. Let P =10

ii

i 0

a x-®¯¦ | a i Z12, *, (3, 9)} b

groupoid built using Z 12.a. Find subgroupoids of P.b Can P have a subgroupoid of order 4?



b. Can P have a subgroupoid of order 4?c. What is the cardinality of P?d. Is P a S-groupoid?

e. Does P have normal subgroupoids?Justify your answers.

49. Can any polynomial groupoid built using Rnormal?

50. Obtain conditions for two polynomial subgrousing Z 45 to be S-semi conjugate subgroupoids.

51. Can G =5

ii

i 0

a x-®¦̄ | a i Z8, *, (2, 8)} the polyn

have S-semi conjugate subgroupoids?

subgroupoids? Is P =20

ii

i 0

a x-®¦̄ , a i {0, 3 6, 9}, *

conjugate with S =20

ii

i 0

a x-®¯¦ , a i {2, 5, 8, 11} where

polynomial subgroupoids of T? Prove T is a S-polynogroupoid.

55. Define S-inner commutative polynomial groupoid and some examples.

56. Is G =8

ii

i 0

a x-®¦̄ | a i Z5, *, (3, 3)} a S-inner comm



polynomial groupoid? Justify your claim.

57. Prove every S-commutative polynomial groupoid is a S-icommutative polynomial groupoid and not conversely.

58. Give an example of a S-Moufang polynomial groupoid busing Z 20.

59. Is G =45

ii

i 0a x-®̄¦ | a i Z10, *, (5, 6)}, the polynomial gr

a S-strong Moufang groupoid? Justify your claim.

60. Is P =20

ii

i 0

a x-®

¯¦ | a i 12 (3, 9)} the polynomial groupo

strong Moufang Groupoid? Justify your claim.

63. Give an example of S-strong P-groupoid built usi

64. Let B =25

ii

i 0

a x-®¦̄ | a i Z6, *, (4, 3)} be a polyn

built using Z 6. Is B a S-strong P-groupoid? Prov

65. Let C =14

ii

i 0

a x-®¦̄ | a i Z4, *, (2, 3)} be a polyn

Is C a S-strong P-groupoid? Justify your claim.

66. Prove if G a S-polynomial groupoid built usinand if G is a S-strong polynomial P-groupoid thin G need not satisfy the P-groupoid identity



in G need not satisfy the P-groupoid identity.

67. Every S-strong polynomial P-groupoid is a S-p

groupoid and not conversely.

68. Define the notion of S-strong right alternative illustrate it by an example.

69. Is G =

20i

ii 0 a x

-®̄¦ | a i Z14, *, (7, 8)} the polyno

S-strong alternative groupoid?

70. Let G = ii

i 0

a xf-

®¦̄ | a i Z12, *, (1, 6)} be a polyn

built using Z12. Is G a S-strong polynomid?

73. Can G =19

ii

i 0

a x-®¦̄ | a i Z19, *, (3, 12)} the poly

groupoid be a strong Bol groupoid?

74. Let G = ii

i 0

a xf-®¦̄ | a i Z19, *, (4, 15)} be a poly

groupoid. Can G be a S-Bol groupoid? Justify your claim.

75. Let T =27

i

ii 0

a x-®̄¦

| ai

Z8, *, (3, 6)} be a poly

groupoid. Can T be a S-strong P-groupoid?



76. Let L = ii

i 0

a xf-

®¦̄ | a i Z9, *, (2, 4)} be a poly

groupoid. Can L be a S-strong right alternative groupoid?

77. Find S-right ideals of L given in problem (76). Does problem (76) have S-ideals? Justify.

78. Let G = {set of all 3 u 5 interval matrices built using(3, 7)} be an interval matrix groupoid. Find subgroupoidG. Is G a S-interval matrix groupoid?

79. Obtain some interesting properties about row matrix integroupoid built using I

nZ .

80 Let P = {all 3 u 1 interval matrices constructed using

82. Give an example of a 3 u 8 matrix intervausing I

15Z , which is a S-groupoid.


7Z which is not a S-groupoid.


12Z which is a S-strong Moufang groupoid

85. Give an example of a S-row matrix interval gusing I

24Z which is a S-alternative row mgroupoid.




23Z which is simple.

87. Does their exists a S-inner commutative mgroupoid constructed using I

43Z ?

88. Obtain some interesting properties about s

interval groupoids built using I pZ ; p an odd pr

89. Construct a class of simple matrix interval groI41Z .

90. Give an example of a 10 u 2 matrix interval nbuilt using IZ

93. Can a 4 u 10 matrix interval groupoid built using Rgroupoid? Justify your claim.

94. Let G = { I9Z [x], *, (3, 7)} be an interval polynomial gro built using the intervals in I

9Z ,a. Does G have subgroupoids?

b. Is G a S-groupoid?c. Is G a S-strong Bol groupoid?

d. Can G be normal?e. Can G have alteast normal subgroupoids?f. Can G be a S-strong alternative groupoid? Justify

answers.



95. Let P =4

ii

i 0

a x-®¦̄

| a iI4Z = {all intervals built using

3}, *, (3, 1)} be a polynomial interval coefficient groupoida. What is the cardinality of P?

b. Find subgroupoids of P.c. Does P satisfy any one of the standard identities?

96. Let W = ii

i 0

a xf-®¦̄ | a i

I12Z = all intervals built using

(1, 3)} be an infinite polynomial interval coeffgroupoids built using I

12Z .a. Prove W is an infinite S-groupoid.

b. Is W a S-strong Moufang groupoid? Prove your answ



102. Obtain some interesting results on polynomial intgroupoids built using IZ .

103. Let S =

9i

ii 0 a x

-®̄¦ / a i

I

11Z = {all intervals built using {…, 10}}, *, (3, 0)} be a interval polynomial groupoid. Csatisfy the Moufang identity?

104. Can the interval polynomial groupoid F = ii

i 0

a xf-

®¯¦ /

intervals built using Z n = {0, 1, 2, …, n – 1} = { InZ ,

be a Bol groupoid?



105. Will T =4

ii

i 0

a x-®¯¦ | a i

I20Z = {all intervals built usin

*, (0, 4)} be an alternative interval polynomial groupoid?

106. Find all subgroupoids of P =3

ii

i 0

a x-®¦̄ | a i

I4Z = {all

built using Z 4 = {0, 1, 2, 3}, *, (0, 3}} the interval polyngroupoid built using intervals from I4Z . Find the ord

Does the order of the subgroupoids divide the order of P?

107. Find all subgroupoids and ideals if any of the int

polynomial groupoid T =

3i

ii 0 a x

-®̄¦ | a i

I

12Z = {all

polynomial groupoid built using coefficients fro

S is a normal groupoid.

109. Prove groupoid S in problem (108) is a P-groupo

110. Prove groupoid S in problem (108) is a groupoid.

111. Can groupoid P in problem (108) be an alternative

112. Let F be the collection of all interval polynomial

where G = ii

i 0

a xf-

®¦̄ | a iInZ = {all intervals u



integers {0, 1, 2, …, n – 1}, n is not a prime, *, (tZn \ {0, 1} and t is such that t 2 = t (mod n). P

only alternative groupoids. Find the number of suwhen n = 128.

113. Let S =7

ii

i 0

a x-®¦̄ | a i

I18Z , *, (9, 9)} be a inte

groupoid built using the intervals from I

18Z

any normal subgroupoid?

114. Is the interval polynomial groupoid T =5

i 0

-®¦̄

d i d 5, *, (17, 17)} a normal groupoid. Is T How many subgroupoids does T have? What is th

e. Can L be a normal groupoid?f. Does L contain any normal subgroupoids?g. Will L be an idempotent groupoid?

116. Does the polynomial interval groupoid V =3

i 0a-®̄¦

I11Z ; *, (4, 7)} satisfy the Bol identity?

117. Can the groupoid V given in problem (116) be a Mougroupoid or an idempotent groupoid? Justify your claim.

118. Does the polynomial interval groupoid M =i 0

af-

®¦̄



I21Z ; *, (3, 7)} have ideals? What is the order of M? Is M

groupoid? Find all right ideals of M?

119. How many semigroups are in the interval polyno

groupoid K =6

ii

i 0

a x-®¦̄ | a i

I7Z ; *, (3, 4)}? Is

groupoid? What is the order of K?

120. Construct a polynomial interval groupoid using I27Z

it is not a S-groupoid.

121. Construct a polynomial interval groupoid T using Z

is an alternative groupoid.

125. What is the order of X given in problem (123)?

126. Let G = ii

i 0

a xf-

®¯¦ | a i

I17Z ; *, (11, 7)} be a poly

groupoid.a. Is G simple?

b. Can G contain semigroups?

127. Obtain some interesting properties about intgroupoids built using intervals from IZ .

128. Let P = {6 u 7 interval matrices built using inte



be a interval matrix groupoid. Is P simsubgroupoids of P. What is the cardinality of P?

129. Let Y = {All 3 u 3 interval matrices with entri

= {Y, *, (3, 5)} be the interval matrix groupoWhat is the cardinality of P? Find all subgroupoiMoufang groupoid? Can P be a normal groupcontain ideals? Find all right ideals of P. If (3, 5

by (4, 4) in P, will P be simple?

130. Let B = {L, *, (3, 9)} where L = {all 7 u 2with entries from IZ } be a matrix interval groua. Can B be a S-groupoid?

b. Find at least 5 subgroupoids of B.c Does B contain right ideal?

132. Let D = {All 2 u 17 interval matrices using I4Z ; *, (3

matrix interval groupoid,a. What is the order of D?

b. How many subgroupoids exists in D?c. Is D simple?d. Is D normal?e. Is D a Moufang groupoid?f. Can D have ideals?g. Is every left ideal of D a right ideal of D? Justify

answer.h. Can we say order the subgroupoids of D will dividorder the groupoid D?

I



133. Let G = {all 2 u 2 interval matrix built using I5Z , *, (

the interval matrix groupoid built using I5Z .

a. Find the number of elements in G. b. Is G a commutative groupoid?c. Is G a normal groupoid?d. Can G have left ideals which are not right ideals and

versa?e. How many subgroupoids does G contain?

134. If in problem (133) Z 5I is replaced by Z nI; n a cnumber. Find the solution from a to e of problem (133).

135. Let G = {1 u 7 interval matrices with entries from Z

11)} be a interval matrix groupoid.a Find subgroupoids of G

136. Let T = {all 3 u 9 interval matrices with entri(0, 2)} be the interval matrix groupoid.a. Prove T is a S-groupoid.

b. Find subgroupoids of G

c. Find the cardinality of T.d. Does T have S-subgroupoids?e. Find ideals in T.f. Prove in T a left ideal in general is not a righ

137. Let X = {all 2 u 2 interval matrices with entrie

(6, 6)}. Derive the important features enjoyed by

138. Let Y = {5 u 5 interval matrices with entries f

4)} be a matrix interval groupoid built using



4)} be a matrix interval groupoid built usinga. Is Y a S-groupoid?

b. Prove Y is of finite order.c. Is Y a S-strong Bol-groupoid?d. Find subgroupoids of Y.e. Will Y be a S-strong Moufang groupoid?f. Can Y contain normal subgroupoids?

139. Construct a S-Moufang groupoid with 3 u 2 built using I

22Z .

140. Find all S-P- 3 u 3 matrix interval groupoids bu

141. Let G = {3 u 3 interval matrices with entries f3)} b i t l t i id Fi d l

143. Let W = {2 u 5 interval matrices with entries from Z

7)} be a matrix interval groupoid built using I19Z . V

interval matrices with entries from I19Z , *, (3, 6)} be

interval groupoid built using I19Z . Give a SG homomo

between W and V. Find all the common properties enjoyedW and V.

144. T = {3 u 2 interval matrices with entries from I19Z , *

be a interval matrix groupoid built using I19Z .

a. Is T a S-Bol groupoid ? b. Find the order of T.c. Find all subgroupoids of T.



c. Find all subgroupoids of T.d. Is T a S-groupoid?e. Is T a normal groupoid?f. Can T have normal subgroupoids?g. Obtain any other interesting property enjoyed by T.

145. Let B = {All 1 u 8 interval matrices with entries from(3, 6)} be an interval matrix groupoid.

a. Find the cardinality of B b. Find all subgroupoids of Bc. Is B a S-strong Moufang groupoid?d. Can B be a S-strong right alternative groupoid?e. Can B have normal subgroupoids?f. Find an ideal in B.

{ l h f

a. Is V a S-inner commutative groupoid? b. Find cardinality of V.c. Find all subgroupoids of V.d. Does V have normal subgroupoids?e. Will V satisfy any one of the special identitie

148. Let G = {4 u 5 interval matrices with entries f5)} be a matrix interval groupoid. Prove every SG is not a S-right ideal of G.

149. Define the notion of S-seminormal matrix intervIllustrate it by some examples.

150. Let G = {all 6 u 6 interval matrices with entri



(2, 6)} be a matrix interval groupoid.a. Prove G is a S-groupoid.

b. Does G have a S-normal subgroupoid?c. Find all S-subgroupoids of G.d. What is the cardinality of G?e. Can G be a S-strong Moufang groupoid?

151. Let T = {Set of all 3 u 6 interval matrices wI12Z , *, (3, 9)} be a interval matrix groupoid. Pr

a S-strong Moufang groupoid. Prove T is only groupoid.

152. Prove every S-strong Moufang groupoid is agroupoid and not conversely.

155. Let S = {all 7 u 7 interval matrices with entries from(7, 7)} be an interval matrix groupoid. Is S a P-groupJustify your claim.

156. Let V = {3 u 3 interval matrices with entries from Z9)} be a interval matrix groupoid;a. Is V a normal groupoid?

b. Does V contain normal subgroupoids?c. Is V an idempotent groupoid?d. Find all subgroupoids of V.e. Is V a S-groupoid?f. Find all S-subgroupoids of V. (Provided V is

groupoid).



157. Let G = {all 3 u 12 interval matrices with entries from(2, 3)} be a matrix interval groupoid.a. Can {0} be an ideal of G?

b. Prove left ideals of G are not right ideals of G.c. Find the cardinality of G.d. Find all subgroupoids of G.e. Is G a S-groupoid?f. Is G a normal groupoid?

158. Let V = {all 3 u 5 interval matrices with entries from(3, 5)} be a matrix interval groupoid.a. Is V a S-strong P-groupoid?

b. Is V a S-P-groupoid?

c. Is V a S-groupoid?d What is the cardinality of V?

e. Can B have ideals?f. Can a right ideal in B in general be a left idea

160. Let R = {All 2 u 2 interval matrices with entri

(3, 13)} be an interval matrix groupoid.a. Find atleast 3 subgroupoids of R. b. Find 3 left ideals of R.c. Find 3 right ideals of R.d. Find 3 ideals of R.e. Can R satisfy special identities?

f. Can R have normal subgroupoids?

161. Obtain some interesting results about the magroupoid G = {all 7 u 7 interval matrices wQ * (1 2)}



IQ , *, (1, 2)}.

162. Let G = {Z 9I, *, (3, 2)} be a neutrosophic gwhether G is a S-neutrosophic groupoid. cardinality of G?

163. Let G = {Z 11I, *, (3, 4)} be a neutrosophic groupwritten as a partition of conjugate groupoids?answer.

164. Find the conjugate groupoids of G = {Z 15I, *,

165. How many neutrosophic groupoids of order 7 exi

166. Find all right ideals of the neutrosophic groupoid

170. Let W = {N(Z +, *, (1+3I, 7+2I)}. Is W a neutrogroupoid? What are the special properties enjoyed by W?

171. Let S = {N(Z + {0}), *, {3I, 2 + 2I)} be a neutrosophic groupoid. Find subgroupoids of S. Dosatisfy any one of the special identities? Justify your answ

172. Does there exist a P-neutrosophic groupoid of order 15?

173. Give an example of a P-neutrosophic groupoid built uZ18I.

174. Find all the ideals of the neutrosophic groupoid W = {N(Z*, (7, 14)}. Find atleast 3 neutrosophic subgroupoids. Is W



idempotent neutrosophic groupoid?

175. Does there exist an infinite neutrosophic groupoid which P-groupoid?

176. Give an example of a S- neutrosophic groupoid of order 32

177. Suppose G is a neutrosophic groupoid of finite order. every neutrosophic subgroupoid of G divide the order oJustify your claim.

178. What is the order of the neutrosophic groupoid G built uZ21 where G = {N(Z 21), *, (2, 3)}?

179. Obtain some interesting results about neutrosophic inte

182. Does there exists a neutrosophic polynomial interof finite order?

183. Let G =12

ii i 4

i 0

[0,a ]x a x I,*, (3,2)- ½® ¾¯ ¿¦ b

polynomial interval groupoid.a. Does G satisfy any of the special identities?

b. Is G normal?c. Is G a Smarandache groupoid?d. Find atleast 2 subgroupoids of G.e. Does G have Smarandache ideals?

184. Let G = ii i 15

i 0

[0,a ]x a Z I,*,(3,13)f- ½

® ¾¯ ¿¦ b



¿ polynomial interval groupoid of infinite order.

a. Find subgroupoids in G. b. Does there exists right ideals in G whic

ideals?c. Is G a Smarandache groupoid?d. Does G satisfy any special identities?

185. Obtain some interesting results about neutrosogroupoids of finite order.

186. Let G = {[0, a i]| a i Z26I, *, (3, 6)} be a neutrogroupoid;a. Find the order of G.

b. Does G have subgroupoids?

188. Let P =

1 6

i i2 7

3 8 i 12

4 9

5 10

a a

a [0, x ];a a

a a x Z I,*,(9,9);

a a 1 i 10

a a

- ½ª º° °« »° °« »° °« »® ¾« »° °d d« »° °« »° °¬ ¼¯ ¿

be a neut

matrix interval groupoid.a. Is P a finite order groupoid?

b. Is P an alternative groupoid?c. Does P satisfy the Moufang identity?d. Find some subgroupoids and ideals in P.e. Is P a Smarandache groupoid?

189. Obtain some interesting results about neutrosol i l i t l g id b ilt i g (NC +)



polynomial interval groupoids built using o(NC ).

190. Does there exists a neutrosophic polynomial interval groupwhich satisfies both the Bol identity and Moufang identity

191. Give an example of a neutrosophic polynomial integroupoid which is a P-groupoid and an alternative groupoi

192. Obtain some interesting properties about neutrosophic mainterval groupoids built using o( I

23Z ).

193. Obtain some interesting results about neutroso polynomial interval groupoids built using o(N(Z p)), p

194 Enumerate the properties enjoyed by neutrosophic polynom

196. Is it always true that in case of finite neutrosgroupoids the order of the subgroupoid dividesthe groupoid? Justify your claim.

197. Let G =n

ii ii 0

[0,a ]x a Z I {0},*,(3-

® ¯ ¦neutrosophic polynomial interval groupoid.a. Find subgroupoids of G.

b. Is G a Smarandache groupoid? Justify.

198. Let G = 8

a aa o(Z I),*, (3,5)

a a- ½ª º° °® ¾« »¬ ¼° °¯ ¿

be

matrix interval groupoid.a What is the order of G?



a. What is the order of G? b. Find subgroupoids of G.c. Does G satisfy any special identities?

199. Let G = 11

a a

a aa o(N(Z ),*, (6,5)

a a

a a

- ½ª º° °« »° °« »® ¾« »° °« »° °¬ ¼¯ ¿

b

matrix interval groupoid.a. Is G a finite order groupoid?

b. Enumerate all the special identities satisfied

200 Let P = 22 i[0 a ]x a Z I * (3 2)- ½® ¾¦ be

202. Let G =12

i i 12i 0

[0,a ] a Z I,*,(9,4)- ½® ¾¯ ¿¦ be a neut

polynomial interval groupoid and

24

a a a aa Z I,*, (12,4)a a a a

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

be a neutrosophic

interval groupoid. Does there exists a grohomomorphism of G to G c?

203. Does there exists a neutrosophic matrix interval grou built using o(ZnI) which has Smarandache consubgroupoids?

204. Give an example of a pseudo simple neutrosophic minterval groupoid



interval groupoid.

205. Give an example of a doubly simple neutrosophic polynominterval groupoid.

FURTHER R EADING

1. Birkhoff. G., and Maclane, S.S., A Brief S Algebra , New York, The Macmillan and Co. (



2. Bruck, R.H., A Survey of binary systems,Verlag, (1958).

3. Castillo, J., The Smarandache SemigroupConference on Combinatorial Methods in MMeeting of the project 'Algebra, Geometria e C

Faculdade de Ciencias da Universidade do Po9-11 July 1998.

4. Nivan, Ivan and Zukerman, H.S., Introdutheory, Wiley Eastern Limited, (1989).

5. Padilla, R., Smarandache Algebraic StructurPure and Applied Sciences Delhi Vol 17 E

7. R.Lidl and G. Pilz, Applied Abstract Algebra, SprVerlag, (1984).

8. Smarandache, Florentin (editor), Proceedings of the International Conference on Neutrosophy, NeutrosoLogic, Neutrosophic set, Neutrosophic probability Statistics, December 1-3, 2001 held at the Universi

New Mexico, published by Xiquan, Phoenix, 2002.

9. Smarandache, Florentin, A Unifying field in Lo Neutrosophic Logic, Neutrosophy, Neutrosophi Neutrosophic probability, second edition, AmResearch Press, Rehoboth, 1999.

10. Smarandache, Florentin, An Introduction to Neutrosohttp://gallup unm edu/~smarandache/Introduction pdf



http://gallup.unm.edu/ smarandache/Introduction.pdf

11. Smarandache, Florentin, Collected Papers II , UnivKishinev Press, Kishinev, 1997.

12. Smarandache, Florentin, Neutrosophic Generalization of the Fuzzy http://gallup.unm.edu/~smarandache/NeutLog.txt

13. Smarandache, Florentin, Neutrosophic Generalization of the Fuzzy http://gallup.unm.edu/~smarandache/NeutSet.txt

14. Smarandache, Florentin, Neutrosophy : A New B Philosophy,

17. Vasantha Kandasamy, W.B., New groupoids using Z n, Varahmihir Journal oSciences, Vol 1, 135-143, (2001).

18. Vasantha Kandasamy, W.B., On Ordered GGroupoid Rings, Jour. of Maths and Com145-147, (1996).

19. Vasantha Kandasamy, W.B., Smarandahttp://www.gallup.unm.edu/~smarandache/Aut

20. Vasantha Kandasamy, W.B., Smarandahttp://www.gallup.unm.edu/~smarandache/Gro

21. Vasantha Kandasamy, W.B., Groupoids anGroupoids , American Research Press, Rehobo



Groupoids , American Research Press, Rehobo

22. Vasantha Kandasamy, W.B., and Smarandach Fuzzy Interval Matrices, Neutrosophic Inteand their Applications, Hexis, Phoenix, (200

INDEX



A

Alternative groupoid, 8-9

B

Bol groupoid, 8Bol identity, 8

C

E

Equal special neutrosophic groupoids, 122-3Extension neutrosophic field, 13

G

Groupoid of finite order, 8Groupoid of infinite order, 8Groupoid, 7-8

I

Ideally simple neutrosophic groupoid, 115-6Idempotent groupoid, 9Indeterminacy, 10



y,Interval groupoid of level four, 57Interval groupoid of level one, 50Interval groupoid of level three, 55Interval groupoid of level two, 52Interval groupoid, 88-92Interval matrix alternative groupoid, 99-100Interval matrix groupoid, 79-86Interval matrix normal groupoid, 97Interval matrix P-groupoids, 98Interval matrix, 83Interval polynomial groupoid of level four, 73Interval polynomial groupoid of level one, 64Interval polynomial groupoid of level three, 70-71Interval polynomial groupoid of level two, 67-8



Neutrosophic interval polynomial subgroupoid, 195 Neutrosophic interval square matrix groupoid, 180- Neutrosophic interval subgroupoids, 170 Neutrosophic matrix alternative groupoid, 165 Neutrosophic matrix P-groupoid, 163-4, 166 Neutrosophic matrix subgroupoid, 156-7 Neutrosophic matrix, 15-6 Neutrosophic polynomial groupoid, 128-9 Neutrosophic polynomial subgroupoid, 132 Neutrosophic row interval matrix groupoids, 176-7 Neutrosophic row matrix groupoid of level five, 154 Neutrosophic row matrix groupoid of level four, 15 Neutrosophic row matrix groupoid of level one, 148 Neutrosophic row matrix groupoid of level three, 15 Neutrosophic row matrix groupoid of level two, 153 Neutrosophic row matrix, 15-6



Neutrosophic semigroup, 121 Neutrosophic subfield, 13 Neutrosophic subgroupoid, 11-2, 111, 119-23 Normal groupoid, 10 Normal matrix groupoid, 26-7 Normal neutrosophic groupoids, 120-2 Normal subgroupoid, 9

P

P-groupoid, 8P-neutrosophic matrix groupoid, 163-7Polynomial groupoid of type I, 39Polynomial groupoid of type II, 40-1

interval groupoids, by w. b. vasantha kandasamy, florentin smarandche, moon kumar chetry

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