interval semigroups, by w. b. vasantha kandasamy, florentin smarandache

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INTERVAL SEMIGROU

W. B. Vasantha KandasamyFlorentin Smarandache



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Copyright 2011 by Kappa & Omega and the Authors 6744 W. Northview Ave.Glendale, AZ 85303, USA

Peer reviewers:Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, RomaniaProf. Mihàly Bencze, Department of MathematicsÁprily Lajos College, Braúov, RomaniaDr. Fu Yuhua, 13-603, LiufangbeiliLiufang Street, Chaoyang district, Beijing, 100028 P. R. China

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

ISBN-10: 1-59973-097-9



CONTENTS

Preface

Dedication

Chapter OneINTRODUCTION

Chapter Two

INTERVAL SEMIGROUPS

Chapter Three

INTERVAL POLYNOMIAL SEMIGROUPS



PREFACE

In this book we introduce the notion of inter

using intervals of the form [0, a], a is real. Sinterval semigroups like fuzzy interval semigsymmetric semigroups, special symmetric intervinterval matrix semigroups and interval polynomare defined and discussed. This book has eight ch

The main feature of this book is that w

problems in the eighth chapter. In this book thdefined 29 new concepts and illustrates thexamples. Certainly this will find several applica

The authors deeply acknowledge Dr. Kan proof reading and Meena and Kama for the designing of the book.

W.B.VASAN

FLORENTIN



~ DEDICATED TO ~

Ayyankali

Ayyankali (1863–1941) was the first leader of DaKerala He initiated several reforms to emancipate t



Chapter One

INTRODUCTION

We in this book make use of special type of ininterval semigroups, interval row matrix semigcolumn matrix semigroups and interval matrix salso introduce and study the Smarandache analog

The new notion of interval symmetric sspecial interval symmetric semigroups are definFor more about symmetric semigroups and theanalogue concepts please refer [9].

Th l i l h f fi i



on. For more about neutrosophy, neutrosophic inter

refer [1, 3, 6-8].Study of special elements like interval zerodiviso

idempotents, interval units, interval nilpotents are their Smarandache analogue introduced [9].



Chapter Two

INTERVAL SEMIGROUPS

In this chapter we for the first time introduceinterval semigroups and describe a few of tassociated with them. We see in general severaltheorems are not true in general case of semig

proceed on to give some notations essential t

new structures.

I (Zn) = {[0, am] | am Zn},I(Z+ {0}) = {[0, a] | a Z+ {0}I(Q+ {0}) = {[0 a] | a Q+ {0



Example 2.1: Let S = {[0, ai] | ai Z6}, under add

interval semigroup. We see S is of finite order and osix.

Example 2.2: Let S = {[0, ai] | ai Z12} be semigroup under addition modulo 12. This is alsosemigroup of finite order.

Now we can define interval semigroup under additi {0}, Q+ {0}, R + {0} and C+ {0}. All thsemigroups are of infinite order.

We will illustrate these situations by some examples.

Example 2.3: Let S = {[0, ai] | ai Z+ {0}}; S issemigroup under addition. Clearly S is of infinite ord

Example 2.4: Let S = {[0, ai] | ai Q+ {0}}; S issemigroup under addition. Clearly S is of infinite ord

Example 2.5: Let S = {[0, ai] | ai R + {0}}; S issemigroup under addition and is of infinite order.

Example 2.6: Let S = {[0, ai] | ai C+ {0}}; S issemigroup under addition and is of infinite order.

Thus we have seen examples of interval semigraddition, these are known as basic interval semigraddition.

We will now define polynomial interval semi



We will illustrate these by some examples. Example 2.7: Let S = {([0, a1], [0, a2], [0, a3], [0 Z12; 1 d i d 5} is a row matrix interval seaddition.

Example 2.8: Let P = {([0, a1], [0, a2], [0, a3][0, a6]) | ai Z+ {0}; 1 d i d 6} is a row semigroup under addition. Clearly P is of infinite

Example 2.9: Let S = {([0, a1], [0, a2], [0, a3], …Q+ {0}; 1 d i d 12}; S is a row matrix inte

under addition and is of infinite order.

Example 2.10: Let S = {([0, a1], [0, a2], [0, a3], R + {0}; 1 d i d 15} be a row matrix inteunder addition; T is of infinite order.

Example 2.11: Let G = {([0, a1], [0, a2]) | ai C2}; be a row matrix interval semigroup of infinite

Now we proceed onto define column matrix inter

DEFINITION 2.3: Let

S =

- ½ª º° °« »° °« » ° °« »® ¾

d d« »° °#

1

2

i m

3

[0,a ]

[0,a ]a Z ;

[0,a ]1 i n

,



We will illustrate these situations by some examples.

Example 2.12: Let

S =

1

2

3 i 5

4

5

[0,a ]

[0,a ]

[0,a ] a Z ;1 i 5[0,a ]

[0, a ]

- ½ª º° °« »° °« »° °« »

d d® ¾« »° °« »° °« »° °¬ ¼¯ ¿

be a column interval matrix semigroup under addit

finite order. Example 2.13: Let

S =

1

2

3 i

15

[0, a ]

[0, a ]

[0, a ] a Z {0};1 i 15

[0, a ]

- ½ª º° °« »° °

« »° °« » d d® ¾« »° °« »° °« »° °¬ ¼¯ ¿

#

be a column interval matrix semigroup under additio Example 2.14: Let

[0 a ]- ½ª º



Example 2.15: Let

S =

1

2

3 i

6

[0, a ][0, a ]

[0, a ] a R {0};1 i

[0, a ]

- ª º° « »° « »° « » d d®

« »° « »° « »° ¬ ¼¯

#

be a column interval semigroup under addition o

Example 2.16: Let

P =

1

2i

3

4

[0, a ][0, a ]

a C {0};1 i[0, a ]

[0, a ]

- ª º° « »° « » d d® « »° « »° ¬ ¼¯

be a column interval semigroup under addition o

Now we will define matrix interval semigroup.

DEFINITION 2.4: Let S = {m u n interval matri

from I(Z n )} be a m u n matrix interval seaddition.

We can replace I (Zn) in definition 2.4 by I (Z+

{0}) or I(Q+ {0}) or I(C+ {0}) and get



be a 3 u 2 interval matrix semigroup under matrmodulo 10 of finite order.

Example 2.18: Let

S =

1 2 3

4 5 6 i 42

7 8 9

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

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

- ª º

° « » d ® « »° « »¬ ¼¯

be a 3 u 3 interval square matrix semigroup of finite interval matrix addition modulo 42.

Example 2.19: Let

S =

1 2 3

4 5 6i

22 23 24

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

[0, a ] [0, a ] [0, a ] a Q {0};1

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

- ª º° « »° « » ® « »° « »° ¬ ¼¯

# # #

be a 8 u 3 matrix interval semigroup under addition

order.

Example 2.20: Let

[0 ] [0 ]-ª º



DEFINITION 2.5:Let S be the matrix interval seaddition. Let M S (M a proper subset of S),

matrix interval semigroup under addition then be a matrix interval subsemigroup of S.

We will illustrate this situation by some example

Example 2.21: Let

S = 1 2i 5

3 4

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

[0, a ] [0, a ]

- ª º° d ® « »

¬ ¼° ¯

be a square matrix interval semigroup under addi

P =[0, a] [0, a]

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

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿ S

P is a square matrix interval subsemigroup of S.

Example 2.22: Let

S =

1 2

3 4 i

5 6

[0, a ] [0, a ]

[0, a ] [0, a ] a Q {0};1[0, a ] [0, a ]

- ª º

° « » ® « »° « »¬ ¼¯

b 3 2 t i i t l i d dditi



Example 2.23: Let

M =

1

2i

12

[0, a ][0, a ]

a R {0}

[0, a ]

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

#

be a 12 u 1 column matrix interval semigroup undChoose

S =

[0,a]

[0,a]a R {0}

[0,a]

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

#

M;

S is a 12 u 1 column matrix interval subsemigroup of

Example 2.24: Let M = {([0, a1], [0, a2], …, [0, a19]){0}; 1 d i d 19} be a 1 u 19 row matrix interval Choose W = {([0, a], [0, a], …, [0, a]) | a R + {0is a 1 u 19 row matrix interval subsemigroup of M.

We can define ideals as in case of usual semigroups.

DEFINITION 2.6: Let S be a matrix interval semigaddition. I a proper subset of S. I is said to be a maideal of the semigroup S if (a) I is a matr



THEOREM 2.1: Let S be a matrix interval se

ideal I of S is a matrix interval subsemigroup ofinterval subsemigroup in general is not a matriof S.

We will illustrate this by some simple examples.

Example 2.25: Let S be a 1 u 5 row matrix inteunder addition with entries from 3Z+ {0}. Ca1], [0, a2], [0, a3], [0, a4], [0, a5]) | ai 6Z+ {01 u 5 matrix interval ideal of S.

We see some matrix interval subsemigroupsgeneral matrix interval ideals. For take J = {([0, a

a]) | a 9Z+

{0}} S. We see J is not a 1 u 5ideal, J is only a 1 u 5 matrix interval subsemigaddition.

Example 2.26: Let S = {([0, a1], [0, a2], …, [0, be a 1 u 10 row interval matrix semigroup. Cons

row interval matrix subsemigroup, we see it cann

For example take V = {([0, a1], [0, a2], …, [0, a Z12} S; V is a 1 u 10 row interval matrix sub

but is not an ideal. For take 2 Z12 we see V + 2= {([0, a1], [0, a2], …, [0, a10]) | ai {0, 6, 8, 2}see V has no 1 u 10 row interval matrix ideal.

We see it is difficult to get ideals in case of rowsemigroups under addition, but however we hav



interval matrix subsemigroups as well as S has matrix ideals. Infact S is a doubly simple interval has no interval matrix ideals and subsemigroups.

Now we proceed onto define interval matrix semigmultiplication.

DEFINITION 2.7:

Let S be a interval matrix semigmultiplication using I(Z n ) or I (Z + {0}) or I (Q+

{0}).

We will illustrate this situation by examples.

Example 2.28: Let V = {([0, a1], [0, a2], …, [0, a9]) d i d 9}; V is an interval matrix semigroup under muClearly V is a finite interval matrix semigroup.

Example 2.29: Let V = {([0, a1], [0, a2], …, [0, a10]){0}; 1 d i d 10} be an interval row matrix semig

multiplication. Clearly V is of infinite order.

Example 2.30: Let V = {([0, a1], [0, a2], …, [0, a12]){0}; 1 d i d 12} be an interval row matrix semigmultiplication.

We can define subsemigroups and ideals in cassemigroups.

DEFINITION 2.8: Let V = {([0, a1 ], [0, a2 ], …, [0, an

{0} ( + {0} + {0}} b



Choose P = {([0, a1], [0, a2], …, [0, a8]) | ai

22}; 1 d i d 8} V; V is a row matrix interval sV.

Example 2.32: Let V = {([0, a], [0, a], [0, a], [{0}} be a row matrix interval semigroup underClearly V has a row matrix interval subsemigrou

In view of this we say a row matrix intervsimple if it has no proper row matrix interval We have a large class of simple row matrix interWe say proper if the row interval matrix semigr0], …, [0, 0])} or {([0, 1], [0, 1], …, [0, 1semigroups will be known as improper row

subsemigroup or trivial row matrix interval subse

Example 2.33: Let V = {([0, a1], [0, a2], …, [0, ad i d 9} be a row matrix interval semmultiplication.

P = {([0, a1], [0, a2], …, [0, a9]) | ai {0, 10

230}; 1 d i d 9} V is a row matrix interval suV.

THEOREM 2.2: Let V = {([0, a], [0, a], …, [0

{0}} p a prime be a 1 u n row interval matrix se

multiplication. V is not a simple 1 u n row

semigroup.

The proof is left as an exercise for the reader.



DEFINITION 2.9: Let V be a 1 u n row inter

semigroup. P V; be a proper subsemigroup. We san row interval matrix ideal of V if for all p P an

and vp are in P.

We will illustrate this situation by some examples.

Example 2.34: Let V = {([0, a1], [0, a2], [0, a3], [0, aai Z12; 1 d i d 5} be a 1 u 5 row interval matrix Choose P = {([0, a1], [0, a2], …, [0, a5]) | ai {0, 2, Z12} V to be a row interval matrix subsemigroup

It is easily verified V is a row interval matrix ideal of

Example 2.35: Let V = {([0, a1], [0, a2], …, [0, a9]){0}; 1 d i d 9} be a 1 u 9 row matrix interval sChoose P = {([0, a1], [0, a2], …, [0, a9]) | ai 5Z+ 9} V is a 1 u 9 row interval matrix ideal of V.

Example 2.36: Let V = {([0, a1], [0, a2], …, [0, an]) | i d n} be a 1 u n row interval matrix semigroup; Clearly V has proper row interval matrix ideals. Howa trivial row interval matrix ideal of V.

If V has no proper row interval matrix ideal theto be a ideally simple row interval matrix semigrouan infinite class of interval matrix semigroups whideally simple row interval matrix semigroups.

THEOREM 2.4: Let V = {([0, a1 ], [0, a2 ], …, [0, an ])



Example 2.37: Let

V =

1

2i 20

10

[0,a ]

[0,a ]a Z ;1 i 10

[0,a ]

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

#

be a 10 u 1 column interval matrix semigroup. finite order for V has only finite number of elem

Example 2.38: Let

V =

1

2

3i

4

5

6

[0, a ]

[0, a ]

[0, a ]a Z {0};1 i

[0, a ]

[0, a ][0, a ]

- ª º° « »° « »° « »°

d d« »® « »° « »° « »° « »° ¬ ¼¯

be a 6 u 1 column interval matrix semigroup Clearly V is of infinite order.

Example 2.39: Let

1[0, a ]- ½ª º° °« »° °



semigroups. So we leave this simple task to the readexamples of these substructures.

Example 2.40: Let

V =

1

2

i 30

12

[0,a ]

[0,a ]

a Z ;1 i 12

[0,a ]

- ½ª º° °« »° °« »

d d® ¾« »° °« »° °¬ ¼¯ ¿

#

be a 12 u 1 column interval matrix semigroup. Take

P =

1

2i 30

12

[0,a ]

[0,a ]a {0,2,4,6,8,10,12,...,28} Z

[0,a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

#

P is a 12 u 1 column interval matrix subsemigroup of

Example 2.41: Let

V =

1

2i

12

[0,a ]

[0,a ]a Z {0};1 i 12

[0,a ]

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

#



W is a 12 u 1 column interval matrix subsemigro

Example 2.42: Let

V =

1

2

i 36

7

[0, a ]

[0,a ]

a Z ;1 i 7

[0,a ]

- ½ª º° °« »° °

« » d d® ¾« »° °« »° °¬ ¼¯ ¿

#

be a 7 u 1 column interval matrix semigroup.Take

I =

1

2i 36

7

[0,a ]

[0,a ]a {0,2,4,...,34} Z ;1 i

[0,a ]

- ª º° « »° « » d® « »° « »° ¬ ¼¯

#

I is a 7 u 1 column interval matrix subsemigroup

Example 2.43: Let

V =

1

2i

9

[0,a ]

[0,a ]a Z {0};1 i

[0,a ]

- ª º° « »° « » d d® « »° « »° ¬ ¼¯

#



I is a 9 u 1 column interval matrix subsemigroup of Vorder.

Example 2.44: Let

V =

1

2i 12

11

[0, a ]

[0, a ] a Z ;1 i 11

[0, a ]

- ½ª º

° °« »° °« » d d® ¾« »° °« »° °¬ ¼¯ ¿

#

be a 11 u 1 column matrix interval semigroup.

I =

1

2i 12

11

[0, a ]

[0, a ]a {0,3,6,9} Z ;1 i 11

[0, a ]

- ½ª º° °« »° °« » d d® ¾« »° °« »° °

¬ ¼¯ ¿

#

is a 11 u 1 column matrix interval subsemigroup of V

Example 2.45: Let

V =

1

2i 3

[0, a ][0,a ]

a Z ;1 i 8

- ½ª º° °« »° °« » d d® ¾« »° °« »° °

#.



W is a 8 u 1 column matrix interval subsemigrouWe can define m u n matrix interval semigro

Let V = {M = (mij) | mij = [0, aij]; 1 d i d n and Z+ {0} (or Zn or R + {0} or Q+ {0}} be a cm interval matrices. Define on V matrix addition

aij]) and N = ([0, bij]) then M + N = ([0, aij + bij])V under interval matrix addition is a semigrou m matrix interval semigroup. If m = n thesemigroup under multiplication as well as additio

We will describe both the operation withmatrices. Let

A =

[0,5] [0,1] [0,3][0, 2] [0, 4] [0,7]

[0,1] [0, 6] [0,5]

[0,0] [0, 2] [0,8]

ª º« »« »« »« »¬ ¼

and

B =

[0,1] [0, 2] [0,3][0, 4] [0,5] [0,6]

[0, 7] [0,8] [0,1]

[0, 2] [0, 4] [0,5]

ª º« »« »« »« »¬ ¼

be interval matrices with entries from Z9. Now

[0,5] [0,1] [0,3]

[0 2] [0 4] [0 7]

ª º« »« »

[0,1] [0

[0 4] [0

ª « «



Clearly the product is not defined. Now [0, a] u [0, b] = [0, ab].

If we take

A =[0,5] [0,7]

[0,1] [0,4]

ª º« »¬ ¼

and

B = [0,3] [0,1][0,5] [0,8]

ª º« »¬ ¼

with entries from Z+ {0}, then

AB = [0,5] [0,7][0,1] [0,4]

ª º« »¬ ¼

[0,3] [0,1][0,5] [0,8]

ª º« »¬ ¼

=[0,5][0,3] [0,7][0,5] [0,5][0,1] [0, 7][0

[0,1][0,3] [0, 4][0,5] [0,1][0,1] [0, 4][0

ª « ¬

=[0,15] [0, 35] [0, 5] [0, 56]

[0, 3] [0, 20] [0,1] [0, 32]

ª º« » ¬ ¼

=

[0,50] [0,61]

[0, 23] [0,33]

ª º

« »¬ ¼ .

Thus interval matrix addition and multiplication are wdefined



be a 4 u 2 interval matrix semigroup under addition.

Example 2.47: Let

1 4 7 10

2 5 8 11 i

3 6 9 12

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

V [0,a ] [0,a ] [0,a ] [0,a ] a Z[0,a ] [0,a ] [0,a ] [0,a ]

- ª º°

« » ® « »° « »¬ ¼¯

be a 3 u 4 interval matrix semigroup under additi

Example 2.48: Let

V =1 2 3

4 5 6 i 30

7 8 9

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

[0, a ] [0, a ] [0, a ] a Z ;1

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

- ª º° « » ® « »° « »¬ ¼¯

be a 3 u 3 interval matrix semigroup under multiThus we can as in case of other interval sem

interval matrix subsemigroups and ideals.This task of defining and giving examples is

exercise for the reader. Now having seen interval matrix semigroups

forth some of the important properties about thesAn interval matrix semigroup V is said to

matrix Smarandache semigroup (interval matri



Example 2.49: Let

V = 1 2i 12

3 4

[0, a ] [0, a ]a Z ;1 i 4

[0, a ] [0, a ]

- ½ª º° ° d d® ¾« »

¬ ¼° °¯ ¿

be the matrix interval semigroup under matrix multipClearly V has atleast one subset

P = 12

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

[0, a] [0, a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿V;

P is a matrix interval commutative subsemigroup of is a weakly commutative matrix interval semigroup.

Example 2.50: Let V = {([0, a1], [0, a2], …, [0, an]){0}; 1 d i d n} be a row matrix interval semigroup. Ca row matrix interval commutative semigroup.

Example 2.51: Let V = {([0, a1], [0, a2], …, [0, an]{0}; 1 d i d n} be a row matrix interval semigroup{([0, a], [0, a], …, [0, a]) | a Z7 \ {0}} V.

W is a row matrix interval group of V under theof V. Hence V is a Smarandache row matrix interval

THEOREM 2.5: Let V = {([0, a1 ], [0, a2 ], …, [0, an ]

{0}; p is a prime; 1 d i d n} be a row matrix interval

Take W = {([0, a], [0, a] , …, [0, a]) | a Z p \ {0}}

i i l H V i



group. Thus V is a row matrix interva

semigroup.

This proof is also left as an exercise for the reade

Example 2.52: Let Z30 = {0, 1, 2, …, 29} be a smultiplication modulo 30. V = {([0, a1], [0, a2], …Z

30; 1 d i d 9} be a row interval matrix semigr

a1], [0, a2], …, [0, a9]) | ai {0, 5, 10, 15, 20, 2W is a row interval matrix ideal of V.

THEOREM 2.7: Let V = {([0, a1 ], [0, a2 ], …, [0,

be a row interval matrix semigroup under multip

no proper ideals.

This proof is also left for the reader.

Let

V =

1 n

1 ni i i n

1 n

[0,a ] [0,a ]

[0,b ] [0,b ] a ,b ,c Z ;

[0,c ] [0,c ]

- ª º

° « »° « » ® « »° « »° ¬ ¼¯

"

"

# #

"

be the collection of all n u n interval square matrsquare matrix interval semigroup under muaddition, or used in the mutually exclusive sense

Example 2.53: Let



A =[0,3] [0,1]

[0,2] [0,2]

§ ·¨ ¸© ¹

and

B =[0,1] [0,2]

[0,2] [0,3]

§ ·¨ ¸© ¹

in V.

A u B =[0,3] [0,1]

[0,2] [0,2]

§ ·¨ ¸© ¹

u[0,1] [0,2]

[0,2] [0,3]

§ ·¨ © ¹

=[0,3][0,1] [0,1][0, 2] [0,3][0, 2] [0,1][0

[0, 2][0,1] [0, 2][0, 2] [0, 2][0, 2] [0, 2][

§ ¨

©

=[0,3] [0, 2] [0, 2] [0,3]

[0, 2] [0, 0] [0, 0] [0, 2]

§ ·¨ ¸ © ¹

= [0,1] [0,1][0,2] [0,2]

§ ·¨ ¸© ¹

.

We can define the notion of Smarandachesemigroup, Smarandache subsemigroup, Smhypersubsemigroup Smarandache p-Sylow Smarandache Cauchy elements of a S-semiSmarandache coset in case of interval matrix semiganalogous way [9].



be a 3 u 3 interval matrix semigroup under multi

1 2 31

4 5 6i

7 8 9

[0,a ] [0,a ] [0,a ]| A | 0;a

W A [0,a ] [0,a ] [0,a ]a 0

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

- ª ºz ° « » ® « » ° « »¬ ¼¯

V is a subgroup. Clearly V is a Smarandachesemigroup.

Example 2.55: Let

V =1 2

i 93 4

[0, a ] [0, a ]

| a Z ;1 i[0, a ] [0, a ]

- ª º°

d ® « »¬ ¼° ¯

be a square matrix interval semigroup.Take

W = 1 2 1 i

3 1

[0,a ] [0,a ] a {1,8},a 0[0,a ] [0,a ] | A | 0;2 i 3

- ª º° ® « » z d d¬ ¼° ¯

V is a square matrix interval group under multiplis a Smarandache square matrix interval semigro

Now we proceed onto give examples of Smarinterval subsemigroup or matrix interval semigroup.



be a 4 u 4 matrix interval semigroup.Take

1 2 3 4

5 1 7 8 1

9 10 1 12

13 14 15 1

[0,a ] [0,a ] [0,a ] [0,a ] | A | 0

[0,a ] [0,a ] [0,a ] [0,a ] a {1,...,W

[0,a ] [0,a ] [0,a ] [0,a ] i 2,3,4,

[0,a ] [0,a ] [0,a ] [0,a ] 10,12,13,

- z ª º° « » ° « » ® « » ° « »°

¬ ¼¯

V; W is a interval matrix group.

Now take P = {All 4 u 4 square matrices with inteform [0, ai]; where ai Z11 \ {0}} V; P is a inte

subsemigroup of V and W P; so P is a matSmarandache subsemigroup of V.

Example 2.57: Let

V = 1 2i 15

3 4[0, a ] [0, a ] a Z ;1 i 4[0, a ] [0, a ]- ½ª º° ° d d® ¾« »

¬ ¼° °¯ ¿

be a interval matrix semigroup.Take

W = 1 2 1 15

3 4 i

[0,a ] [0,a ] a {0,3,6,9,12} Z ;

[0,a ] [0,a ] a 0;2 i 4

- ª º° ® « » d d¬ ¼° ¯



Now we proceed onto give examples ofSmarandache interval matrix subsemigroup.

Example 2.58: Let V = {set all 5 u 5 intervaintervals of the form [0, ai] with ai Z43} be asemigroup.

Take W = {A / all 5 u 5 interval matrices wthe form [0, a1] with a1 Z43 \ {0} such that |A| zdiagonal interval matrix} V; W is a groupmatrix multiplication. So V is a interval matrisemigroup.

Further if we take P = {all 5 u 5 diagonal intervintervals of the form [0, ai] with ai Z43 \ {0}}interval matrix subsemigroup of V.

We see W P and W is the largest interv present in P. Thus P is a matrix intervsubsemigroup of V.

Example 2.59: Let

V = 1 2i 11

3 4

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

[0,a ] [0,a ]

- ª º° d d® « »

¬ ¼° ¯

be a matrix interval semigroup.

Let



subgroup of V. Infact V has no proper matrsubsemigroup containing P.

Example 2.61: Let V = {[0, ai] / ai Z6} be the masemigroup. Take W = {[0, 1], [0, 5]} V is a intervof V.

Take P = {[0, 1], [0, 5], [0, 0]} V isSmarandache subsemigroup we see P is a interval Smhyper subsemigroup of V.

It is left for the reader to prove the following theorem

THEOREM 2.8: Let V = {all n u n diagonal interv

with intervals of the form [0, a1 ], a1 Z p; all diagonare the same} is a Smarandache simple inter semigroup which is a Smarandache interval matrix s

THEOREM 2.9: Let V be a Smarandache matr semigroup.

Every Smarandache matrix interval hyper subsea Smarandache matrix interval subsemigroup

Smarandache matrix interval subsemigroup in geneS-matrix interval hyper subsemigroup.

Now we proceed onto give examples of Smarand

interval Lagrange semigroup (S-matrix intervalsemigroup).

Example 2.62: Let V = {[0, a] | a Z4} be asemigroup A = {[0 1] [0 3]} V is a interval subg



Example 2.64: Let V = {[0, a] | a Z10}semigroup. V is a S-weakly Lagrange interval se

The proof of the following theorem is left as an reader.

THEOREM 2.10: Every S-interval Lagrange seinterval weakly Lagrange semigroup.

Next we proceed onto illustrate S-p-Sylow intera S-interval semigroup.

Example 2.65: Let V = {[0, a] | a Z16} semigroup. A = {[0, 1], [0, 9]} V is a interval2/ o(V) but 22 / o(V), but V has S-2-Sylow intervorder 4 given by B = {[0, 6], [0, 2], [0, 4], [0, 8]}

We see in case of S-interval semigroup V w prime such that p / o(V) then we can have intervorder pD; where pD / o(V), we call such intervathe S-interval semigroup to be S-p-Sylow intervV.

We give examples of S-Cauchy elementsemigroup.We see a S-Cauchy element of a interval se

is such that xt = 1 and t / o(V).



THEOREM 2.11: Let V = {[0, a] / a Z p }; (p a priminterval semigroup under multiplication. No elementCauchy element of V.

The proof is obvious from the fact that no integer nthe prime p. Hence the claim.



Chapter Three

INTERVAL POLYNOMIAL SEMIGR

In this chapter we introduce the notion of intersemigroups. We call a polynomial in the variainterval polynomial if the coefficients of x are

form [0, ai] / ai Z p (or Zn or Z+ {0} or Q+

{0}.[0, 5] + [0, 7]x + [0, 2] x3 + [0, 14] x9 = p

polynomial in the variable x.



p (x) + q (x) = ([0, 2] + [0, 3] x2 + [0, 7] x7 + [0, 12] + [0, 7] x + [0, 14] x3 +

[0, 5] x8 + [0, 12] x9 + [0, 5] x20)

= ([0, 2] + [0, 12]) + [0, 7] x + [014]x3 + ([0, 7] x7 + [0, 10] x7) +([0, 11] x9 + [0, 12] x9) + [0, 5] x

= [0, 14] + [0, 7]x + [0, 3] x2 + [0, 17] x7 + [0, 5] x8 + [0, 23] x9 + [

Now we will just define interval polynomial multipli p (x) = [0, 3] + [0, 5] x2 + [0, 11] x5

and

q (x) = [0, 8] + [0, 1] x + [0, 9] x3.

p(x).q(x) = ([0, 3] + [0, 5] x2 + [0, 11] x5) ([0, 8][0, 9] x3).

= [0, 3] [0, 8] + [0, 5] [0, 8] x2 + [0, 1

+ [0, 3] [0, 1] x + [0, 5] x2 [0, 1] x [0, 1] x + [0, 3] [0, 9] x3 + [0, 5] x2

[0, 11] x5 [0, 9] x3.

= [0, 24] + [0, 40] x2 + [0, 88] x5 + [05] x3 + [0, 11] x6 + [0, 27]x3 + [0,

99] x8.

= [0, 24] + [0, 3]x + [0, 40]x2 + [0, 45]x5 + [0, 11]x6 + [0, 99] x8.



indeterminate} S under addition of interval po

semigroup defined as interval polynomial semigr


Example 3.1: Let

S =

9

ii ii 0

[0, a ]x a Z {0}

- ½ ® ¯ ¿¦

be a interval polynomial semigroup under additnumber of elements in S is infinite so S is aninterval polynomial semigroup.

Example 3.2: Let

S =3

ii i 11

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

be a interval polynomial semigroup. Clearly S is

We see clearly the interval polynomial semigrouabove examples are not compatible under multip

Example 3.3: Let

S =7

ii i

i 0

[0, a ]x a R {0}- ½ ® ¾

¯ ¿¦



is itself an interval polynomial semigroup under ad

we define W to be an interval polynomial subsemigro

We will illustrate this situation also by some exampl

Example 3.4: Let

S =

8

ii ii 0

[0, a ]x a Z {0}

- ½ ® ¾¯ ¿¦

be a interval polynomial semigroup under addition. T

W =

8i

i ii 0 [0, a ]x a 3Z {0}

- ½ ® ¾¯ ¿¦ S

W is a interval polynomial subsemigroup of S under

Example 3.5: Let

S =20

ii i 12

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦

be a interval polynomial semigroup under addition. T

W =10

ii i 12

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦ S;



We now proceed onto define polynomial inteunder multiplication.

DEFINITION 3.3: Let

V =f

- ®

¯¦ i

i i

i 0

[0,a ]x a Z {0 }

(or Z n or R+ {0}, or Q+ {0}, x a variable} beinterval polynomials. If product is defined on interval polynomial semigroup under multiplicat


Example 3.6: Let

S = ii i 8

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a polynomial interval semigroup underClearly S is of infinite order.

Example 3.7: Let

S =i

i ii 0 [0, a ]x a R {0}

f

- ½

® ¾¯ ¿¦ be a interval polynomial semigroup underCl l S i f i fi i d



be a interval polynomial semigroup.Take

P = ii i

i 0

[0, a ]x a Zf

- ½® ¾

¯ ¿¦ S;

P is a interval polynomial subsemigroup of S. Clearlas interval polynomial ideal of S.

Example 3.9: Let

S = ii i 30

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a interval polynomial semigroup under multiplicaTake

T = ii i 3

i 0

[0, a ]x a {0, 2, 4,6, ..., 26, 28} Zf

- ®

¯ ¦

T is a interval polynomial subsemigroup of S. Tinterval polynomial ideal of S.Thus we can have interval polynomial sub

which are not interval ideals of the polynomial semig

Example 3.10: Let

S = ii i

i 0

[0, a ]x a Q {0}f

- ½ ® ¾¯ ¿¦ S;

T is only a polynomial interval subsemigroup an



We will illustrate this by some examples.

Example 3.11: Let

S =2

ii i 3

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦

be a polynomial interval semigroup under additio

S = {0, [0, 1] x, [0, 1] x2, [0, 2] x, [0, 2] x[0, 1] + [0, 1]x, [0, 1] + [0, 1] x2, [[0, 1] + [0, 2] x2, [0, 2] + [0, 1] x, [0,[0, 2] + [0, 1]x2, [0, 2] + [0, 2]x, [0[0, 1] x2, …}.

We see[0, 1] + [0, 1] + [0, 1] = 0

[0, 1] x + [0, 1]x + [0, 1]x = 0[0, 2]x + [0, 2]x + [0, 2]x = 0

Thus we have several Cauchy elements,

polynomial interval semigroup.

It is left as an exercise for the reader to find thefind out whether the elements are S-Cauchy elem

Example 3.12: Let

S =5

ii i 2

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦



Example 3.13: Let

S =i

i i 7i 0

[0, a ]x a Z

f

- ½® ¾¯ ¿¦

be a polynomial interval semigroup. We see S isorder (S be under addition or multiplication).We cannot in this case define S – Cauchy element.

is a S-polynomial interval semigroup under additionS-polynomial interval semigroup under multiplicatio

Example 3.14: Let

S =8

ii i 8

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿

¦

be a interval polynomial semigroup under additioninterval polynomial semigroup. S is a S-commutat

polynomial semigroup.

Further it is easily verified S is a S-weak polynomial semigroup. For [0, 1]xi in S generatgroup under addition where 1 d i d 8.

Example 3.15: Let

S =6 i 7

ii 0

[0, a ]x x 1,

- ®¯¦ x8 = x, so on; ai

b l i l i l i d l i li



It is easily verified S is a S-interval polynomiasemigroup.

This simple result can be proved by the reader.

Example 3.16: Let

S = ii i 12

i 0

[0, a ]x a Zf

- ½® ¾

¯ ¿

¦

be interval polynomial semigroup under multiinterval polynomial ideals, for take

P =i

i ii 0 [0, a ]x a {0, 2, 4,6,8,10}

f

-

® ¯ ¦

is an interval polynomial ideal of S.

Example 3.17: Let

S = ii i 12

i 0

[0, a ]x a Zf

- ½® ¾¯ ¿¦

be a interval polynomial semigroup under addhas only interval polynomial subsemigroups and

P = ii i

i 0

[0, a ]x a {0, 2, 4, 6,8,10} Zf

- ®

¯ ¦



Example 3.19: Let

S =3

i 4 0i i 2

i 0

[0,a ]x a Z ; x x 1,

- ®

¯¦ x5 =

be polynomial interval semigroup under multiplicatio

Clearly S = {[0, 1]x, 0, [0, 1], [0, 1]x2

, [0, 1]x3

,1]x [0, 1] + [0, 1]x2, [0, 1] + [0, 1]x3, [0, 1]x + [0, 1]x[0, 1]x3, [0, 1]x2 + [0, 1] + [0, 1]x + [0, 1]x3, [0, 1] [0, 1]x3, [0, 1]x + [0, 1]x2 + [0, 1]x3, [0, 1] + [0, 1]x [0, 1]x3}, and o (S) = 16.

T = {[0, 1]x, [0, 1]x2, [0, 1]x3, [0, 1]} S i

polynomial subgroup of S.P = {[0, 1]x2, [0, 1]} S is also a interval

subgroup of S. Thus S is a S-interval polynomial Infact S is a commutative interval polynomial semidentity [0, 1]. Further I = {0, [0, 1] + [0, 1]x + [01]x3} S is a interval polynomial ideal of S.

Now having seen examples of polynomsemigroups we now proceed onto define symmetsemigroups or permutation interval semigroup

permutative semigroup in the following chapter.



Chapter Four

SPECIAL INTERVAL SYMMETRIC

SEMIGROUPS

In this chapter we for the first time introducemapping of n row intervals ([0, a1], …, [0, an]

forms the semigroup under the composition of misomorphic with the symmetric semigroup S(n).

We also define special interval symmetric group i l d i h h



Example 4.1: Let X = {[0, a1], ]0, a2]} be the interva

The set of all maps of X to X are as follows:K1: X o X given by

K1 ([0, a1]) = [0, a1] and K1 ([0, a2]) = [0, a2

K2 : X o X is given byK2 ([0, a1]) = [0, a2] and K2 ([0, a2]) = [0, a1

K3 : X o X is defined byK3 ([0, a1]) = [0, a1] and K3 ([0, a2]) = [0, a1

K4 : X o X is such thatK4 ([0, a1]) = [0, a2] and K4 ([0, a2]) = [0, a2

Thus S (X) = {K1, K2, K3, K4}; and S (X) under com

maps is an interval symmetric semigroup.Clearly |S (X)| = 22 = 4.

Example 4.2: Let X = {[0, a1], [0, a2], [0, a3]}; ai z a

0; 1 d i d 3. The maps of X to X is S(X) = {K1, K2, K3

K7, …, K26, K27}.

K1([0, ai]) = [0, ai]; i = 1, 2, 3;K2 ([0, a1]) = [0, a2], K2 ([0, a2]) = [0, a3],K2 ([0, a3]) = [0, a1], K3 ([0, a1]) = [0, a1];K3 ([0, a2]) = [0, a3], K3 ([0, a3]) = [0, a2];K4 ([0, a1]) = [0, a2], K4 ([0, a2]) = [0, a1];

K4 ([0, a3]) = [0, a3], K5 ([0, a1]) = [0, a3];K5 ([0, a2]) = [0, a2], K5 ([0, a3]) = [0, a1];K6 ([0, a1]) = [0, a3], K6 ([0, a2]) = [0, a1];K ([0 a ]) = [0 a ] K ([0 a ]) = [0 a ];



We have the following interesting theorem anwhen we say the interval [0, a] is mapped on to

the continuous interval segment 0 to a is macontinuous interval segment 0 to b. We seecontract or extend the interval for instance [0, 5

[0, 2 ] then certainly a contraction has taken prealize the map is not an embedding.

On the other hand if [0, 2 ] interval is mapped

can realize it as expansion. All these maps will we use the concept of finite element methodsmatrices or in any other applications.

However we see we have an isomorphismand S(X) where X = {[0, a1], [0, a2], …, [0, an]} distinct ai > 0 and ai z a j, if i z j; 1 d i, j d n. Aft

o [0, a p] are only maps. 1 d t, p d n.

Keeping this in mind we have the following.

THEOREM 4.1: Let S (n) be the symmetric semig{1, 2, …, n} and S (X) be the interval symmetri

the set X = {[0, a1 ], [0, a2 ], …, [0, an ]}, ai z a j , i, j d n. Then S (n) is isomorphic with S (X).

Proof : We know the symmetric semigroup S(n

0 0 05 2



Example 4.3: Let S(X) = {K1, K2, K3, K4} where Ki

{1, 2}; i = 1, 2, 3, 4. K1 (1) = 1, K1 (2) = 2, K2 (1) = 1

K3 (1) = 2, K3 (2) = 2, K4 (1) = 2 and K4 (2) = 1 is thesemigroup of order 22 = 4.

Now X = {[0, a1], [0, a2]} be the interval set a1 z0; i = 1, 2. S(X) = V1, V2, V3, V4} where VI : X o X4.

V1 ([0, a1]) = [0, a1], V1 = ([0, a2]) = [0, a2]V2 ([0, a1]) = [0, a1], V2 = ([0, a2]) = [0, a1]V3 ([0, a1]) = [0, a2], V3 = ([0, a2]) = [0, a2]

and V4 ([0, a1]) = [0, a2], V4 = ([0, a2]) = [0, a1].

S(X) is the interval symmetric semigroup of order 2

define a map P : S (2) o S (X) as follows.P (K1) = V1, P (K2) = V2

P (K3) = V3 and P (K4) = V4.

It is easily verified P is a semigroup homomorphismisomorphism. Hence the claim.

We will enumerate some of the properties enjointerval symmetric semigroup.

Example 4.4: Let S(X) be the set of all maps fromelement interval set X = {[0, a1], [0, a2], [0, a3]} to itsS (X) is the semigroup under the operation of com

map. Thus S (X) is the symmetric interval semigroup= 27.We see S (X) is S-symmetric interval semigroup

interval subgroups. For take P1 = {K1, K2} where



K2

:1 1

2 3

3 2

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ]

-°

®°¯

6

6

6

P2 = {K1, K3} where K is given by and

K3 :1 3

2 2

3 1

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ]

-°®°¯

6

6

6

P3 = {K1, K4} where K1 is given above

K4 :1 2

2 1

3 3

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ]

-°®°¯

6

6

6

P4 = {K1, K5, K6} where K1 is the identity map and

K5 :1 2

2 3

3 1

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ]

-°®°¯

6

6

6

and

K6 :1 3

2 1

[0, a ] [0, a ]

[0, a ] [0, a ]

[0 a ] [0 a ]

-°®°¯

6

6

6



THEOREM 4.2: Let S (X) be a interval symmetric

where X = {[0, a1 ], [0, a2 ], …, [0, an ]}, ai z a j; i z j

S(X) is a S-weakly cyclic interval symmetric semigro

THEOREM 4.3: Let S (X) be a interval symmetric

where X = {[0, a1 ], [0, a2 ], …, [0, an ]}, ai z a j , i z jS(X) is a S-interval symmetric semigroup.

THEOREM 4.4: Let S (X) be a interval symmetricwhere X = {[0, a1 ], [0, a2 ], …, [0, an ]}, ai z a j if i z j

i, j d n. S(X) is only a Smarandache weakly cinterval symmetric semigroup.

Proof : Take in S (X), P the collection of all one to o

of X to itself, then P is a interval symmetric subgro but is not a commutative interval symmetric group.

Hence S (X) is only a S-weakly commutativsymmetric semigroup.

We first proceed onto give the basic definition symmetric group.

DEFINITION 4.2: Let X = {[0, a1 ], [0, a2 ], …, [0,interval set S X denote the set of all one to one m

interval set X. S X under the composition of mappings

which will be known as the interval symmetric group

Example 4.5: Let SX = {K1, K2, K3, …, K6} where X[0 a2] [0 a3]} is the interval set SX is the interval



It is easily verified SX under the composition of m

called the interval symmetric group.

We have the following theorems which are for the reader to prove.

THEOREM 4.5: Let S X be the interval symmetric

{[0, a1 ], [0, a2 ], …, [0, an ]}, of n distinct intervalwhere S n is the symmetric group of degree n.

THEOREM 4.6: The S – interval symmetric semigits largest interval group S X to be contained

interval subset A = S X { V 1 , V 2 , …, V n } where V

ai ] for all j = 1, 2, …, n. true for i=1, 2, …,interval symmetric subsemigroup of S (X).

We will illustrate this situation by an example.

Example 4.6: Let X = {[0, a1], [0, a2], [0, a3

interval set of cardinality four. S (X) be the intesemigroup. Consider

A = SX 1 2 3

2 2 2

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

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

-§ °®¨ °© ¯

1 2 3 4

1 1 1 1

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

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

§ ·¨ ¸© ¹

Cl l A i b t d i th i t l



Clearly A is a proper subset and is the intervalsubsemigroup of S(X). Further A is a S-hyp

symmetric subsemigroup of S(X).

COROLLARY 4.1: S(X) the S-interval symmetric senot a S-simple symmetric semigroup.

We will be using the definitions of S-Lagrange sem

S-weakly Lagrange semigroup [9].

Example 4.7: Consider the interval symmetric semiwhere X = {[0, a1], [0, a2], [0, a3], [0, a4]} of 4 distinai > 0; i = 1, 2, 3, 4. Clearly order of S(X) is 44. We sSX to be interval subgroup of order 24. Clearly 24

S(X) is not a S-Lagrange interval symmetric However S(X) has interval subgroups of order twwhich divides 44. Hence S(X) is a S-interval symmeLagrange semigroup.

THEOREM 4.7: Let S(X) be a interval symmetric se

n-distinct intervals, i.e. X = {[0, a1 ], [0, a2 ], …, [0, a(1) S(X) is a S-interval symmetric semigroup(2) S(X) is not a S-interval symmetric

semigroup.(3) S(X) is a S-interval symmetric weakly

semigroup.

(4) S(X) has Smarandache interval symmet subgroups provided X has p numberintervals and p is a prime. If the numbeintervals in X is a composite number say

Pl f [9] f S d h C h



Please refer [9] for Smarandache Cauchy semigroup.

Example 4.8: Let S(X) be a interval symmetric X = {[0, a1], [0, a2], [0, a3], [0, a4], [0, a5]} wdistinct intervals, o(S(X)) = 55. We have K S(= identity map and 5 | 55. Thus S(X) has Smaraelements.

In view of this we have the following results.

THEOREM 4.9: Let S(X) be a interval symmetrorder nn where X = {[0, a1 ], [0, a2 ], …, [0, an Cauchy elements.

The proof is left as an exercise for the reader.

However we will illustrate this situation by an ex

Example 4.9: Let S(X) be a interval symmetri

order 66. S(X) has S-Cauchy elements.For take

Kt = 1 2 3 4 5

2 3 4 5 6

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

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

§ ¨ ©

in S(X). Clearly (Kt)6 = identity element of S(X)

S-Cauchy elements.

THEOREM 4 10: Let S(X) be a interval symmetric se



THEOREM 4.10: Let S(X) be a interval symmetric se

order nn; n a composite number S(X) has S-Cauchy e

Proof: Take

Ki = 1 2 n 1 n

2 3 n 1

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

[0, a ] [0, a ] [0, a ] [0, a ]§

¨ ©

"

"

in S (X). Clearly (Ki)n = identity element of S (X). Ta

prime then

1 2 p 1 p p 1

t2 3 p 1 p 1

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

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

§ K ¨

©

"

"

in S(X) is such that (Kt) p = identity element of S(X)

has S-Cauchy elements.Cayley’s theorem for S-semigroups can also be

case of S-interval semigroups.

Please refer for S-semigroup homomorphissemigroup automorphism [9]. Since S(X) # S(n) wha permutation of (1, 2, …, n) and S(X) is a intervalsemigroup with n interval set X = {[0, a1], [0, a2], …> 0 and ai z a j if i z j; 1 d i, j d n.

We can have Cayley’s theorem for S-interval sSeveral interesting results in classical group theo proved for S-interval semigroups with modifications.

We will first illustrate special intervals



We will first illustrate special intervals.

Example 4.10: Let X = {[7, 12], [5, 10], [3, 8]}interval set. Suppose X = {[5, 10], [6, 7], [54, 5not a special interval set as 7 < 9 and 6 < 5 so afor every ai and b j.

Let X = {[a1, b1], [a2, b2]} be the special inteinterval set generated by X denoted by ¢X² is {

[a2, b1], [a2, b2]} is an interval set.Let X = {[a1, b1], [a2, b2], [a3, b3]} be the spe

then the interval set generated by X denoted by ¢[a2, b2], [a3, b3], [a1, b2], [a1, b3], [a2, b1], [a2,

b2]}.Thus we see if X = {[a1, b1], [a2, b2], …, [an,

interval set then the interval set generated by X is {[a1, b1], …, [an, bn], [a1, b2], [a1, b3], …, [a1, [an, b2], …, [an, bn-1]}. Clearly the number of elen2.

Now we proceed onto define the notion of

symmetric semigroup or interval special symmet

DEFINITION 4.3: Let X be a special interval set

set generated by X. S( ¢ X ² ) set of all mappings

S( ¢ X ² ) is defined as the special interval symmetrinterval special symmetric semigroup.

The order of S ( ¢ X ² ) = 2

2 n

n .

W ill ill t t thi it ti b l

We can as in case of interval semigroups build



We can as in case of interval semigroups build of special interval groups also.

S¢X² = {all one to one maps of ¢X² to itself} interval symmetric group. Clearly SX S¢X²; likewS(¢X²).

SX is a proper subgroup of S¢X² and |SX| = | X |

X . Further |S (X)| = |X||X| and |S(¢X²)| = | ¢X²| | ¢X² |.

The following results are important.

THEOREM 4.11: Let X = {[a1 , b1 ], [a2 , b2 ], …, [a

special interval set. ¢ X ² denotes the intervals gene

Let S ( ¢ X ² ) be the special symmetric interval semigr

is a S-special interval symmetric semigroup.

Proof : We know S(X) is a interval symmetric semigS(X) so S(X) is a S-interval symmetric semigroup interval symmetric group. Similarly S¢X² is tsymmetric interval group of S(¢X²) as the set of all mto ¢X² and S¢X² is the set of all one to one maps of ¢S(¢X²) is a S-special symmetric interval semigroup.

We have several interesting results associated with thsymmetric interval semigroups which we describe be

THEOREM 4.12: Let S( ¢ X ² ) be a special symme

semigroup P # S(X) is a S-special symmetr

subsemigroup of S( ¢ X ² ) and is not a S-specia

intervals {[a1, b2], [a1, b3], …, [an, bn 2] [an, bn 1]}



intervals {[a1, b2], [a1, b3], …, [an, bn-2] [an, bn-1]}fixes intervals if the intervals are mapped to itsel

Clearly P # S(X) and P is a special symsubsemigroup of S(¢X²). Now if we take all oneT = {[a1, b1], [a2, b2], …, [an, bn], [a1, b2], …, [the interval elements [a1, b2], …, [an, bn-1]. We sesymmetric group and is isomorphic with the sym

as |X| = n. Thus T # SX # Sn. Hence T is a integroup isomorphic to SX. Thus P is a S-speinterval subsemigroup of S(¢X²). Now if P is symmetric interval hyper subsemigroup then P mlargest special interval symmetric group of S(¢Xlargest special interval symmetric group B of S(¢all one to one maps of ¢X² to ¢X². Thus clearly Ba special symmetric interval hyper subsemigHence the claim.

THEOREM 4.13: The special interval symme

S( ¢ X ² ) is not a S-Lagrange special inte semigroup but is a S weakly Lagrange symmetric semigroup.

Proof is left as an exercise for the reader.

We will however illustrate this situation by an ex

Example 4.12: Let X = {[a1, b1], [a2, b2], [a3, b3]set with 3 distinct elements. ¢X² = {[a1, b1], [a2, b ] [ b ] [ b ] [ b ] [ b ] [ b ]}

clearly number of elements of P is 9 and 9 | (9)9. H



clearly number of elements of P is 9 and 9 | (9) . His only a S-weakly Lagrange symmetric interval sem

The following theorems are expected to be proreader.

THEOREM 4.14: Let S( ¢ X ² ) be the special interva

semigroup S( ¢ X ² ) has S-p-Sylow subgroups.

THEOREM 4.15: Let S( ¢ X ² ) be the special interva semigroup. S( ¢ X ² ) has S-Cauchy elements.

THEOREM 4.16: Let S(X) be the S-symme semigroup where the interval set X has p distinct el

prime. Then S(X) has S-non p-Sylow subgroup for al

p.

We will illustrate this situation by an example.

Example 4.13: Let X = {[a1, b1], [a2, b2], [a3, b3], b5]} be the interval set with five distinct intervals.

interval symmetric semigroup. Clearly o(S(X)) = 55. 3 is a prime. S(X) has a subgroup P of order three g

K = 1 1 2 2 3 3 4 4 5

2 2 3 3 1 1 4 4 5

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

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

§ ¨ ©

P = {identity element of S(X), K, K2, as K3 = identityS(X)}. Thus o(P) = 3 and 3 / 55.Consider

[ b ] [ b ] [ b ] [ b ] [§



Chapter Five

NEUTROSOPHIC INTERVAL SEMI

In this chapter we for the first time introduceneutrosophic interval semigroups and pure neutrsemigroups and discuss several interesting pr

with them.We just briefly introduce the notations whusing in this chapter and as well as in the followi

PN (C + I) = {[0, tI] | tI C + I {0}} is the pure n



( ) {[ , ] | { }} pcomplex intervals.

N (Zn) = {[0, a] | a = x + yI; x, y Zn and I the intethe neutrosophic modulo integers intervals,

N(Z + {0}) = {[0, a] | a = x + yI where x, y Z + neutrosophic integer intervals;

N (Q + {0}) = {[0, a] | a = x + yI where x, y Q + neutrosophic rational intervals;

N (R + {0}) = {[0, a] | a = x + yI where x, y R + neutrosophic real intervals and

N (C + {0}) = {[0, a] | a = x + yI where x, y C + neutrosophic complex intervals.

It is to be noted that even if the mention of wh

interval is not made it can be easily understood by th Now we proceed onto define the notion of n

interval semigroup.

DEFINITION 5.1: Let S = {[0, tI] | tI Z n I}; S undmodulo n is a semigroup called the interval pure n

semigroup or pure neutrosophic interval semigroup.

We can replace ZnI by Z+I or Q + I or R + I or C + I.We will illustrate this by examples.

Example 5.1: Let S = {[0, aI] | aI Z5I} be a pure ninterval modulo integer semigroup under addition.

Example 5.2: Let S = {[0, aI] | aI Z + I {0}}t hi i t l i t i d dditi

Example 5.4: Let S = {[0, aI] | aI R + I {



neutrosophic interval real semigroup under add

order.

Example 5.5: Let S = {[0, aI] | aI C + I {neutrosophic complex interval semigroup undinfinite order.

Now having seen neutrosophic interval seaddition of finite and infinite order we caneutrosophic interval semigroup under mureplacing addition by multiplication of intervals way [0, aI], [0, bI] = [0, abI] and [0, aI] + [0, bI]In the case of multiplication we do not include C

–1 and [0, i] [0, i] = [0, -1], not defined.

We give only examples of pure neutrosophic intunder multiplication.

Example 5.6: Let P = {[0, aI] | aI Z8I} be a pu

interval modulo integer semigroup under multipl

Example 5.7: Let W = {[0, aI] | aI Q+ I neutrosophic rational interval semigroup underClearly W is of infinite order.

Example 5.8: Let T = {[0, aI] | aI Z+ I neutrosophic integer interval semigroup under mis also of infinite order.

We will illustrate these situations before we



notion of zero divisors, nilpotents and idempoten

neutrosophic interval semigroups.

Example 5.10: Let S = {[0, aI] | aI Z9I} neutrosophic modulo integer interval semigromultiplication. Take P = {[0, aI] | aI {0, 3I, 6I}}

pure neutrosophic modulo integer interval ideal of S.

Example 5.11: Let M = {[0, aI] | aI Z+ I {0}neutrosophic interval semigroup, clearly P = {[0, aI] {0}} M is a pure neutrosophic interval subsemiP is also a pure neutrosophic interval ideal of M.

Example 5.12: Let S = {[0, aI] | aI Z12I} neutrosophic interval semigroup under addition mod{[0, aI] | aI {0, 2I, 4I, 6I, 8I, 10I} Z12I} neutrosophic interval subsemigroup of S. Clearly S ineutrosophic interval ideal of S.

Example 5.13: Let T = {[0, aI] | aI Z+ I {0}neutrosophic interval semigroup under addition. Clno ideals only pure neutrosophic interval subsemigro

Example 5.14: Let W = {[0, aI] | aI Q+I {0}neutrosophic interval semigroup under addmultiplication). W has only pure neutrosophsubsemigroups and has no ideals.

E l 5 15 L N {[0 I] | I R+I {0}

aI Q+I {0}} or N = {[0, aI] | aI R +Iddi i ( l i li i ) i i h



addition (or multiplication) is torsion free. Thus

not contain any element of finite order.Before we proceed to define neutrossemigroups which are not pure we discuss abouidempotents and nilpotents in these structuneutrosophic interval semigroup built using Z+I{0} or R +I {0} under addition or multip

interval idempotents or interval zero divisonilpotents. However all pure neutrosophic inter

built using ZnI have zero divisors, nilpotents anda composite number).

Example 5.16: Let S = [0, aI] | aI Z1

neutrosophic interval semigroup under multiplicx = [0, 6I] in S. Clearly [0, 6I] [0, 6I] = [0, 36I] =has non trivial interval idempotent. Also y = [0, that y2 = [0, 10I] [0, 10I] = [0, 100I] = [0, 10I],an interval idempotent.

Also t = [0, 11t] in S is such that t2 = [0, 11

121I] = [0, I] is a torsion element in S as [0, I] unit. w = [0, 4I] in S is such that w2 = [0, 4I] [0, [0, I] is a torsion element and is not a S-Cauchy 2 u 15.

We see [0, 3I]. [0, 5I] = [0, 15I] = 0 is th

divisor in S. Also [0, 5I] [0, 6I] = [0, 30I] = 0 is zero divisor in S. Further [0, 3I] [0, 10I] = [0, 30interval zero divisor in S.

Example 5.18: Let S = {[0, aI] | aI Z30I} t hi i t l i d lti li ti



neutrosophic interval semigroup under multiplicati

o(S) = 30. [0, I] is the identity element of S.S is a S-pure neutrosophic interval semigroupelements which are units or self inversed. For [0, 29I[0, I] where [0, 29I] S.

[0, 6I] S is an idempotent of S as [0, 6I] [0, 6I

= [0, 6I]. We have [0, 6I] [0, 10I] = [0, 0] = 0 and [0,= [0, 0] = 0 to be zero divisors. Several other zero

present [0, 2I], [0, 15I] = 0 and so on.Also [0, 10I] [0, 10I] = [0, 100I] = [0, 10I] is an

of S.[0, 16I [0, 16I] = [0, 256I] = [0, 16I] is also an

of S. This S has no nilpotents.

Now we proceed onto give examples of pure ninterval semigroups which has non trivial nilpotents.

Example 5.19: Let S = {[0, aI] | aI Z27I}

neutrosophic interval semigroup. Consider [0, 3I] [0, 3I]3 = [0, 27I] = 0 thus [0, 3I] is a nontrivielement of S.

Also [0, 9I]2 = [0, 0] is again a nontrivial nilpotof S. Further [0, 18I] S is nilpotent of order two18I]2 = 0. But [0, 26I] in S is such that [0, 26I]2 = [0

in S. Further [0, 4I] [0, 7I] = [0, 28I] = [0, I], [0inverse of [0, 7I]. Likewise [0, 2I] is the inverse of[0, 2I] [0, 14I] = [0, I], [0, 11I] [0, 5I] = [0, I] is agaS

{[0, 3I], [0, 9I]}; P5 = {0, I], [0, 5I]} and P6 = {[7I] [0 11I]}



7I], [0, 11I]}.

Example 5.21: Let S = {[0, aI] | aI Zneutrosophic interval semigroup. It is easily prweakly Lagrange pure neutrosophic interval sem

Example 5.22: Let S = {[0, aI] | aI Z

neutrosophic interval semigroup. It is easily vsimple pure neutrosophic interval semigroup.

Example 5.23: Let P = {[0, aI] | aI Zneutrosophic interval semigroup. P is a S-neutrosophic interval semigroup.

Example 5.24: Let T = {[0, aI] | aI Zneutrosophic interval semigroup. T is not a Sneutrosophic interval semigroup.

Example 5.25: Let W = {[0, aI] | aI Z

neutrosophic interval semigroup. W does not cneutrosophic interval hyper subsemigroup.

Example 5.26: Let T = {[0, aI] | aI Z2

neutrosophic interval semigroup. A = {[0, I]subgroup of order 2 in T. Thus T has S-non

neutrosophic interval subgroup.

THEOREM 5.1: Let S = {[0, aI] | aI Z p; p a pneutrosophic interval semigroup S has no pur

THEOREM 5 2: Let W = {[0 aI] | aI Z I; p a prime



THEOREM 5.2: Let W = {[0, aI] | aI Z p I; p a prime

neutrosophic interval semigroup. W has no S-Cauchy

THEOREM 5.3: Let W = {[0, aI] | aI Z p I; p a primeneutrosophic interval semigroup. Clearly W has

subgroup which cannot be properly contained in a pneutrosophic interval subsemigroup.

Proof: Given W = {[0, [0, I], [0, 2I], [0, 3I], …, [0,S-pure neutrosophic interval semigroup of order p,The pure neutrosophic interval subgroups of W are A[0, (p-1)I]} and A2 = {[0, I] [0, 2I], …, [0, (p-1)I]}. {0} is a pure neutrosophic interval subsemigr

Hence A1 {0} is a S-pure neutrosophic interval suof W. But A2 cannot be strictly contained in any prof W. So A2 cannot be contained in a pure neutrosopsubsemigroup of W. Hence the claim.

It is left for the reader to prove the following theo

THEOREM 5.4: Let S = {[0, aI] | aI Z n } be a S-Lagneutrosophic interval semigroup then S is a S-weakl

pure neutrosophic interval semigroup.

Proof : Clear from the very definition. Now having seen properties about pure

interval semigroup we now proceed onto define ninterval semigroup.

DEFINITION 5 2: Let S = {[0 a + bI] | a b Z }

Example 5.29: Let T = {[0, a + bI] | a, b Qneutrosophic interval semigroup under addition o



neutrosophic interval semigroup under addition o

Example 5.30: Let W = {[0, a + bI] | a, b Zneutrosophic interval semigroup under additioninfinite semigroup.

It is important and interesting to note

neutrosophic interval semigroups are alwsubsemigroups of a neutrosophic interval semigr

Example 5.31: Let S = {[0, a + bI] | a, bneutrosophic interval semigroup. T = {[0, bI] | ba neutrosophic interval subsemigroup. We se

neutrosophic interval semigroup, which is a subs

Example 5.32: Let W = {[0, a + bI] | a, b Qneutrosophic interval semigroup. Take S = {[0, {0}} W, S is a neutrosophic interval subsemig

We can as in case of pure neutrosophic intedefine interval subsemigroups and ideals ointerval semigroups, which is left as an exercise

Example 5.33: Let S = {[0, a + bI] | a, bneutrosophic interval semigroup. Now T = {[0, {0, 2, 4, 6, 8, 10, 12, 14, 16, 18} Z20} neutrosophic interval subsemigroup of S. C = {[ {0, 5, 10, 15} Z20} S is a proper neutro

Example 5.35: Let S = {[0, aI] / aI Z19I} bneutrosophic interval semigroup Clearly S has n



neutrosophic interval semigroup. Clearly S has n

neutrosophic interval hyper sub semigroup. W = {[0 b Z19}, the neutrosophic interval semigroup contneutrosophic interval subsemigroup. Clearly Sneutrosophic interval semigroup.

The properties of S-coset in case of pure neutrosop

semigroup and neutrosophic interval semigroup can by the reader. For the notion of S-coset refer [9].It is interesting to note that every neutrosoph

semigroup contains an interval semigroup whneutrosophic.

In view of this we make the following definition

DEFINITION 5.3: Let S = {[0, a+bI] / a, b Z +

or Q+ {0}, or R

+ {0}) be a neutrosoph

semigroup. Take W = {[0, a] / a Z + {0} (or (or

{0}, or R+ {0})} S; W is a interval semigroup whneutrosophic interval semigroup. We call W to b

neutrosophic interval subsemigroup of S.

We will illustrate this by examples.

Example 5.36: Let S = {[0, a + bI] |a, b Z8} be a ninterval semigroup . Clearly W = {[0, a] | a Z

pseudo neutrosophic interval subsemigroup of S. Fuisomorphic to Z8. P = {[0, bI] | b Z8I} Sneutrosophic interval subsemigroup of S.

(0, 1 + I) (0, 2+3I) = [0, 2 + 2I + 3I + 3I]= [0, 2 + 3I]



[0, 2 3I]

(0, 2 +3I) (0, 3I+2I) = [0, 6 + 9I + 4I + 6I]= [0, 1 + 4I]

(0, 1 + 4I) (0, I+4) = [0, I + 4I + 4 + 16I]= [0, I + 4]

(0, 3 + 3I)2 = [0, 9 + 9I + 18I]= [0, 4 + 2I]

[0, 2 + 3I]2 = [0, 4 + 9I + 12I]= [0, 4 + I]

and so on.

As in case of pure neutrosophic interval semfind interval zero divisors, interval idempotentand interval nilpotents in case of neutros

semigroups.

Example 5.38: Let S = {[0, a + bI] |a, b Z6} beinterval semigroup. S has interval zero divisors g

[0, 3I] [0, 2I] = [0, 0] = 0.

[0, 3] [0, 2] = [0, 0] = 0.[0, 3I]2 = [0, 3I] and[0, 3]2 = [0, 3]

We see [0, 5] [0, 5] = [0, 1] and [0, 5I] [0, 5I] = [



[ , ] [ , ] [ , ] [ , ] [ , ] [

The reader can study these special elements neutrosophic interval semigroups. Compare (i) elements in neutrosophic interval semigroups and t

pure neutrosophic interval semigroups. (ii) substructures like ideals, hyper subsemigroups in

structures.



Chapter Six

NEUTROSOPHIC INTERVAL MAT

SEMIGROUPS AND FUZZY INTERSEMIGROUPS

This chapter has three sections. In section one

neutrosophic interval matrix semigroups and puinterval matrix semigroups are introduced. In sneutrosophic interval semigroup polynomials aninterval polynomial semigroups are introduce

We will be using the notations given in chapter book.



DEFINITION 6.1.1: Let A = {([0, a1 ], [0, a2 ], …, [0

Z n I (or Z + I {0} or Q+ I {0}, R+ I {0})} be a rneutrosophic matrix. Define usual addition on Abecomes a semigroup. A is defined as the pure ninterval row matrix semigroup under addition. Wh

(multiplication) is used instead of addition then we a pure neutrosophic interval row matrix semigrmultiplication.

We will illustrate both situations by some examples.

Example 6.1.1: Let A = {([0, a1], [0, a2], …, [0, a9]) {0}} be a pure neutrosophic interval row matrix undeIf

x = ([0, a1], [0, a2], …, [0, a9])and

y = ([0, b1], [0, b2], …, [0, b9])

are in A then

x + y = ([0, a1], [0, a2], …, [0, a9]) + ([0, b1], [0, b2], = ([0, a1 + b1], [0, a2 + b2], …, [0, a9 + b9]

is in A so A is a pure neutrosophic interval row m

addition. Clearly A has infinite number of elements i

Example 6.1.2: Let X = {([0, a1], [0, a2], [0, a3], [0, aai Z3 I; 1 d i d 5} be a pure neutrosophic interval

Example 6.1.4: Let S = {([0, a1], [0, a2], [0, a3]



a12]) | ai R +

I {0}) be a pure neutrosophmatrix semigroup under addition (or multiplicais of infinite order we can define subsemigroups case of other pure neutrosophic semigroups.

Here we only describe them by some examples.

Example 6.1.5: Let T = {([0, a1], [0, a2] , …, [0, {0}} be a pure neutrosophic interval row maunder multiplication. Take W = {([0, a1], [0, a2] 3Z+I {0}} T; W is a pure neutrosophmatrix subsemigroup of T. It can also be verif

pure neutrosophic interval row matrix ideal of T.

Example 6.1.6: Let S = {([0, a1], [0, a2] , [0, a3])d 3} be a pure neutrosophic row matrix inteunder multiplication.

Consider T = {([0, a], [0, a] , [0, a]) | a Z

pure neutrosophic row matrix interval subsemig pure neutrosophic row matrix interval subsem{([0, a], [0, a] , [0, a]) | a Z7I \ {0}} neutrosophic row matrix interval group under mu

Example 6.1.7: Let P = {([0, a1], [0, a2] , [0, a3]

Z+I {0}; 1 d i d 8} be a pure neutrosopinterval semigroup. Take S = {([0, a1], [0, a2], …5Z+ I {0}} P; S is a pure neutrosophic

b i ll hi

ai Z+I {0}, 1 d i d 12} T is a pure neutromatrix interval subsemigroup of T. Clearly W is



neutrosophic row matrix interval ideal of T. T ineutrosophic row matrix interval semigroup for S = a2], …, [0, a12]) | ai Q+I, 1 d i d 12} T is a pure nrow matrix interval group.

Example 6.1.9: Let V = {([0, a1], [0, a2], …, [0, a18

{0}, 1 d i d 18} be a pure neutrosophic row masemigroup. V is a S-pure neutrosophic row matsemigroup. V has no pure neutrosophic row matideal but has pure neutrosophic row matrisubsemigroup.

Now having seen pure neutrosophic row-ma

semigroups we now proceed onto define pure ncolumn matrix interval semigroups. However it is imention that no pure neutrosophic row matrsemigroup under addition has non trivial pure neutromatrix interval ideals, built using ZnI or Z+I {0} oor Q+I {0}.

We can define pure neutrosophic column matsemigroup under addition.

However multiplication cannot be definedneutrosophic column matrix interval semigroup acompatible.

We give examples of them. We cannot define multi pure neutrosophic column matrix interval semigroup

Example 6 1 10: Let

Example 6.1.11: Let



Q =

1

2i

6

[0,a ][0,a ]

a Z I {0};1 i

[0,a ]

- ª º° « »° « » d d® « »° « »° ¬ ¼¯

#

be a pure neutrosophic column matrix interval saddition. Q is of infinite order.

Example 6.1.12: Let

T =

1

2i

18

[0,a ][0,a ]a Q I {0};1 i

[0,a ]

- ª º° « »° « » d d® « »° « »° ¬ ¼¯

#

be a pure neutrosophic column matrix interval saddition.

S =

1

2i

18

[0,a ]

[0,a ]a Z I {0}

[0,a ]

- ½ª º° °« »° °« » ® ¾« »° °

« »° °¬ ¼¯ ¿

# T

S is a pure neutrosophic column matrix interva

be a pure neutrosophic column interval matrix semigaddition.



T =

1

2i

25

[0, a ][0, a ]

a Q I {0};1 i 25

[0, a ]

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

#

T is a pure neutrosophic column interval matrix suunder addition. T is a pure neutrosophic column intesubsemigroup and is not a pure neutrosophic columatrix ideal.

Example 6.1.14: Let

X =

1

2i 8

8

[0, a ]

[0, a ]a Z I;1 i 8

[0, a ]

- ½ª º° °« »° °« » d d® ¾« »° °« »° °

¬ ¼¯ ¿

#

be a pure neutrosophic column interval matrix semigaddition. Take

W =

1

2i 8

[0, a ]

[0, a ]a {0, 2I, 4I, 6I} Z I;1 i 8

[0 ]

- ª º° « »° « » d d® « »° « »°¬ ¼

#

Now we proceed onto define pure neutromatrix semigroup using addition.



DEFINITION 6.1.2: Let P = {Set of all n u m in

with intervals of the form [0,ai ] where ai Z n I (

R+ I {0} or Z + I {0}} (m z n), P under

addition is a semigroup called the pure neutrinterval matrix semigroup.

Clearly when m z n we cannot define semultiplication. When m = n we can have purinterval matrix semigroup under additiomultiplication.


Example 6.1.15: Let

X =1 2

3 4 i 18

5 6

[0,a ] [0, a ]

[0, a ] [0, a ] a Z I;1 i

[0, a ] [0, a ]

- ª º° « » d ® « »° « »¬ ¼¯

be a pure neutrosophic 3 u 2 interval matrix seaddition. Clearly X is of finite order.

Example 6.1.16: Let

1 2 3[0, a ] [0, a ] [0, a ]Q {

- ª º°®

Example 6.1.17 : Let



S =

1 2

3 4

5 6i

7 8

9 10

11 12

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

[0, a ] [0,a ]a R I {0};1 i

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ]

- ª º° « »° « »° « »°

d d« »® « »° « »°

« »° « »° ¬ ¼¯

be a pure neutrosophic 6 u 2 interval matrix semigaddition. Clearly S is of infinite order. However define product semigroup as product is undefined in

We can define only one substructure in this c pure neutrosophic interval matrix subsemigroup.

We will illustrate this situation by some examples.

Example 6.1.18: Let

S =

1 2

3 4i 20

5 6

7 8

[0, a ] [0, a ]

[0, a ] [0, a ]a Z ;1 i 8

[0, a ] [0, a ]

[0, a ] [0, a ]

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

be a pure neutrosophic 4 u 2 interval matrix semigrouTake



Example 6.1.20 : Let



V = 1 2

3 4

[0, a ] [0, a ]

[0, a ] [0, a ]

-ª º°®« »°¬ ¼¯

where a1, a2, a3, a4 Z

be a pure neutrosophic interval square matrix semigaddition. V is of finite order.

Take

U = 5

[0, a] [0, b]a, b Z I

[0, a] [0, b]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿V,

U is a pure neutrosophic interval square matrix subseV under addition.

Clearly V is a S-pure neutrosophic interval sqsemigroup for

P =0 0 [0, I] [0, I] [0,2I] [0,2I]

, ,0 0 [0, I] [0, I] [0,2I] [0,2I]

-ª º ª º ª °®

« » « » « °¬ ¼ ¬ ¼ ¬ ¯

[0,3I] [0,3I] [0, 4I] [0, 4I],

[0,3I] [0,3I] [0, 4I] [0, 4I]

½ª º ª º°¾« » « »°¬ ¼ ¬ ¼¿

V is a pure neutrosophic interval square matrix groaddition. Hence the claim.

d fi d f

Take W = {All 5 u 5 interval neutrosophiintervals of the form [0, ai]; ai 5Z+ I {0}}



a pure neutrosophic interval square matrix subsInfact W is also a pure neutrosophic interval squof V.

Thus every pure neutrosophic interval subsemigroup built using Z+I {0} is a purinterval square matrix ideal of Z+I {0}.

Example 6.1.22: Let T = {6 u 6 interval matriceof the form [0, ai] where ai R +I {0}} be a puinterval square matrix semigroup under multip

pure neutrosophic interval square matrix subsem pure neutrosophic interval square matrix righ

ideals. Let P = {6 u 6 intervals matrices with form [0, ai] where ai Q+I {0}} T is a puinterval square matrix subsemigroup and is cleneutrosophic interval square matrix ideal of T.

Example 6.1.23: Suppose T given in example 6.

a pure neutrosophic square matrix interval seaddition. Then we see T has only subsemigroups

Now we proceed onto define the notion of restric

DEFINITION 6.1.3: Let S be a pure neutrosophic

semigroup and P S be a pure neutrosophic subsemigroup of S. Suppose T P be a

neutrosophic interval matrix subsemigroup of P d l f h d f b



Example 6.1.26: Let



S = 19

[0, a] [0, a]a Z I \ {0}

[0, a] [0, a]

- ª º° ® « »

¬ ¼° ¯

be a pure neutrosophic interval matrix semigroupideally relatively simple pure neutrosophic semigroup.

Also the class of all pure neutrosophic semigroups built using ZnI or Z+I {0} or R +I{0} under addition are ideally relativelyneutrosophic interval matrix semigroups.

COROLLARY 6.1.1: Let

ª -« °°« ®«

°« °« ̄¬

"

"

# # " #

"

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

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

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

be n u n interval matrices with a Z p I, p a

ideally relatively simple pure neutrosophic semigroup.

Thus we see the ideally relatively simple purmatrix interval semigroups depends also on the oi d fi d th t hi t i i t

Example 6.1.27: Let

-ª º°



V =

1

2i 9

3

4

[0,a ][0,a ]

a {x yI / x, y Z };1 i[0, a ]

[0,a ]

- ª º° « »° « » d d® « »° « »° ¬ ¼¯

be a neutrosophic interval matrix semigroup. Choose

1

2i 9

3

4

[0,a ]

[0,a ]W a {x yI}| x, y {0,3,6} Z ;1

[0,a ]

[0,a ]

- ª º° « »° « » d® « »°

« »° ¬ ¼¯

W is a neutrosophic column interval matrix subsemof finite order.

Example 6.1.28: Let

V =

1

2i

10

[0, a ]

[0, a ]a {x yI / x, y Z {0}};1 i

[0, a ]

- ª º° « »° « » d d® « »°

« »° ¬ ¼¯

#

be a neutrosophic column interval matrix semigroup

P is a neutrosophic column interval matrix subsof infinite order.



We see P is not a neutrosophic column intervalV of infinite order.

Example 6.1.29: Let

P =

1

2i

3

4

[0, a ][0, a ]

a {x yI / x, y R {0}[0, a ]

[0, a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

be a neutrosophic column interval matrix semigrTake

W =

1

2

i3

4

[0, a ]

[0, a ]a {x yI / x, y Z {

[0, a ][0, a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

W is a neutrosophic column interval matrix subInfact W is not an ideal. The neutrosophic c

matrix semigroups are always only under additnot possible for them to contain ideals.

Example 6 1 30: Let

T =1

2 i 12

[0, a ] 0

0 [0, a ] a {x yI / x, y Z };1 i

[0 ] 0

- ª º° « » d® « »°« »¬ ¼



3[0, a ] 0° « »¬ ¼¯

T is a neutrosophic matrix interval subsemigroup of an ideal of S. However S is of finite order. Take

B =1 2

3 4 i 12

5 6

[0, a ] [0, a ][0, a ] [0, a ] a Z I;1 i 6

[0, a ] [0, a ]

- ½ª º° °« » d d® ¾« »° °« »¬ ¼¯ ¿

B is an interval matrix subsemigroup however

neutrosophic interval matrix ideal of S, thus B is onneutrosophic interval matrix subsemigroup of S.

However S has no ideals.

Example 6.1.31: Let V =

1 2 3 4 5 i

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] a {x yI | x,

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] 1 i

- ª º ° ® « » d d¬ ¼° ¯

be a neutrosophic interval matrix semigroup. Take W

1 2 3 4 5i

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]a Z I {0}

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]- ª º°

® « »¬ ¼° ¯



neutrosophic matrix interval semigroup under additV has no neutrosophic matrix interval ideals.

Remark: If V i t hi i t l t i i



Remark: If V is a neutrosophic interval matrix semigmultiplication certainly V has non trivial neutrosopmatrix ideals, V built using Z+ {0}.

We give examples of neutrosophic interval matrneutrosophic interval matrix semigroup under multip

Example 6.1.34: Let V = {([0,a1], [0,a2], …, [0,an]) | | x, y Q+ {0}} be a neutrosophic interval semigroup. Take W = {([0,a1], 0, …, 0) | a1 = x1 + yQ+ {0}} V; W is a neutrosophic interval row m

of V. Example 6.1.35: Let V = {([0,a1], [0,a2], [0,a3], [0,a4

N (Z+ {0})} be a neutrosophic interval rsemigroup.

S1 = {([0,a1], 0, 0, 0, 0) | a1 N Z+ {0})

neutrosophic interval row matrix ideal of V.S2 = {(0, [0,a2], 0, 0, 0) | a2 N Z+ neutrosophic interval row matrix ideal of V.

Thus we have 5 ideals with only one coordinarest zero.

H1 = {([0,a1], [0,a2], 0, 0, 0) | a1, a2 N Z+ {0

a neutrosophic interval row matrix ideal of V.Likewise we have 5C2 = 10 ideals with only two coorzero and three of the coordinates zero.

L t P {([0 ] [0 ] [0 ] 0 0) | N Z+

In view of this we can have the following theoexcept the reader to prove.



THEOREM 6.1.2: Let V = {([0, a1 ], [0, a2 ], …, [

+ yi I} where xi , yi Z t (or Z + {0} or Q+ {0}

d i d n} be a neutrosophic interval row matrix seatleast (nC 1 + nC 2 + nC 3 + … + nC n-2 + n C

distinct neutrosophic interval row matrix ideals.

It is important to note that neutrosophic intervalsemigroups have no ideals. Likewise neutrosoph(s z t) matrix semigroups have no proper ideals.t = m the neutrosophic interval m u m matrix sem

trivial proper ideals.

It is pertinent to mention here that neutrosophicsemigroup in general may or may not satisfy thusual S-semigroups.

We define the determinant of neutrosophic

square matrix as follows:

Let |A| = 1 2

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]

ª º« »¬ ¼

= [0, a1] [0, a4] - [0, a2] [0, a3]

= [0, a1, a4] – [0, a2, a3]= [0, |a1 a4 – a2 a3|]If |a1 a4 – a2 a3| z 0 then we define A to be non sin



Let A = {The group generated by all n u

interval matrices in V} V. V is a S-neutro

matrix semigroup.



matrix semigroup.

Proof is direct and is left as an exercise for the reHowever we can have S-neutrosophic interv

semigroup under addition m z n.

Example 6.1.36: Let

V =1 2

3 4 i i i i i 10

5 6

[0,a ] [0,a ]

[0,a ] [0,a ] a x y I;x y Z };

[0,a ] [0,a ]

- ª º° « » ® « »° « »¬ ¼¯

be a neutrosophic interval matrix semigroup undTake

P =

1 2i i i i i

3 4

5 6

[0,a ] [0,a ]a x y I;x y {0, ,

[0,a ] [0,a ] 1 i 6[0,a ] [0,a ]

- ª º ° « »

® « » d d° « »¬ ¼¯

!

V; P is a neutrosophic interval matrix group Thus P is a S-neutrosophic interval matrix semig

Example 6.1.37: Let A = {All 10 u 10 neutromatrices with intervals of the form [0, ai] with en {0}} be a neutrosophic interval matrices sem

under multiplication (p a prime). V is a Smarandachinterval semigroup.

Example 6.1.39: Let A = {([0 a] [0 a] [0 a] [0



Example 6.1.39: Let A {([0, a], [0, a], [0, a], [0Z10I} be a S-neutrosophic row matrix interval semigmultiplication. M = {([0, a], [0, a], [0, a], [0, a]) | a 8I} Z10I} be the neutrosophic row matrix interval A.

([0, 6I], [0, 6I], [0, 6I], [0, 6I]) is the identity multiplication modulo 10I. Clearly xA = Ax = A for(0, 0, 0, 0), ([0, 5], [0, 5], [0, 5], [0, 5]). For [0, 0, 0, and ([0, 5], [0, 5], [0, 5], [0, 5]) M = (0, 0, 0, 0). TSmarandache normal neutrosophic matrix interval sthe S-neutrosophic matrix interval semigroup A.

Several other properties enjoyed by semigroups can by any interested reader in case of neutrosophic intesemigroups and pure neutrosophic interval matrix sem

6.2 Neutrosophic Interval Polynomial Semigro

In this section we for the first time introduce notioneutrosophic interval polynomial semigroups and ninterval polynomial semigroups and discuss a f

properties related with them. We will be using th

described in chapter five of this book.Throughout this book x will denote a variindeterminate.

When we say a polynomial with interval coef

f(x) = [0, 3 + I] x3 + [0, 5I] x2 + [0, 2/9] x +and

g(x) = [0, 8I] x9

+ [0, 9I + 8] x8

+ [0, 7 ] x7

+ [



g(x) [0, 8I] x + [0, 9I + 8] x + [0, 7 ] x + [+ [0, 9I] x2 + [0, 8] x2 + [0, 8/9 +

are neutrosophic interval coefficient polynomials

Now we will proceed onto give the defin

DEFINITION 6.2.1: Let S =f

-®

¯¦ i

i i

i 0

[0,a ]x a

{0} Q+ I {0}, R+ I {0} }, S under polynomial ma semigroup; S is defined as the pure neutro

polynomial semigroup under multiplication.

We will just illustrate how the product of two puinterval polynomials are done.

Suppose p(x) = [0, 3I] x2 + [0, 2I] x + [0, 7I] andx3 + [0, 3I]x + [0, 2I] be two pure neutro

polynomials in the variable x, then

p(x) g(x) = ([0, 3I] x2 + [0, 2I]x + [0, 7I]) (3I] x + [0, 2I])

= [0, 3I] x2 . [0, 12I] x3 + [0, 3I] x2

3I] x2 [0, 2I] + [0, 2I] x [0, 12I] x3I] x + [0, 2I] x [0, 2I] + [0, 7[0, 7I] [0, 3I]x + [0, 7I] [0, 2I]

= I is also used in every multiplication of pure nintervals. [10]

We will now give examples of pure neutrosoph



We will now give examples of pure neutrosoph polynomial semigroups under multiplication.

Example 6.2.1: Let

S = ii i n

i 0

[0, a ]x a Z If

- ½® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial semigmultiplication. Clearly S is of infinite order.

Example 6.2.2: Let

P = ii i

i 0

[0, a ]x a Z I {0}f

- ½ ® ¾

¯ ¿¦

be a neutrosophic interval polynomial semigmultiplication of infinite order.

Example 6.2.3: Let

T = ii i

i 0

[0, a ]x a Q I {0}f

- ½ ® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial semigmultiplication of infinite order.

E l 6 2 4 L t

Example 6.2.5: Let

P =3

ii i 12[0, a ]x a Z I;- ®¯¦



i 0[ , ] ;

-®¯¦

x4 = 1, x5 = x, x6 = x2, x7 = x3, x8 = 1 and soneutrosophic interval polynomial semimultiplication. P is of finite order.

Thus we have infinite number of pure neutro polynomial semigroups under multiplication. them are built using only ZnI; n < f. All other puinterval polynomial semigroups constructed usinQ+I {0} or R +I {0} are of infinite order.

Example 6.2.6: Let P =8

i 9i

i 0

[0, a ]x x 1

-®

¯¦ , x

…, x16 = x7 x17 = x8, x18 = 1 and so on where ai a pure neutrosophic interval polynomial semultiplication. Clearly P is also of infinite order.

Now we can construct pure neutrosophic intesemigroups using the operation of interval polyn

Example 6.2.7: Let

S =7

ii i[0, a ]x a Q I {0}

½- ° ® ¾

°¯ ¿¦

be a pure neutrosophic polynomial interval semigaddition of finite order.



Example 6.2.9: Let

W =3

ii i 31

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial semigaddition. W is of finite order.

Example 6.2.10: Let

B =2

ii i

i 0

[0, a ]x a R I {0}

- ½ ® ¾

¯ ¿

¦

be a pure neutrosophic interval polynomial semigaddition, B is of infinite order.

Now we will proceed onto study the substructures s

neutrosophic interval polynomial subsemigroup neutrosophic interval polynomial ideals.

Example 6.2.11: Let

P =i

i i 17i 0 [0, a ]x a Z

f

- ½

® ¾¯ ¿¦

b t hi i t l l i l i

Example 6.2.12: Let

W = ii ii 0

[0, a ]x a R I {0}

f

- ½ ® ¯ ¿¦



i 0¯ ¿¦

be a pure neutrosophic interval polynomial smultiplication. Clearly

B = ii i

i 0

[0, a ]x a Q I {0}f

- ½ ® ¾¯ ¿¦

is a pure neutrosophic polynomial interval subsand is not an ideal of W.

From these examples it is clear that we cneutrosophic interval polynomial subsemigroupideals.

Example 6.2.13: Let

W = ii i

i 0

[0, a ]x a Z I {0}f

- ½ ® ¯ ¿¦

be a pure neutrosophic interval polynomial saddition. Consider

T = 2ii i

i 0

[0,a ]x a 2Z I {0}f

- ½ ® ¾

¯ ¿¦



Example 6.2.16: Let

W =8

ii i 40

i 0

[0, a ]x a Z I

- ½® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial semig



be a pure neutrosophic interval polynomial semigTake

P =8

ii i

i 0

[0, a ]x a {0,10I, 20I,30I}

- ½® ¾

¯ ¿¦

P is a pure neutrosophic interval polynomial ideW is taken as a pure neutrosophic intervsemigroup under multiplication. (Here x9 = 1, on). Clearly W is of finite order. Infact Wneutrosophic polynomial interval semigroup.

Example 6.2.17: Let

S = ii i 2

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial smultiplication. Clearly S has non trivial purinterval polynomial subsemigroups.

Now having seen examples of substructuonto study more properties related with them.

Example 6.2.18: Let

P = i[0 a ]x a Q I {0}f

- ½ ® ¾¦

In view of this we now proceed onto study these

Example 6.2.19: Let P = {[0, t] xi | t Z6I; 0 d i d1} be a pure neutrosophic interval polynomial semigm ltiplication



multiplication.P = {0, [0, I], [0, 2I], [0, 3I], [0, 4I], [0, 5I], [0, I

[0, 3Ix], [0, 4Ix], [0, 5Ix], [0, Ix2], [0, 2Ix2], [0, 3Ix2

[0, 5Ix2], [0, Ix3], [0, 2Ix3], [0, 3Ix3], [0, 4Ix3], [0, 5I[0, 2Ix4], [0, 3Ix4], [0, 4Ix4], [0, 5Ix4], [0, Ix5], [03Ix5], [0, 4Ix5], [0, 5Ix5], [0, Ix6], [0, 2Ix6], [0, 3Ix6

[0, 5Ix6]} is of order 36.Consider W = {0, [0, 2I], [0, 2Ix], [0, 2Ix2], [0

2Ix4], [0, 2Ix5], [0, 2Ix6], [0, 4I], [0, 4Ix], [0, 4Ix2], [4Ix4], [0, 4Ix5], [0, 4Ix6]} P is a pure neutrosophicinterval subsemigroup of P. W is also a pure ninterval polynomial ideal of P.

Consider B= {[0, 3I], [0, 3Ix], [0, 3Ix2], [0, 3Ix3

[0, 3Ix5], [0, 3Ix6], 0} P, is both a pure neutrosop polynomial subsemigroup as well as ideal. Ho

/ o(P). If C = B \ {0} P is taken. C is o

neutrosophic interval polynomial subsemigroup anideal of P. By change of conventition in notation waIx] = [0, aI]x.

Example 6.2.20: Let V= {0, [0, I], [0, 2I], [0, 3I], [0,[0, 6I], [0, 7I], [0, Ix], [0, 2Ix], [0, 3Ix], [0, 4Ix], [

6Ix], [0, 7Ix], [0, Ix2

], [0, 2Ix2

], [0, 3Ix2

], [0, 4Ix2

], [6Ix2], [0, 7Ix2] | aI Z8I; 0 d a d 7 and x3 = Ineutrosophic interval polynomial semigroup of ord

l i l lti li ti

Example 6.2.21: Let V = {0, [0, I], [0, 2I], [0, 3[0, I]x, [0, 2I]x, [0, 3I]x, …, [0, 16I]x | x2 = I anda pure neutrosophic interval polynomial semigrZ17 I. Clearly o(V) = 33. V has pure neutrosopinterval subsemigroups whose order does not div



interval subsemigroups whose order does not div

Example 6.2.22: Let S = {[0, a]x, [0, a], [0, a]xI} = { 0, [0, I] [0, 2I], [0, 3I], [0, 4I], [0, I]x, [0, [0, 4I]x, [0, I]x2, [0, 2I]x2, [0, 3I]x2, [0, 4I]neutrosophic interval polynomial semigroup. ClBut S has proper pure neutrosophic intervsubsemigroups.

Example 6.2.23: Let S = {[0, a] xi | i = 0, 1, 2Z8I} = {0, [0, I], [0, 2I], ..., [0, 7I] [0, I] x, [0, 2I][0, I] x2, [0, 2I] x2, ..., [0, 7I] x2, [0, I] x3, [0, 2Ix3} be a pure neutrosophic interval polynomiaorder 29. Clearly S is a S- pure neutro

polynomial semigroup.Consider T = {0, [0, 2I], [0, 4I], [0, 6I], [0,

[0, 6I] x, [0, 2I] x2, [0, 4I] x2, [0, 4I] x3, [0, 2I] x6I] x3} S; T is also a pure neutrosophic intersubsemigroup of S. o(T) = 13, 13 / 29. Further 4I], [0, 4I]x, [0, 4I] x2, [0, 4I] x3} S; Y neutrosophic interval polynomial subsemigroup o(Y) = 5, 5 / 13 , 5 / 29.

Consider M = {[0, I], [0, 7I], [0, I]x, [0, 7I] I] x2, [0, I] x3, [0, 7I] x3} S. M is also a puinterval polynomial subsemigroup of order 8. W

V = {0, [0, I], ..., [0, 12I], [0, I] x, [0, 2I] x, ...,and o (V) = 25. Consider T = {0, [0, I], [0, 12I], [0, x} V is a pure neutrosophic interval polynomial suof V. Further o(T) = 5 and 5 | 25. Also T is neutrosophic interval polynomial subsemigroup as



neutrosophic interval polynomial subsemigroup as [0, 12 I], [0, I] x, [0, 12I] x} T is a group under prV is a S pure neutrosophic polynomial interval semig

Example 6.2.25: Let M = {[0, ai] xi | x3 = I, ai Z11

I], [0, 2I], ..., [0, 10 I] , [0, I] x, [0, I] x, [0, 2I] x, ...[0, I] x2, [0, 2I] x2, ..., [0, 10I] x2} be a pure ninterval polynomial semigroup. o(M) = 31. M ineutrosophic interval polynomial semigroup, as A =10 I]} M is a subgroup.

Also B = {[0, I], [0, 10I], [0, I] x, [0, 10 I] x , [0, 1 I] x2} M is a subgroup. Further T = {0, [0, I M is a S-pure neutrosophic interval polynomial suof M. Likewise P = { [0, I], [0, 10 I], [0, I] x, [0, 10x2, [0, 10 I] x2, 0} is a S-pure neutrosophic interval subsemigroup of M. But both T and P are not id

Clearly M has no pure neutrosophic interval polynom Example 6.2.26: Let S = {[0, ai] x

i | x3 = I ; ai Z12

neutrosophic interval polynomial semigroup of Clearly S is a S pure neutrosophic interval semigroup. S has nontrivial pure neutrosoph

polynomial ideal. For take T = {0, [0, 3I], [0, 6I],3I]x, [0, 6I] x, [0, 9 I] x2, [0, 3I] x2, [0, 6I] x2, [0, 9I] pure neutrosophic interval polynomial subsemigroid l f S T k K {0 [0 4I] [0 8 I] [0 4 I] [0

2 I], [0, I]x, [0, 2 I]x, [0, I]x2, [0, 2 I] x2, [0, I]x3

pure neutrosophic interval polynomial semigroup{0, [0, I], [0, 2 I]} M; is a pure neutro

polynomial subsemigroup of M.Clearly P is not an ideal of M Also T = {0 [



Clearly P is not an ideal of M. Also T {0, [I]x2, [0, I]x3} M is again a S – pure neutro

polynomial subsemigroup of M which is not anhas no proper ideals.

Example 6.2.28: Let P = {[0, ai] xi | ai Z2 I, x5

[0, I]x, [0, I ]x2, [0, I]x3, [0, I]x4} be a pure neutr polynomial semigroup of order six. Suppose instead of x5 = I we get a pure neutrosophic intersemigroup M of order seven. If we take x4 = I

pure neutrosophic interval polynomial semigrofive. But this W has nontrivial pure neutrosopinterval subsemigroup of order three given by x2}; 3 / 5.

Now one natural question is how many pr

neutrosophic interval semigroups of this type weThe answer is infinite, for when even xn = get either a prime order semigroup or an odd ord

Similarly when xm = I with m odd we getsemigroup. This is true when we take the basic s

pure neutrosophic polynomial interval semigrou

Z2I

Example 6.2.29: Let M = {[0, ai] xi | x2 = I, ai

t hi i t l l i l i f t

using Z3I is of order 5, 7, 9, 11, 13 and so on and (2n + 1) ; n = 2, 3, ..., f.

By this method we get very many special pure ninterval polynomial semigroups of varying orde



interval polynomial semigroups of varying ordeindependent of the order of the basic set (Z p I) on

built.

Example 6.2.30: Let K = {[0, ai] xi | x3 = I, ai Z4 I

I], [0, 2 I], [0, 3 I], [0, I]x, [0, 2 I]x, [0, 3 I]x, [0, I]x[0, 3 I]x2, [0, I]x3, [0, 2 I]x3, [0, 3 I] x3} be a pure ninterval polynomial semigroup of order 13. If x3 = Iorder of the semigroup would be, 10. If x2 = I then the semigroup is 7 and if x8 = I then order of thewould be 25 and so on. Thus using Z4 I wneutrosophic interval polynomial semigroups of orde16, 19, 22, 25, 28 and so on.

Thus if Zn I is used we will get pure neutrosop polynomial semigroups of order n+(n–1), n+2 (n–1)and so on.

All properties like S-Lagrange semigroup, semigroup, S-Cauchy element etc can be studied incspecial polynomial pure neutrosophic interval semigr

Now we proceed on to define neutrosophic interval semigroups.

DEFINITION 6.2.2: Let

V =f-

®¦ i

i i[0,a ] x a = xi + yi I N (Z n )

Example 6.2.31: Let

S = ii i 5i 0

[0,a ]x a N(Z )

f

- ½® ¾¯ ¿¦



be a neutrosophic interval polynomial smultiplication. Clearly S is of infinite order.

Example 6.2.32: Let

P = 2ii i

i 0

[0, a ]x a N(Z {0})f

- ®

¯ ¦

be a neutrosophic interval polynomial smultiplication. Clearly P is of infinite order.

Example 6.2.33: Let

W =i

i ii 0 [0, a ]x a N(R {0}

f

-

® ¯ ¦ be a neutrosophic interval polynomial smultiplication. Clearly W is also of infinite order

Example 6.2.34: LetS =

7i

i i 40i 0

[0, a ]x a N(Z )

- ½® ¾

¯ ¿¦

be a neutrosophic polynomial interval semigroup undof infinite order.

Example 6.2.36: Let



T =4

ii i

i 0

[0,a ]x a N(R {0})

- ½ ® ¾

¯ ¿¦

be a neutrosophic polynomial interval semigaddition. T is of infinite order.

Now we have seen finite and infinite neutrosophic interval semigroups we will proceed to illussubstructures.

Example 6.2.37: Let

T =9

i 10i i 7

i 0

[0, a ]x x 1 a N(Z )

- ½ ® ¾

¯ ¿¦

be a neutrosophic interval polynomial semigmultiplication for x11 = x, x12 = x2 and so on.

Example 6.2.38: Let

M =3 i 4

i ii 0

[0, a ]x a N(Z {0}), x 1

- ® ¯ ¦

be a neutrosophic polynomial interval semigroup(or under multiplication); W is a finite order sem

Now we proceed to give examples of substructur



Example 6.2.40: Let

K = ii i

i 0

[0, a ]x a N(Z {0})f

- ®

¯

¦

be a neutrosophic polynomial interval smultiplication. Consider

T =2i

i ii 0 [0, a ]x a N(Z {0})

f

- ½

® ¾¯ ¿¦

T is a neutrosophic polynomial interval subsemig

Example 6.2.41: Let

P = ii i

i 0

[0, a ]x a N(Z {0})f

- ®

¯ ¦

be a neutrosophic polynomial interval s

multiplication. Consider

T = i[0 a ]x a N(3Z {0})f

- ½ ® ¾¦

be a neutrosophic interval polynomial semigrouporder under multiplication. Consider

C i[0 ] {0 2I 4I 6I 8I 10I} (Zf-

®¦



C = ii i

i 0

[0,a ]x a {0,2I,4I,6I,8I,10I} (Z

® ¯ ¦

C is a neutrosophic interval polynomial subsem

infinite order under multiplication.

Example 6.2.43: Let

T =8

ii i 11

i 0

[0, a ]x a N(Z )

- ½® ¾

¯ ¿

¦

be a neutrosophic polynomial interval semigroup undof finite order

P =

4i

i i 11i 0 [0,a ]x a N(Z )

- ½

® ¾¯ ¿¦is a neutrosophic polynomial interval subsemigroufinite order.

Clearly T has no ideals.

Example 6.2.44: Let

- ½

M is a neutrosophic interval polynomial subseClearly M is not an ideal of W.

Example 6.2.45: Let



S =9

i 10i i 11

i 0

[0, a ]x x 1;a N(Z

- ®

¯ ¦

be a neutrosophic interval polynomial smultiplication. Take

T =9

ii i 11

i 0

[0,a ]x a Z I

- ½® ¾

¯ ¿

¦ S,

is a neutrosophic interval polynomial subsemigris also a neutrosophic interval polynomial ideafinite order.

Example 6.2.46: Let

M =12

ii i 3

i 0

[0, a ]x a N(Z )

- ½® ¾

¯ ¿¦

be a neutrosophic interval polynomial saddition. Clearly M is of finite order M hainterval polynomial subsemigroup but has no ide

W =12

i 13i i 3

i 0

[0, a ]x a Z with x 1

- ½ ® ¾

¯ ¿

¦

W is a interval polynomial subsemigroup b



W is a interval polynomial subsemigroup bneutrosophic. Thus W is a pseudo neutrosoph

polynomial subsemigroup of T. Further

A =12 i 13

i i 3i 0

[0, a ]x a Z I; x 1

- ½ ® ¾¯ ¿¦

is a pure neutrosophic interval polynomial subsemiA is an ideal of T but W is not an ideal of T.

Example 6.2.48: Let

M = ii i

i 0

[0, a ]x a N(Q {0})f

- ½ ® ¾

¯ ¿¦

be a neutrosophic interval polynomial semigmultiplication. Take

N = ii i

i 0

[0, a ]x a Q I {0}f

- ½ ® ¾

¯ ¿¦ M,

N is a neutrosophic interval polynomial subsemigrois an ideal of M



be a neutrosophic interval polynomial semigmultiplication. Consider

C = ii i 12

i 0

[0, a ]x a Z If

- ½® ¾¯ ¿¦ M,



¯ ¿

C is a neutrosophic interval polynomial ideal of M.

Example 6.2.51: Let

T = ii i

i 0

[0, a ]x a N(Z {0})f

- ½ ® ¾

¯ ¿¦

be a neutrosophic interval polynomial semigaddition. Choose

B = ii i

i 0

[0, a ]x a Z I {0}f

- ½ ® ¾

¯ ¿¦ T,

B is only a neutrosophic interval polynomial subseT. Infact

Sn = ii i

i 0

[0, a ]x a N(nZ {0}) N(Z {0f

- ®

¯ ¦

for n = 2, 3, …, f are neutrosophic interval subsemigroups of T and none of them are ideals of T



subsemigroup of V. T = {0, [0, 1]x [0, 1]x2 [0, 1]} a special neutrosophic interval polynomial subsemig

M = {0, [0, I], [0, I]x, [0, I]x2} S is a s

neutrosophic interval polynomial subsemigroup of [0, I], [0, 1] [0, I]x, [0, I]x2, [0, 1]x, [0, 1]x2} special neutrosophic interval polynomial subsemigro



special neutrosophic interval polynomial subsemigroB = {0, [0, 1 + I], [0, 1 + I]x, [0, 1 + I]x2}

special pure neutrosophic interval polynomial subseV.

C = {0, [0, 1 + I], [0, 1 + I]x, [0, 1 + I]x2, [0, 1]}a special neutrosophic interval polynomial subsemiand o(C) | o(V).

Take A = {0, [0, 1 + I], [0, 1 + I]x, [0, 1 + I]x2, is a special neutrosophic interval polynomial subsemand o(A) | o(V). P = {0, [0, 1], [0, I], [0, I]x, [0, I]xspecial neutrosophic interval polynomial subsemigand o(P) | o(V).

Example 6.2.55: Let V = {[0, a] xi | x2 = 1, a Nspecial neutrosophic interval polynomial semigr

multiplication.V = {0, [0, 1], [0, 2], [0, 3], [0, 1]x, [0, 2]x, [0,[0, 2I], [0, 3I], [0, I]x, [0, 2I]x, [0, 3I]x, [0, 1 + I] [0,1 + 3I], [0, 2 + I] [0, 2 + 2I], [0, 2 + 3I], [0, 3 + I] [0,3 + 3I], [0, 1 + I]x [0, 1 + 2I]x [0, 1 + 3I]x, [0, 2 + 2I]x, [0, 2 + 3I]x, [0, 3 + I]x [0, 3 + 2I]x, [0, 3 + 3I

order 31.Clearly o(V) is a prime. But V has seveneutrosophic interval polynomial subsemigroup. T1 =[0 2] [0 3] [0 1] [0 2] [0 3] } V i

V is a S special neutrosophic interval polynofor X = {[0, 1], [0, 3]} V is a group under mul

Example 6.2.56: Let V = {[0, a] xi | x2 = 1, aspecial neutrosophic interval polynomial semmultiplication V = {0 [0 1] [0 2] [0 3] [0



multiplication. V {0, [0, 1], [0, 2], [0, 3], [0,2]x, [0, 3]x, [0, 4]x, [0, I], [0, 2 I], [0, 3 I], [0,2I]x, [0, 3I]x, [0, 4 I]x, [0, 1 + I], [0, 1 + 2 I], [+ 4 I], [0, 1 + I]x, [0, 1 + 2 I]x, [0, 1 + 3 I]x, [0,

+ I], [0, 2 + 2 I], [0, 2 + 3 I], [0, 2 + 4 I], [0, 2 + I[0, 2 + 3 I]x, [0, 2 + 4 I]x, [0, 3 + I], [0, 3 + 2 I],3 + 4 I], [0, 3 + I]x, [0, 3 + 2 I]x, [0, 3 + 3 I]x, [0+ I], [0, 4 + 2 I], [0, 4 + 3 I], [0, 4 + 4 I], [0, 4 I]x, [0, 4 + 3 I]x, [0, 4 + 4 I]x}, that is order of Vwill give some of its substructures.

Take K 1 = {0, [0, 1], [0, 2], [0, 3], [0, 4]} special neutrosophic interval subsemigroup of oK 1 is not an ideal of V. Consider K 2 = {0, [0, I], [0, 4 I]} V, K 2 is a special pure neutrosubsemigroup of V and is not an ideal of V. K 32], [0, 3], [0, 4] [0, I], [0, 2 I], [0, 3 I], [0, 4neutrosophic interval subsemigroup of V and isV. K 4 = {0, [0, 1] [0, 2], [0, 3], [0, 4], [0, 1]x, [0,4]x} V is a special neutrosophic intervsubsemigroup of V and is not an ideal of V.

K 5 = {0, [0, 1 + I], [0, 2 + 2 I], [0, 3 + 3 I], [0+ I]x, [0, 2 + 2 I]x, [0, 3 + 3 I]x, [0, 4 + 4 I]x} neutrosophic interval polynomial subsemigroupalso not an ideal of V.

(b) V has interval subsemigroups

(c) V has neutrosophic interval subsemigroups(d) V is a S semigroup.

Example 6.2.57: Let V = {[0, a]xi | 0 d i d f and a{0}} be the special neutrosophic interval polynomia



{0}} be the special neutrosophic interval polynomiaunder multiplication of infinite order. Clearly V semigroup.

In view of this we have the following theorem.

THEOREM 6.2.4: Let V = {[0, a] xi | 0 d i d f , a

{0}), (or N(Q+ {0}) or N(R+ {0}))} beneutrosophic interval polynomial semigroumultiplication of infinite order. Then

(a) V has neutrosophic interval subsemigrouporder.

(b) V has interval subsemigroup of infinite order(c) V has no S –p –Sylow subgroup.(d) V has no S – Lagrange subgroup.

(e) V has no S- Cauchy elements.

Now we see in case of interval semigroups polynomial semigroups when we use N(Z+ {0}) {0}) or N (R + {0}) are of infinite order. Howevthem are S semigroups.

Several interesting properties related with thstudied.

DEFINITION 6.3.1: Let S = {[0, a] | 0 d a d 1} bof fuzzy intervals. For any x = [0, b] and y = [0{x, y} = min {[0, b], [0, a]} = {[0, min {a, b}]}.

Clearly S with min. operation on fuzzy semigroup known as the fuzzy interval semigrou semilattice with min operation.



p If we replace the min. operation on S by the

still we see S with max operation is a semigroup interval semigroup with max operation.

Example 6.3.1: Let S = {[0, 1/2n] | n = 0, 1, 2fuzzy interval semigroup under min. operation.

Example 6.3.2: Let T = {[0, (7/10)n] | n = 0, 1, 2fuzzy interval semigroup under max operation.

Example 6.3.3: Let W = {[0, 1/n] | 1 d n d 29interval semigroup under min. operation.

It is interesting and important to note that thesemigroups in examples 6.3.1 and 6.3.2 are o

where as the fuzzy interval semigroup given in eof finite order.

We can define fuzzy interval sub semigroups.

DEFINITION 6.3.2: Let S = {[0, a] | 0 d a d

interval semigroup with min. operation Suppose

d bd 1} S, and if W itself is a fuzzy interval min operation then we call W to be a



I = ^ `m10, m 20,21,...,40

5ª º « »¬ ¼

is a fuzzy interval subsemigroup. It is easily veriof T.



Example 6.3.8: Let

S =

^ `n

10, n 0,1,2,...,60

7

ª º « »¬ ¼

be a fuzzy interval semigroup under min. operati

W = ^ `m10, m 25,26,...,60

7ª º « »¬ ¼

W is a fuzzy interval semigroup under min. opemin. operation is fuzzy interval ideal of S under m

W under min. operation is fuzzy interval idmin. operation.

Example 6.3.9: Let

S = ^ `m10, m 0,1,2,...,

8ª º f« »¬ ¼

be a fuzzy interval semigroup under max operati

W = ^ `n10, n 0,1,2,...,20

8ª º « »¬ ¼

be a fuzzy interval semigroup under max operation.

T = m40, m 0,1, 2,...,1209- ½ª º ® ¾« »¬ ¼¯ ¿

M



is a fuzzy interval subsemigroup under max operafuzzy interval ideal under max operation.

Now having seen fuzzy interval semigroups.

We now proceed onto define the concept of fuzmatrix semigroup and fuzzy interval polynomial sem

We will give examples of these structures.

Example 6.3.11: Let V = {([0, a1], [0, a2], …, [0, an

1, 1 d i d n}, V is a fuzzy interval row matrix. V operation (or min. operation) or used in the mutuallsense (V, min) is defined as a fuzzy interval rsemigroup.

We will describe how the operation is carried out; s([0, a1], [0, a2], …, [0, an]) and Y = ([0, a1], [0, a2],are in V then

min {X, Y} = min {([0, a1], [0, a2], …, [0, an]) b2], …, [0, bn])}

= (min {[0, a1], [0, b1]}, min. {[0, b2]}, …, min {[0, an], [0, bn]})= ([0, min {a1, b1}], [0, min {a2, b

bm]}) = ([0, max. {a1, b1}], [0, max {a2, b2}] , … bm}]) is in W.

We illustrate it by an example if X = ([0, 1],

[0, 1/7] and Y = ([0, 1/3], [0, 1], [0, 2/5], [0, 5/9= {[0, 1], [0, 1], [0, 2/5], [0, 5/9]).

All i di d i f i



All properties studied in case of intervsemigroup can be derived in case of fuzzy intersemigroup with appropriate modifications.

Example 6.3.13: Let

X =

1

2 i

n

[0, a ]

[0, a ] 0 a 1

1 i n

[0, a ]

- ½ª º° °« » d d° °« »® ¾« » d d° °

« »° °¬ ¼¯ ¿

#

be a collection of all column interval matriceconstructed using the fuzzy set [0, 1]. We define

x =

1

2

n

[0, a ]

[0, a ]

[0, a ]

ª º

« »« »« »« »¬ ¼

#

and

y =

1

2

[0, b ]

[0, b ]

ª º

« »« »« »« »

#

=

1 1

2 2

n n

max[0,a ],[0, b ]

max[0,a ], [0,b ]

max[0,a ], [0,b ]

ª º« »

« »« »« »¬ ¼

# =

1 1

2 2

n n

[0, max{a , b }]

[0,max{a ,b }]

[0,max{a ,b }]

ª «

« « « ¬

#



max {x, y} X.

Thus X is a fuzzy interval column matrix uoperation.

Example 6.3.14: Let

Y =

1

2 i

m

[0, a ][0, a ] 0 a 1

i 1,2,...,m

[0, a ]

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

#

be a fuzzy interval column matrix under min. operatiIf

x =

1

2

m

[0, a ]

[0, a ]

[0, a ]

ª º« »« »« »

« »¬ ¼

#and y =

1

2

m

[0, b ]

[0, b ]

[0, b ]

ª º« »« »« »

« »¬ ¼

#

i Y th i { }

=

1 1

2 2

m m

min[0,a ],[0, b ]

min[0,a ], [0, b ]

min[0, a ],[0,b ]

ª º« »« »

« »« »¬ ¼

#=

1

2

m

[0, min{a ,

[0,min{a ,

[0, min{a ,

ª « «

« « ¬

#



is in Y.

Substructures can be defined for these fuzzy i

(row) matrix semigroups.

Example 6.3.15: Let M = {set of all m u n matrices with fuzzy intervals of the form [0, ai];n}, M under max or min operation is a fuzzy semigroup.

Properties related with these fuzzy isemigroups can be analysed as in case of usual m

If m = n we have the fuzzy interval semigroup. In this case apart from max or min woperation of max, min. Under all these three ointerval square matrices form a semigroup. Finadefine special fuzzy interval matrix semigroufuzzy interval semigroups.

DEFINITION 6.3.3: Let (S, .) be a interval sem

P : S o [0, 1] is called a special fuzzy interv

P (x.y) = min { P (x), P (y)} for all x, y S.


K is a special fuzzy interval semigroup.

Example 6.3.17: Let S = {[0, x] | x Z45} be

semigroup.

1if x 0

- z°®



Define P : S o [0, 1] by P ([0, x]) =if x 0

x1if x 0

z°®° ¯

P is a special fuzzy interval semigroup.

Now for any interval row matrix semigroup S = {([0…, [0, an]) | ai Zn or Z+ {0} or Q+ {0} or R + Ks : S o [0, 1] is such that each interval [0, ai]

interval matrix is mapped on to a fuzzy interval [0, b1 so that the resultant row interval matrix is a fuzzy rmatrix.

Then Ks is a fuzzy row interval semigroup of row matrix semigroup S.

Similarly we define the special fuzzy colum

matrix semigroup and special fuzzy m u n intesemigroup.


Example 6.3.18: Let

S = 1 2

3 4

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

-ª º°®« »°¬ ¼¯

where ai z 0 if ai = 0 we replace the interval bspecial fuzzy interval square matrix semigroup.

Example 6.3.19: Let

1 2[0, a ] [0, a ]- ª º°« »



V = 3 4i

5 6

7 8

[0, a ] [0, a ]a Q {0};1

[0, a ] [0, a ]

[0, a ] [0, a ]

° « »° « » ® « »°

« »° ¬ ¼¯

be a interval 4 u 2 matrix semigroup. Define Kfollows:

(We wish to state that be it any interval matrixai] is always mapped on to the some fuzzy intervai] is present in which ever matrix in S, S thinterval matrices.)

K

1 2 1

3 4 3

5 6 5

7 8 7

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

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

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

§ · -ª º ª

¨ ¸ °« » « °¨ ¸« » « ®¨ ¸« » « °¨ ¸« » « ¨ ¸ °¬ ¼ ¬ © ¹ ¯

0 d bi d 1 if ai z 0 and 1 if ai = 0}.Thus this map can be realized as a map f

intervals built using Zn or Z+ {0} or Q+ {into the set of f inter al b ilt sing [0 1]

S under min (or max) operation is a fuz polynomial semigroup.

Example 6.3.20: Let S = {0, [0, 1]x [0, .5]x2

, [0, .3][0, 0.7]x5} is not a interval fuzzy polynomial semigro

Example 6.3.21: Let S = {[0 ai]xi | 0 < ai < 1; x8 = 1



Example 6.3.21: Let S {[0, ai]x | 0 < ai < 1; x 1interval polynomial semigroup. Clearly S is of infThis semigroup can be formed by min or max

operation.For if min {[0, 0.5]x8, [0, .7]x3} = [0, min {0.5, 0.7} m

= [0.0.5] x3.

Now we give an illustration of special interval fuzzy

Example 6.3.22: Let V = ii

i 0

[0,a ]xf

-®¯¦ | ai Z+} b

polynomial semigroup.Define K : V o [0, 1] as follows

K ii

i 0

[0,a ]xf

§ ·¨ ¸© ¹¦ = i

i 0 i

10, x

a

f

ª º« »¬ ¼

¦ ;

K(V) or (K, V) is a special fuzzy interval polynomial

Interested reader can study this concept as in casesemigroups of all types.



Chapter Seven

APPLICATIONS OF INTERVAL SE

In this chapter we just indicate the applications ointerval semigroups.

We can think of the challenging problem oalphabets in finite automation by intervals. Thisfor each bit can be visualized as an interval.

H d f li i



Chapter Eight



SUGGESTED PROBLEMS

In this chapter we suggest 241 number of prreader to solve and some of them are innovative.

1. What is the order of the interval semigroup Z12}. Find at least 2 interval subsemigroup

2. Obtain some interesting properties semigroups.

+

7. Find the order of V = {([0, a1], [0, a2], …, [0, a10

1 d i d 10}, the row matrix interval semigroup.

8. Let V = {([0, a1], [0, a2], [0, a3]) | ai Z+

{0}}matrix interval semigroup. Find two ideals of V.

9. Let V = {([0, a1], [0, a2], [0, a3], [0, a4], [0, a5])



d i d 5} be the row matrix interval semigroup. order of V? Does V have row matri

subsemigroups which does not divide the order atleast three row matrix interval subsemigroups.

10. Let V = {([0,a], [0,a], [0,a], [0,a]) | a Z19} brow matrix semigroup. Can V have interval subsemigroups?

11. Let V = {([0,a1], [0,a2], [0,a3], [0,a4]) | ai Z19} brow matrix semigroup. Find atleast five intervalsubsemigroups of V. Can V have interval ideals?

12. Prove in any interval row matrix semigroup of

the order of row matrix interval subsemigroudivide the order of V.

13. Let V =

1

2i 12

9

[0, a ]

[0, a ]a Z ;1 i 9

[0, a ]

- ½ª º° °« »° °« » d d® ¾

« »° °« »° °¬ ¼¯ ¿

#

be a colu

ideals? Find at least five distinct msubsemigroups of V.

16. Let V = {All 7 u 2 interval matrices with form [0, ai] where ai Z16} be a 7 u 2 semigroup. Can V have proper ideals? Findsubsemigroups of V.



17. Let V =i 11

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

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

- ½ª º°

® « »¬ ¼° ¯ ¿interval semigroup under addition. Can interval subsemigroups? Can V have noninterval ideals? Justify your answer.

18. Obtain some interesting results about m u nmatrix semigroups built using Z p, p a prime.

19. Let V = {5 u 5 interval matrices with entrithe interval matrix semigroup under multipli

a. Find 3 ideals of V. b. What is the order of V?

c. Does V have interval matrix subsemorder does not divide the order of V?

20. Give an example of a interval matrix semigno matrix interval ideals?

21. Does there exist a matrix interval semigrouinterval matrix subsemigroups?

26. Find the order of the 6 u 3 interval matrix semusing Z17.

27. What is the order of 1 2 3 4 5

i6 7 8 9 10

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

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]1 i

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

- ª º° « »

® « » d d° « »¬ ¼¯



11 12 13 14 15[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]°« »¬ ¼¯ V have ideals? Justify.

28. Find ideals in V = {3 u 3 interval matrices with eZ+ {0}}; V a matrix interval semigrmultiplication.

29. Does a matrix interval subsemigroup alway

interval matrix ideal? Prove your claim.30. Does there exists a matrix interval semigrou

every matrix interval subsemigroup in an ideal?

31. Does there exist a matrix interval semigroup imatrix interval subsemigroup is an ideal?

32. Give an example of a simple matrix interval sem

33. Give an example of a doubly simple matrsemigroup.

34. Give an example of a S-matrix interval semigrou35. Give an example of a matrix interval semigrou

S i i l i

39. Does there exists a matrix interval semigrou proper S-matrix interval subsemigroup.

40. Define S-coset in case of matrix interval illustrate the situation by some examples.

41. Give an example of S-Cauchy matrix interva



42. Is every matrix interval semigroup a S-Cmatrix semigroup?

43. Let V = 1 2i 10

3 4

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

[0, a ] [0, a ]

- ª º° d d® « »

¬ ¼° ¯ interval semigroup.

a. Is V a S-Lagrange matrix interval se b. Does V contain a S-p-Sylow subgrouc. Can V have S-Cauchy elements?d. What is the order of V?e. Can V have S-ideals?f. Can V have S interval matrix hyper

g. Can V have matrix interval subsemthat o(W) / o(V) ?

44. Give an example of matrix interval semigrouS-Cauchy elements?

45. Let V = 1 2 3i 7

4 5 6[0, a ] [0, a ] [0, a ] a Z[0, a ] [0, a ] [0, a ]- ª º° ® « »

¬ ¼° ¯

47. Obtain some interesting properties about interval semigroups of finite order under adusing Zn (n < f).

48. Find examples of polynomial interval semigroupnot S-polynomial interval semigroups.

49. Find examples of polynomial interval semigroup



49. Find examples of polynomial interval semigroupS-polynomial interval semigroups.

50. Let S =4

ii i 5

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦ be a polynom

semigroup under addition.a. Find the order S.

b. Is S a S-polynomial interval semigroup?

c. Can S have elements of finite order?d. Does S have S-Cauchy element?e. Find at least 2 polynomial interval subse

S.f. Can S have polynomial interval ideals?

51. Let S =9 i

i i 7i 0

[0, a ]x a Z

- ½® ¾¯ ¿¦ be a polynom

semigroup under addition.a. Is S a S-polynomial interval semigroup?

b. Can S have polynomial interval ideals?

c. Does S have a polynomial interval suwhich divides the order of S?d. Does S have a polynomial interval su

53. Let S = ii i

i 0

[0, a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦ be

interval semigroup under addition. Can S polynomial interval ideals?

54. Let S = ii i[0,a ]x a Z {0}

f- ½

® ¾¯ ¿¦



i ii 0

® ¾¯ ¿¦

polynomial semigroup under multiplicati

trivial interval polynomial ideals. Foi

i ii 0

[0, a ]x a 3Z {0}f

- ½ ® ¾

¯ ¿¦ S; T

polynomial ideal of S, prove. Can S polynomial subsemigroups that are not inte

ideals? Justify your claim.

55. Prove S = ii i

i 0

[0, a ]x a R {0}f

- ½ ® ¾

¯ ¿¦ h

polynomial ideals.

56. Can the polynomial interval semi

ii i

i 0

[0, a ]x a Zf

- ½® ¾

¯ ¿¦ have polynomial i

Justify your claim.

57. Prove some interesting results about polysemigroup.

61. Give examples of polynomial interval semigroupno S-hyper polynomial interval subsemigroups.

62. Give an example of a polynomial interval sewhich every S-polynomial interval subsemigro

polynomial interval hyper subsemigroup.

63. Does a polynomial interval semigroup S have



element? Justify.

64. Give an example of a polynomial interval sewhich has S-Lagrange polynomial interval semig

65. Does there exist a polynomial interval semigroupa p-Sylow interval subgroup?

66. Can S = 9 ii i 10

i 0

[0, a ]x a Z

- ®¯¦ in which x10 = 1

= x and so on} be a polynomial interval semigrhave p-Sylow subgroups? Justify your claim.

67. Let S =

3

ii i 3i 0

[0, a ]x a Z

- ®¯¦ ; x4 = x0 = 1 and x

polynomial interval semigroup.a. What is the order of S?

b. Does S have p-Sylow subgroups?c. Is S a S-Langrange polynomial interval s

d. Can S have polynomial interval ideals?e. Can S have polynomial interval sub

which are not polynomial interval ideals

71. Find at least 5-S-Cauchy elements of S(X) w66.

72. Is S(X) where o(S(X) = 77 a S-weakly intesemigroup?

73. Find the S-interval symmetric hyper subsemwhere X has 8 distinct intervals.



74. Can the concept of S-coset be define

symmetric semigroup S(X)?

75. Obtain some interesting properties about S(X

76. Can S(X) ever be simple?

77. The S-semigroup S(X) is a S-p-Sylow semig

78. Find S-Cauchy elements of S(X).

79. Can the concept of S-double coset by defined

80. Does S(X) have S-normal interval subgroup

answer.81. Find some interesting properties associated w

82. Let X be the interval set containing 19 interv

a. Find the order of S(X).

b. How many S-p-Sylow subgroups doc. Find S-interval symmetric subsemigd D S(X) h S i t l

84. Can S(¢X²) have S-interval special symmesubsemigroup? Justify your answer.

85. Can S(¢X²) have S-p-Sylow special symmetric the cardinality of the interval set X is a composit

86. Let S(X) and S(¢X²) be interval symmetric semspecial interval semigroup related with the inter



p g p{[a1, b1], [a2, b2], [a3, b3]}.

a. Obtain all results for S(X) and S(¢X²) athem.

b. Are S(X) and S(¢X²) S-Lagrange weaksymmetric semigroups?

c. Does S(X) have S-p-Sylow subgroups?

87. Obtain some interesting properties about pure ninterval semigroups built using Q+ {0}.

88. Determine the properties enjoyed by the pure ninterval semigroup built using Zn, n a composite

89. Can ever a pure neutrosophic interval semigroup

{0} be a S-pure neutrosophic interval semigroyour claim.

90. Give an example of a S-pure neutrosophsemigroup.

91. What is the order of the neutrosophic interval s= {[0, a + bI] | a, b Z2? Can S be a S-neutrosopsemigroup?

a. What is the order of S? b. Is S a S-Lagrange neutrosophic interc. Find some neutrosophic interval idea

d. Does S have S-Cauchy elements?e. Find some proper S-neutrososubsemigroups of S.

f. Is S a S-neutrosophic interval hyper



95. Let W = {[0, a + bI] | a, b Z24} be a neutrosemigroup. Find all S-neutrosophic intervalof W.

96. Find some interesting properties about neutrsemigroups built using Z p, p a prime.

97. Let S = {[0, a + bI] | a, b Z7} be a neutro

semigroup;a. Can S have zero divisors?

b. Does S have S-Cauchy elements?c. Is S a S-Lagrange neutrosophic interd. Is S a S-weakly Lagrange neutro

semigroup?

e. Does S have S-p-Sylow neutrosubgroups?

f. Can S have idempotents?g. Does S contain units?h. Can S have non trivial nilpotents?

98. Let S = {[0, a + bI] | a, b Z10} be a neutrosemigroup.

a Find all units in S

d. Does A contain S-neutrosophic intesubsemigroups?

e. Does A contain S-cauchy elements?

100. Characterize the zero divisors in S = {[0, a + bI]n a composite number}; a neutrosophic interval s

101. Can A = {[0, a + bI] | a, b Z p; p a phi i l i h di i



neutrosophic interval semigroup have zero diviso

102. Find all units in A described in problem (101).

103. Can A in problem (101) have neutrosophic interv

104. Obtain some interesting properties about neutromatrix interval semigroups under addition.

105. Can pure neutrosophic column matrix interval shave non trivial pure neutrosophic column matideals in S?

106. Let V =

1

2 i

6

[0, a ]

[0, a ] a R {0}1 i 6

[0, a ]

- ½ª º° °

« » ° °« »® ¾« » d d° °« »° °¬ ¼¯ ¿

#be a pure n

interval matrix semigroup. Can V have non neutrosophic interval matrix ideals? Justify your

107. Obtain some important properties enjoyed by n5 u 3 interval matrix semigroups built using N (Z

110. Give an example of a pure neutrosophicinterval semigroup which has no S-purnormal subgroups.

111. Define some new properties on pure neutromatrix semigroups.

112. Define Smarandache inverse of pure neutrot i i



matrix semigroups.

113. Give an example of a Smarandache neutrosophic matrix interval semigroup.

114. Give an example of a Smarandache Lagranmatrix interval semigroup.

115. Give an example of a S-weakly Lagrange puinterval matrix semigroup.

116. Give an example of a finite Smaranneutrosophic interval matrix semigroup.

117. What is the order of A = {All 2 u 2 intervaspecial intervals of the form [0, ai], ai = xi + the neutrosophic interval matrix semigroup?

a. Is A a S-neutrosophic interval matrix b. Can A have S-neutrosophic interval

118. Find the order of S wh

1

2

[0, a ]

[0, a ]

- ª º° « »°« »

119. Let V =

1

2i 15

9

[0, a ]

[0, a ]a Z I;1 i 9

[0, a ]

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

#b

neutrosophic interval matrix semigroup undeWhat is the order of V? Does the order of ne trosophic inter al matri semigro p di ide t



neutrosophic interval matrix semigroup divide tV?

120. Does a pure neutrosophic matrix interval seorder 31 exist?

121. Let P = 43

[0,a]

[0,a]a Z I

[0,a]

- ½ª º° °« »

° °« » ® ¾« »° °« »° °¬ ¼¯ ¿

# be a pure neutrosop

column interval matrix semigroup. What is the Can P have proper pure neutrosophic 15 u 1 colu

matrix subsemigroups?122. Obtain some interesting properties about

neutrosophic interval matrix semigroups which hideals in them.

123. Give a class of pure neutrosophic inter

semigroups which have no proper subsemigroup

124 Give a class of pure neutrosophic inter



c. Prove A cannot have ideals!d. Can A have S-Cauchy elements?

131. Enumerate the special properties enjoyedneutrosophic column interval matrix semigrouorder.

132. Let V = {5 u 2 interval matrices with special inteform [0 ai]; ai Z7I} be a pure neutrosophic in



form [0, ai]; ai Z7I} be a pure neutrosophic inmatrix semigroup.

a. What is the order of V? b. Can V be a S-semigroup? Justify.c. Does V have ideals?

133. Let V = {all 2 u 7 interval matrices with interform [0, ai] where ai Z+I {0}} be a pure neu

u 7 interval matrix semigroup matrix under addita. Can V be a S-pure neutrosophic mat

semigroup? Justify. b. Prove V cannot have proper ideals.c. Find atleast 3 pure neutrosophic inte

subsemigroups.d. What is the special property enjoye

semigroup regarding special elements?

134. Let G =

1

2i

12

[0, a ]

[0, a ]a R I {0};1 i 12

[0, a ]

- ½ª º° °« »° °« » d d® ¾

« »° °« »° °¬ ¼¯ ¿

#

135. Let S = {5 u 5 neutrosophic matrices with form [0, ai] | ai Z2 I} be a pure neutromatrix semigroup under multiplication.

a. Find the order of S. b. Find ideals in S.c. Is S a S-Lagrange semigroup?d. Prove S has zero divisors.e. Does S contain idempotents?f C S h S C h l ?



f. Can S have S-Cauchy elements?

g. Is S a S-semigroup?h. What is the structure enjoyed by thdiagonal matrices?

136. Let T = {All 7 u 5 neutrosophic intervaintervals of the form [0, ai] where ai = xi + y

Z3} be a neutrosophic interval matrix seaddition.a. Find the order of T.

b. Is T a S – semigroup?c. Find atleast 3 subsemigroups of T.d. Prove T has pure neutrosophic

subsemigroups.137. Let W = {All 3 u 3 neutrosophic interva

intervals of the form [0, ai]; ai Z1

neutrosophic interval matrix semigroup.a. Find the order of W.

b. Find ideals in W.c. Does there exists a pure neutromatrix subsemigroup in W which is

i. Will the set of all non singular interval W be a pure neutrosophic interval matrix

138. Let V = {all 6 u 6 neutrosophic interval maintervals of the form [0, ai] where ai Z+ I {0}neutrosophic interval matrix semigroup undmatrix multiplication.

a. Is V a S-semigroup?b Find atleast two ideals in V



b. Find atleast two ideals in V.c. Does V have zero divisors?d. Can V have idempotents?e. Can V have nilpotents?f. Does the set of all non singul

neutrosophic matrices form a group? Jusg. Does V have pure neutrosophic inte

subsemigroups which are not ideals?h. Can V be S-normal?

139. Let P =4

ii i 10

i 0

[0, a ]x a Z

- ½® ¾

¯ ¿¦ ; x5 =1, x6 = 1 an

a pure neutrosophic polynomial interval semig

the order of P.

140. Let S = 3i i 19

i 0

[0,a ]x a Z ;f

-®

¯¦ ; x4 =1} b

neutrosophic polynomial interval semigroup. Fi

S. Can S have pure neutrosophic polynomsemigroups? Justify.

f. Is T a S-semigroup?

142. Let V = {All n u n interval neutrosophicentries from Z45I} be a pure neutrosophic semigroup.

a. Can V have S-pure neutrosophic subsemigroup?

b. Can V have S-zero divisors?c Can V have S-nilpotents?



c. Can V have S nilpotents?d. Can V have S-idempotent?

143. Give an example of a neutrosophic isemigroup in which every neutrosophic subsemigroup is a neutrosophic interval matr

144. Give an example of a neutrosophic i

semigroup in which there exists no neutromatrix ideals.

145. Give an example of a neutrosophic isemigroup which has no zero divisors.

146. Give an example of a neutrosophic isemigroup which is a S-Lagrange semigroup

147. Give an example of a neutrosophic matrix seis a S-Cauchy semigroup.

148. Does there exists a pure neutrosophic

semigroup S of finite order such that the ordneutrosophic matrix ideal subsemigroup divS?

151. Give an example of a pure neutrosophic intesemigroup which has no proper pure neutrosopmatrix subsemigroup.

152. Obtain some interesting properties about ninterval matrix semigroups under multiplication.

153. Can we have a neutrosophic matrix interval sewhich no element is of finite order?



154. Does there exists a neutrosophic interval matrixin which every element is of infinite order?

155. Can we say every matrix in the pure neutrosointerval semigroup constructed using Z+I infinite order? Justify your answer.

156. Enumerate some interesting properties abouneutrosophic interval matrices.

157. Give some interesting applications of neutrosopmatrices.

158. Let V be any pure neutrosophic interval semigroup of order p, p a prime, prove V canhave non trivial pure neutrosophic subsemigroups.

159. Construct a pure neutrosophic polynomia

semigroup of order 25.

160 How many such polynomial interval semigroups

c. Is S a S-pure neutrosophic intersemigroup?

d. Can S have ideals?e. Does S contain atleast one pur

interval polynomial subsemigroup Po(S)? Justify your answer.

162. Prove using Z pI (p-a prime) one can neutrosophic interval polynomial semigrou



p p y gorder.

163. Prove every pure neutrosophic intervsemigroup constructed using ZnI (n prime orS-pure neutrosophic interval polynomial sem

164. Can any pure neutrosophic interval polyno

constructed using Z+

I {0} or Q+

I {0} oS-pure neutrosophic interval polynomiSubstantiate your claim.

165. Construct a pure neutrosophic polynsemigroup which has S- Lagrange subgroup.

166. Give an example of a pure neutrosophic polysemigroup which has S-Cauchy elements.

167. Study the problem (166) in case of neutrosopinterval semigroup.

168. Does there exist a pure neutrosophic intersemigroup which has S-p-Sylow subgroups?

171. Give an example of a Smarandache Lagrneutrosophic interval polynomial semigroup.

172. Give an example of a pure neutrosophic interval

semigroup in which no element is a Smarandacelement of S.

173. Give an example of a pure neutrosophic intervalsemigroup S in which every element is a S-Cauc

f S



of S.

174. Can a S-Lagrange pure neutrosophic interval semigroup be constructed using Z18I?

175. Study the concept of Smarandache cosets in cneutrosophic polynomial interval semigroup.

176. Give an example of a Smarandache coset Hneutrosophic interval polynomial semigroup usin

177. Give an example of a Smarandache weakly Lagneutrosophic interval polynomial semigroup usin

178. Can the same problem be true if Z10I is replaced

179. Study problems (177) and (178) in case Z10I areplaced by N (Z10) and N (Z11) respectively.

180. Obtain some interesting applications of the

semigroups.181. Can interval semigroups be used in finite automa

185. Study the problem (184) in case of S-symsemigroups.

186. Does there exist a S-weakly Lagrange

semigroup?

187. Study problem (186) in case of pure neutromatrix semigroup.

188 Find all Smarandache p Sylow subgroups o



188. Find all Smarandache p-Sylow subgroups o

interval semigroup using {[a1, b1], [a2, b2], [a189. Find all Smarandache p-Sylow subgroups

symmetric interval semigroup ¢{[a1, b1], [a2, b4]}².

190. Find S-Cauchy elements of the semigroup(188) and (189).

191. Give an example of pure neutrosophic intesemigroup of order 51 using Z11I which haelements.

192. Can S([a1, b1], [a2,b2], [a3,b3], [a4,b4], [a5,b5]interval semigroup contain a subgroup of ord

193. Study problem (192) in case of the speinterval semigroup S(¢[a1, b1], [a2, b2], …, [an

194. Find all subgroups of the special symsemigroup S(¢[a1, b1], [a2, b2], …, [a9, b9]}²).

198. Let S = {[0, ai] xi / x2 = I, ai Z19I} be a pure ninterval polynomial semigroup.

a. Prove S has no S-Cauchy element.

b. Prove S is a S-pure neutrosop polynomial semigroup.c. Prove S has non trivial S-pure neutrosop

polynomial subsemigroups.

199. Study problem (198) in which Z19I is replaced b



{x + yI / x, y Z19}.

200. Prove H = S ([a1, b1], [a2, b2], [a3, b3]) u S ([a1, bis a S-interval symmetric semigroup and is Lagrange semigroup.

201. Find all S-Cauchy elements of H given in proble

202. Find all S-p-Sylow subgroups of H given problem

203. Let M = {[0, a] xi / x3 = 1 a N (Z6)} bneutrosophic interval polynomial semigroup.

a. Find the number of elements in M.

b. Is M a S-special neutrosophic intervalsemigroup?c. Find atleast 3 special neutrosoph

polynomial subsemigroups of M.d. Does M contain ideals?

204. Let G = {[0, a] x

i

/ x

3

= 1 a N (Z7)} bneutrosophic interval polynomial semigroup.a. Find the order of G.

a. Find special neutrosophic intervsubsemigroups which are ideals?

b. What is the order of W?c. Does W have S-Lagrange semigroupd. Can W have S-Cauchy elements?

206. Obtain some interesting properties about thinterval polynomial semigroup

i[0 a ] a N(Z )f- ½

® ¾¦ nder addition



ii i 10

i 0

[0, a ]x a N(Z )

® ¾

¯ ¿¦ under addition.

a. What is order of S? b. Can S have ideals?c. Can S have S-Cauchy elements?d. Is S a S-neutrosophic interv

semigroup?

207. Prove a neutrosophic interval polynomial saddition cannot have ideals.

208. Let S = ii i

i 0

[0, a ]x a N(Z I {0})f

- ½ ® ¾

¯ ¿¦ be

interval polynomial semigroup under multipS has non trivial neutrosophic interval polyn

209. Let S =5

i 6i i 4

i 0

[0, a ]x x 1;a N(Z )

- ½ ® ¾

¯ ¿¦ be

interval polynomial semigroup under multipla. Find the order of S.

a. Find the order of S. b. Is S a S-neutrosophic interval

semigroup?c. Does S contain proper ideal?d. Find zero divisors and idempotents in S.

211. Obtain some interesting properties abom

i m 1i i n

i 0

[0, a ]x a N(Z ); m and x 1

- ½ f ® ¾

¯ ¿¦ ,



i 0¯ ¿

neutrosophic interval polynomial semigroup.

212. Does there exist a neutrosophic interval semigroup which is a S-Lagrange semigroup?

213. Does there exist a neutrosophic interval

semigroup which is a S-weakly Lagrange semigr214. Obtain some interesting applications of n

interval polynomial semigroups.

215. Give an example of a neutrosophic interval semigroup which has S-p-Sylow subgroups.

216. Give an example of a neutrosophic interval semigroup which has no S-p-Sylow subgroups.

217. Give an example of a neutrosophic polynomsemigroup which has no S-hyper subsemigroup.

218. Does there exists a neutrosophic polynomsemigroup which has S-hyper subsemigroup?

222. Does there exist a special neutrosophic polysemigroup in which every element is a S-Cau

223. Obtain some interesting properties

neutrosophic interval polynomial semigroup

224. Can we always prove a special neutro polynomial semigroup built using N(Z p),always of a prime order?



225. Does there exist a special neutrosophic intesemigroup which is a S-Smarandache semigr

226. Study Smarandache coset properties neutrosophic interval polynomial semigroupx3 = 1, a N (Z8)}.

227. Does there exist a Smarandache Cneutrosophic interval polynomial semigroup

228. Enumerate some interesting properties abousemigroups.

229. Obtain some interesting properties about matrix semigroups.

230. Analyse all the properties related with polynomial semigroups.

231. Suppose S = {all 10 u 10 interval matricintervals of the form [0, ai] ai Q+ {0}} u 10 matrix semigroup

233. Define special fuzzy interval row matrix ideal o problem (232)

234. Let S = ii ii 0

[0,a ]x a R

f

- ½® ¾¯ ¿¦ be a interval

semigroup. Define K : S o [0,1] so that SK orspecial fuzzy interval polynomial semigroup.

235. For S given in problem (234) define fuz



g p ( )

polynomial semigroup.236. Determine conditions for special fuzzy interval

semigroup to be a S-special fuzzy interval semigroup.

237. Given S = ii i

i 0[0, a ]x a Z

f

- ½® ¾¯ ¿¦ is a interval

semigroup. Find the special fuzzy interval semigroup SK or (S, K).

238. Give some interesting properties about fuzzy inte

semigroups.

239. Enumerate the properties enjoyed by S-fuz polynomial semigroups.

240. Does there exist a finite fuzzy interval semig

some examples.7 1 1 1 1 1 1-

FURTHER READING



1. Ashbacher, C., Introduction to NeutAmerican Research Press, Rehobhttp://www.gallup.unm.edu/~smarandache/In.pdf

2. Kuperman, I.B., Approximate Linear AlgebThe new University Mathematics Series, Reinhold Company, London (1971).

3. Liu, F., and Smarandache, F., Logic: Concept. A Contradiction Study toward Ag

Proceedings of the First International Neutrosophy, Neutrosophic Logic, Ne Neutrosophic Probability and StatSmarandache editor, Xiquan, Phoenix, ISBN1, 147 p., 2002, also published in "LibertasUniversity of Texas at Arlington, 22 (2

http://lanl.arxiv.org/ftp/math/papers/0211/02

4 Si j A B id S M d V h R B



INDEX

C

Column interval matrix semigroup, 11-2

F

Fuzzy column matrix interval semigroup, 126-7

Fuzzy interval ideal, 118-122Fuzzy interval semigroup, 104-8, 115-6

Interval matrix Smarandache semigroup (S-semigrouInterval nilpotents, 65-6Interval polynomial semigroup, 37-9Interval polynomial subsemigroup, 39-41Interval pure neutrosophic semigroup, 61-3Interval row matrix semigroup, 10-11Interval semigroup under addition modulo n, 9-10Interval symmetric semigroup, 52-3Interval units, 65-6Interval zero divisors 65-6



Interval zero divisors, 65 6

M

m u n interval matrix semigroup, 13-4m u n matrix interval semigroup, 13-4Matrix interval subsemigroup, 15-6

N

Neutrosophic complex integer intervals, 61-2 Neutrosophic integer intervals, 61-2 Neutrosophic interval polynomial semigroup, 94-6 Neutrosophic interval semigroup, 61-3 Neutrosophic modulo integer intervals, 61-2 Neutrosophic rational integer intervals, 61-2 Neutrosophic real integer intervals, 61-2

P

Pseudo neutrosophic interval matrix subsemigroup, 8

Pure neutrosophic interval square matrix semigroPure neutrosophic modulo integer interval, 61-2Pure neutrosophic rational interval, 61-2Pure neutrosophic real interval, 61-2

R

Row interval matrix subsemigroup, 18-9Row matrix interval semigroup, 10-11



S

S- normal neutrosophic matrix interval subgroupS- weakly cyclic interval symmetric semigroup, S-Cauchy elements, 54-6S-Cauchy interval element, 100-106S-interval matrix subsemigroup, 30-4S-interval polynomial semigroup, 39-46S-interval symmetric semigroup, 50-52S-interval symmetric subsemigroup, 50-52S-interval symmetric weakly cyclic semigroup, 5S-interval symmetric weakly Lagrange semigrouS-Lagrange pure neutrosophic interval semigroupS-Lagrange semigroup, 53-5Smarandache Cauchy element of a semigroup, 54Smarandache interval matrix semigroup, 27-8Smarandache p-Sylow interval symmetric semigSmarandache weakly commutative symmetric seS-neutrosophic interval matrix semigroup, 92-4S-non 2-Sylow pure neutrosophic interval semigr

S-special interval symmetric subsemigroup, 57-9S-special neutrosophic interval polynomial semigrou115-7S-symmetric interval semigroup, 50-52S-weakly Lagrange pure neutrosophic interval semigS-weakly Lagrange semigroup, 53-5



ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is an Associate Department of Mathematics, Indian Institute Madras, Chennai. In the past decade she has gscholars in the different fields of non-assocalgebraic coding theory, transportation theory, fuapplications of fuzzy theory of the problems fa

industries and cement industries. She has to research papers. She has guided over 68 M.Sprojects. She has worked in collaboration projects



Space Research Organization and with the Tamil NControl Society. She is presently working on a rfunded by the Board of Research in Nu

Government of India. This is her 52nd book.On India's 60th Independence Day, Dr

conferred the Kalpana Chawla Award for CouraEnterprise by the State Government of Tamil Nad

of her sustained fight for social justice in the InTechnology (IIT) Madras and for her contribution

The award, instituted in the memory of Iastronaut Kalpana Chawla who died aboard Columbia, carried a cash prize of five lakh rupeprize-money for any Indian award) and a gold meShe can be contacted at vasanthakandasamy@gmWeb Site: http://mat.iitm.ac.in/home/wbv/public_

Dr. Florentin Smarandache is a Professor of the University of New Mexico in USA. He publisheand 150 articles and notes in mathematics, physpsychology, rebus, literature.

In mathematics his research is in numbe

Euclidean geometry, synthetic geometry, algebstatistics, neutrosophic logic and set (generalizlogic and set respectively) neutrosoph

interval semigroups, by w. b. vasantha kandasamy, florentin smarandache

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