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

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INTERVAL SEMIRIN

W. B. Vasantha KandasamyFlorentin Smarandache



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Prof. Gabriel Tica, Bailesti College, Bailesti, Jud. Dolj, Romania

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ISBN-13: 978-1-59973-033-2



CONTENTS

Preface

Chapter One

BASIC CONCPETS

Chapter Two

INTERVAL SEMIRINGS ANDINTERVAL SMIFIELDS

Chapter ThreeNEW CLASSES OF INTERVAL SEMIRINGS

3.1 Matrix Interval Semirings

3 2 Interval Polynomial Semirings



PREFACE

In this book the notion of interval s

introduced. The authors study and analy

algebraically. Methods are given for the c

non-associative semirings using loops

semirings or interval loops and semirings. A

non-associative semirings are introduced us

and interval semirings or interval groupoids

Examples using integers and modulo integ

Also infinite semirings which are semifie

using interval semigroups and semirings oand interval semirings or using groups

semirings.

I l i d d



semirings like matrix interval semirings an

polynomial semirings. Chapter four for theintroduces the notion of group interval

semigroup interval semirings, loop interval sem

groupoid interval semirings and these stru

studied. Interval neutrosophic semirings are int

chapter five. Applications of these structures ar

chapter six. The final chapter suggests ar

problems for the reader. We thank Dr. K.Kand

proof reading.

W.B.VASANTHA FLORENTIN SM



Chapter One

BASIC CONCEPTS

In this book we introduce several types of s

intervals, mainly of the form [0, a]; a real. Study

interval semirings is carried out for the first time

A semiring is a nonempty set S endowed w

operations “+” and “.” such that (S, +) is an asemigroup and (S, .) is a semigroup and both ri

distributive laws are satisfied.

Example 1.1: The set of positive integers w



A semiring S is said to be a strict semiring if for a an

b = 0 implies a = 0 or b = 0.Semirings mentioned in the above examplessemirings. A semiring S is said to be a semifield if

commutative semiring with the multiplicative

Semirings given in examples 1.1 and 1.2 are sem

semiring is said to be of infinite cardinality if it

number of elements in them. A finite semiring is onefinite number of elements in them. All distributive

semirings. If the distributive lattices are of finite o

finite semirings. All Boolean algebras are semiring

Boolean algebras of order greater than two

commutative semirings and are not strict semirings,

are not semifields. All chain lattices are semifields.

For more about semirings and semifields please refer

I(Z+ � {0}) denotes the collection all intervals [

in Z+ � {0}. I(Q+ � {0}) = {[0, a] | a � Q+ � {0} }

{0}} denotes the collection of all [0, a] with a in R +

�

I denotes the indeterminancy and I2 = I. All inte

form [0, a+ bI] denotes the neutrosophic interval w

reals.

For more about neutrosophic concepts please refer [5



Chapter Two

INTERVAL SEMIRINGS

AND INTERVAL SEMIFIELDS

In this chapter for the first time we define the

interval semirings and interval semifields we

structures with examples. These structures willlater chapters to construct neutrosophic interva

other types of interval semialgebras. We wil

notations given in chapter one.

DEFINITION 2.1: Let S = {[0, a], +, . } be + +



We will give examples of interval semirings.

Example 2.1: Let S = {[0, a] | a � Z5}; (S, +, .) is

semiring of finite order.

Example 2.2: Let S = {[0, a] | a � Z+ � {0}}; (S

interval semiring of infinite order.

Example 2.3: Let P = {[0, a] | a � Q+ � {0}} is

semiring of infinite order.

We can as in case of usual semirings defi

subsemirings.

DEFINITION 2.2: Let V = {[0, a] / a � Z n or Z +

� {{0} or R+ � {0}} be an interval semiring under a

multiplication. Let P � V, if (P, +, .) is an interval se proper subset of V) then we define P to be subsemiring of P.

If in an interval semiring (S, +, .), the interval semiis commutative then we define S to be a commutat

semiring. If (S, +, .) is such that (S, .) is not a co

interval semigroup, then we define (S, +, .) tocommutative interval semigroup.

We will give examples of them.

Example 2.4: Let (S, +, .) = {[0, a] | a � Z15, +, .} be

semiring. Consider (P, +, .) = {[0, a] | a � {0, 3, 6, 9

+, .} � (S, +, .); P is an interval subsemiring of S. B

f fi it d



Example 2.7: Let (S, +, .) = {[0, a] | a � R + �

interval semiring. Consider (W, +, .) = {[0, a] | a.} � (S, +, .), W is an interval subsemiring of S.

All the examples of interval semirings givcommutative semirings. If an interval semiring S

[0, 1] such that [0, 1] acts as identity that is [0, 1

. [0, 1] = [0, a] for all [0, a] � S. then we callsemiring with identity.

All interval semirings in general need not

semiring with identity. To this effect we w

examples.

Example 2.8: Let (S, +, .) = {[0, a] | a � 2Z+

an interval semiring. (S, +, .) is not an interva

identity.

Example 2.9: Let (S, +, .) = {[0, a] | a � Q+ �

interval semiring . S is an interval semiring wicall an interval semiring S to be of characteristic

implies n = 0, where a z 0 and a � Z+ � {0} or

� {0}. If in case n [0, a] = 0 still n z 0 and a z 0

interval semiring S is of characteristic n, with m

m < n.

We will illustrate this situation before wedefine different types of interval semirings.

Example 2.10: Let S = {[0, a], +, . | a � R +

i t l i i f h t i ti



We say an interval semiring S = {[0, a], +,

divisors if [0, a] . [0, b] = [0, ab] = [0, 0] = 0 where z 0.

We will give some examples of interval semiring

divisors.

Example 2.14: Let S = {[0, a], +, . | a � Z18} be

semiring. Consider [0, 6], [0, 3] in S; [0, 6] [0, 3] = [0] is a zero divisor in S. Also for [0, 2], [0, 9]�S; w2].[0, 9] = [0, 18] = [0, 0] = 0 is again a zero diviso

several other zero divisors in S.

Example 2.15: Let S = {[0, a], +, . | a � Z+ � {0} be

semiring. S has no zero divisors.

Example 2.16: Let S = {[0, a], +, . | a � Z41} be

semiring S has no zero divisors.

THEOREM 2.1: All interval semirings S = {[0, a], +

p a prime} have no zero divisors.

The proof is straight forward and hence is left as an

the reader.

THEOREM 2.2: All interval semirings S = {[0, a], +

n a composite number} have zero divisors.

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

T 2 3 L S [[0 ] + | Z+ {0}



Example 2.17: Let S = {[0, a], +, . | a � Z12}

semiring [0, 4] � S is such that [0, 4] . [0, 4] = [= [0, 4]; so [0, 4] is an idempotent of S. [0, 9]

idempotent for [0, 9].[0, 9] = [0, 81(mod 12) ] =

Thus S has nontrivial idempotents.

Example 2.18: Let S = {[0, a], +, . | a � Z p}wh

be interval semiring S has no idempotents anHowever S has units.

In view of this we have the following theorem.

THEOREM 2.4: Let S = {[0, a] | a � Z p , p a prim

interval semiring, S has no idempotents and zero

The proof is straightforward and is left as an e

reader.

Example 2.19: Let S = {[0, a] | a � Z23, +, .}

semiring. S has non trivial units. For take [0, 8](mod 23)] = [0, 1] is a unit of S. [0, 12] [0, 2] =

23)] = [0, 1] is a unit of S. Consider [0, 6] [0, 4

23)] = [0, 1] is a unit of S. Now [0, 22] [0, (mod23)] is a unit in S. Thus S has several u

idempotents or zero divisors.

THEOREM 2.5: Let S = {[0, a] / a � Z p , p a pr

interval semiring. S has units.

The proof is also left as an exercise to the reader



DEFINITION 2.3: Let S = {[0, a] | a � Z + � {0} or

with operation + and . so that S is an interval c semiring. If (a) [0, a] + [0, b] = [0 0] = 0 if and only[0, 0] and [0, b] = [0, 0]. That is the S is a str

semiring and (b) in S, [0, a] . [0, b] = [0, 0] then [0,or [0, b] = [0, 0] then we define the interval semiri

interval semifield. If in the interval semifield 0. [0,

and for no integer n, n [0, x] = 0 or equivalently if {0} = [0, 0], n [0, x] = [0, x]+ [0, x] + [0, x] + . . .times equal to is zero is impossible for any positivethen we say the semifield S is of characteristic 0.

We will illustrate this by some examples.

Example 2.20: Let S = {[0, a] , +, . | a � Z+ � {0}} b

semifield of characteristic zero.

Example 2.21: Let S = {[0, a], a � R + � {0}; +

interval semifield of characteristic zero.

Example 2.22: Let S = {[0, a]; a � Q+ � {0}, +

interval semifield of characteristic zero.

Now having seen examples of interval semifiel proceed on to show that all interval commutative se

not interval semifields.

Example 2.23: Let S = {[0, a], a � Zn, n < f,

interval semiring. Take n = 6, clearly S is not semifield For consider



interval semiring. So S is not an interval semifi

has no zero divisors. For in S we do not have [0,0] if a, b � Z11 \ {0}.

Example 2.25: Let S = {[0, a] | a � Z15, +, .}

semiring. We see S is not an interval semifield.

11] + [0, 4] = [0, (11+4) mod (15) ] = [0, 0] so

the definition 2.3 is not true.Further [0, 6] . [0, 5] = [0, 6.5 (mod 15)]

nontrivial zero divisor in S. Infact we have sever

in S. Thus S is not an interval semifield.

Example 2.26: Let S = {( [0, a], [0, b], [0, c]) |

{0}, + , .} be an interval semiring under caddition and component wise multiplication. Th

a], [0, b], [0, c] ) and y = ([0, t], [0, u], [0, w] )

and a, b, c, u, t, w � Z+ � {0}.

We see

x + y = ([0, a], [0, b], [0, c] ) + ([0, t], [0, u],

= ([0, a + t], [0, b + u], [0, c + w]).

x.y = ([0, a], [0, b], [0, c]) ([0, t], [0, u], [0

= ([0, a] [0, t], [0, b] [0, u], [0, c] [0, w

= ([0, at], [0, bu], [0, cw]).

We see S has zero divisors for take p = ([0, 0], [0q = ([0, a] [0, 0], [0, c]) ; We have

p.q = ([0, 0] [0, b], [0, 0]. ([0, a], [0, 0], [0

= ([0 0] [0 a] [0 b] [0 0] [0 0] [0 c



THEOREM 2.6: Let S = {([0, a1 ], [0, a2 ], …, [0, an ])

{0} or ai � R+

� {0} or ai � Q+

� {0}; 1 d i d n;interval semiring. S is not an interval semifield.

This proof is left as an exercise for the reader.

Now we can as in case of other algebraic struc

the notion of interval subsemifield, this task is also

reader. However we give examples of them.

Example 2.27: Let S = {[0, a], +, ‘.’ where a � Q+ �interval semifield. Take T = {[0, a], +, . where a � ZS; T is an interval subsemifield of S.

Example 2.28: Let P = { [0, a], + , ‘o’ where a � R +

an interval semifield. Take W = {[0, a], +, ‘o’ wher

{0}} � P; W is an interval subsemifield of P. Infact

interval subsemifields.

Now having seen examples of interval subse

now proceed on to define the notion of direct producsemirings.

DEFINITION 2.4: Let S 1 and S 2 be two interval sem

direct product S 1 u S 2 = {([0, a], [0, b]) | [0, a] � S

� S 2 } is also an interval semiring with comp

operation. Similarly if S 1 , S 2 , . . . , S t are t intervaThe direct product of these interval semirings denote

u … u S t = {([0, a1 ], [0, a2 ], ..., [0, at ] ) / [0, ai ] � Sis an interval semiring.



We see S = {([0, a1], …, [0, an]) | a � Zm, +,

semiring; This interval semiring can be visualiz product of n interval semirings. P = {[0, a] | a �

n times

P P P S

u u u ! .

We can define ideals in an interval semiring S.

DEFINITION 2.5: Let S = {[0, a] | a � Z n or Z + �

interval semiring. Choose I = {[0, t] | t � Z + � {is an ideal of S if

1. I is an interval subsemiring and

2. For all [0, t] � I and [0, s] � S [0, t] . [0

[0, t] are in I.

We can have the notion of left ideals and rig

interval semiring. When both right ideal and lef

for a non empty subset I of S we call it as an idea

We will illustrate this situation by some example

Example 2.30: Let S = {[0, a] | a � Z+ � {0

interval semiring. Let I = {[0, a] | a � 3Z+ �{0},

an interval ideal of the interval semiring S.

Example 2.31: Let S = {[0, a] | a � Q+ � {0}interval semirings. It is easily verified S has no

but S has nontrivial interval subsemirings. For T

nZ+ � {0}, +, ‘o’} � S for n = 1, 2, …, t, t <

b i i f S hi h id l



Example 2.33: Let W = {[0, a] / a � Z19, +, ‘�’} besemiring. Clearly W has no nontrivial ideals

subsemiring. It is interesting to note that W is n

semiring for we see

[0, 10] + [0, 9] = [0, 10 + 19 (mod 19) = [0,

[0, 12] + [0, 7] { 0 (mod 19)

[0, 5] +[0, 140 { 0 (mod 19)and so on. However W has no zero divisors. Infact W

ring. All rings are semirings.

Example 2.34: Let W = {[0, a] | a � Z25, +, ‘�’ } be

semiring. W has interval ideals as well as interval su

For consider T = {[0, a] | a � {0, 5, 10, 15, 20} � ZW is an interval ideal so is also an interval subsemir

no other ideals or subsemirings.

Example 2.35: Let M = {[0, a] / a � Z16; +, ‘�’} be

semiring. N = { [0, a] / a � {0, 2, 4, 6, 8, 10, 12, 14

‘.’} � M is an interval ideal of M.

Example 2.36: Let S = {[0, a] / a � Z24, +, ‘�’} be

semiring. Take T = {[0, a] / a � {0, 2, . . . , 22} � Z

S, T is an interval subsemiring as well as an interva

V = {[0, a] / a � {0, 6, 12, 18} � Z24, +, ‘�’} � S is

subsemiring, as well as an ideal of S . R = { [0, a] / a

16} � Z24, +, ‘�’} � S, R is an interval subsemir

interval ideal of S.

Now having seen ideals and subsemirings



Example 2.38: Let V = V1 u V2 u V3 u V4 u V5a � Q

+ � {0}, +, ‘�’} u {[0, a] / a � Z12, +, ‘�’}Z19, +, ‘�’} u {[0, a] | a � Z3, +, ‘�’} u {[0, a] | a

{[0, a] | a � Z43, +, ‘�’} be an interval semiri

nontrivial interval ideals as well as nontrivial su

is clear none of these are interval semifields.

Now we can as in case of other algebraic shomomorphisms.

DEFINITION 2.6: Let S and S' be any two interva

interval mapping I : S o S' is called the in

homomorphism if I ([0, a]+[0, b] ) = I ([0, a]) I ([0, a] .[0, b] ) = I ([0, a]) * I ([0, b])

operation on S c ; this is true for all [0, a] and [0,


Example 2.39: Let S = {[0, a] | a � 3Z+ � {0}, {[0, a] | a � Q

+ � {0}, +, ‘�’} be any two inte

Define I : S o Sc by I ([0, a]) = [0, a] [0, a]

easily verified I is an interval semiring homom

I is an embedding of S in to Sc as clearly S � Sc.is an embedding of S in to Sc as S � S1. We sother homomorphisms, we can in case of in

homomorphisms K also define kernel of K.

Let K: S o Sc where S and Sc are interval s

an interval semiring homomorphism. Ker K = {[



Example 2.40: Let S = {[0, a] | a � Q+ � {0}} be

semiring. Consider P = {[0, a] | a � Z

+

� {0}} �interval semifield, hence S is a S-interval semiring.

It is interesting to note that all interval semirings

general be S- interval semirings.

Example 2.41: Let S = {[0, a] | a � Z+ � {0}} be

semiring. Clearly S is not a S- interval semiring ascontain any proper interval subset which is

semifield.

We can as in case of S- semirings define

subsemirings. We call a non empty subset T of

semiring A to be a S-interval subsemiring if T is

subsemiring of A and T has a proper subset B, (B �B is an interval semifield.

In view of this we have the following interesting

proof of which is left to the reader.

THEOREM 2.7: Let A be an interval semiring. Supp proper subset B � A such that B is a S- interval sthen A is a S- interval semiring.

We will illustrate this situation by an example.

Example 2.42: Let V = {[0, a] | a � R +

� {0}} besemiring. Take P = {[0, a] | a � Q+ � {0}} � V;

interval subsemiring of V as P contains a proper sub

a] | a � Z+ � {0}} � P such that T is an interval sem

N V i S i t l i i T P V



Example 2.43: Let S = {[0, a] | a � Q+ � {0}}

semiring. Take W = {[0, a] | a � 3Z

+

� {0}} �interval subsemiring and is not an S- interval sub

Example 2.44: Let T = {[0, a] | a � R + � {0}} b

semiring. Take V = {[0, a] | a � Z+ � {0}}�interval subsemiring which is not an S- interval s

We can define S- interval ideals of an interval usual way [13].

Example 2.45: Let V = {[0, a] | a � R + � {0}

semiring. Take P = {[0, a] | a � Q+ � {0}} � V

subsemiring. Consider A = {[0, a] | a � Z+

� {0interval semifield of P. We see for every q � P

and qa are in A. Hence P is a S-interval ideal of VWe wish to state here that it is an interest

study whether the interval semiring constructed

or Q+ � {0} or R + � {0} have no S-ideals. How

direct product we have S-interval ideals.

Example 2.46: Let V = {[0, a] u [0, b] | a, b � Q

interval semiring.

Take P = {[0, a] u [0 b] | a, b � Z+ � {0}

interval subsemiring. For A = {[0, a] u {0} | a �is an interval semifield contained in P. We

interval ideal of V.

Example 2.47: Let V = {([0, a], [0, b], [0, c], [0



We will only illustrate this situation by an examp

Example 2.48: Let S = {[0, a] / a � Q+ � {0}} be

semiring. Consider A = {[0, a] / a � 3Z+ � {0}} � S

of S. Consider P = { [0, a] / a � Z+ � {0}} � S, weand P is a S-interval subsemiring of S, so A is a Sm

pseudo interval subsemiring (S- pseudo interval subs

S. Infact S has infinitely many S- pseudo interval subThe interesting factor is we have interval se

which has no S- pseudo subsemiring.

This is illustrated by the following example.

Example 2.49: Let V = {[0, a] / a � Z+ � {0}} be

semiring. V has no S- pseudo interval subsemirings.

This can be easily verified by the reader.

We can define Smarandache pseudo right (lef

pseudo (left) right ideal) of the interval semiring a

semirings [13].

We will only illustrate this situation by some exa

Example 2.50: Let V = {[0, a] / a � Q+ � {0}} be

semiring. P = {[0, a] | a � 3Z+ � {0}} is a S- pseu

subsemiring of V. Consider A = {[0, a] | a � Z+ �clearly P � A and pa and ap are in P for every p � P

Thus P is a Smarandache pseudo interval ideal of V.Likewise we can define Smarandache dual in

and Smarandache pseudo dual ideal of an interval sem

Th d i l ft ith th t k f d fi i th




Example 2.51: Let S = {( [0, a1], [0, a2], [0, a3]

[0, a6] ) | ai � Q+� {0}; 1 d i d 6} be an interval

a = ([0, a1], [0, a2], 0, 0, 0, 0) and b = (0, 0, [0, b

in S. Clearly a.b = (0, 0, 0, 0, 0, 0). Now consider x = (0, 0, 0, 0, [0, x

1], [0, x

2])

[0, y2], 0, 0, 0 [0, y3]) � S, we see ax = 0, by

Thus a, b in S is a Smarandache interval zero

interval semiring S.

DEFINITION 2.8: Let S be an interval semiring.

S is said to be a Smarandache anti – interval zanti interval zero divisor) if we can find a y such

a, b� S \ {0, x, y} such that

i. ax z 0 or xa z 0

ii. by z 0 or yb z 0iii. ab = 0 or ba = 0.


Example 2.52: Let S = {[0, a1], [0, a2] [0, a3], . .

R + � {0}; 1 d i d 9} be an interval semiring. Let

a2] [0, a3], 0, 0, 0, 0, 0, 0) and y = (0, [0, a] [0, b

0, 0) in S.We see x.y z (0, 0, 0, …, 0) that is x.y = (0

b], 0, 0, 0, 0, 0, 0) z (0, 0, …, 0). Consider a = (

0) and b = (0, 0, 0, [0, y1], 0, …, 0) in S. We see



However it is important and interesting to n

interval semifield will not have S- interval zero div

interval anti zero divisors.

Further if x � S, S an interval semiring and x

interval zero divisor then x need not be an interval ze

We now proceed on to define Smarandac

idempotent of an interval semiring.

DEFINITION 2.9: Let S be an interval semiring . An

[0, a] � S is a Smarandache idempotent (S- idempote

(i) a2 = a that is ([0, a])2 = [0, a2 ] = [0, a]

(ii) There exists [0 , b] � S \ [0, a] such th

a. ([0, b])

2

= [0, a]b. [0, a] [0, b] = [0, b] ([0, b] [0, a] [0, b] [0, a] = [0, a] or ([0, a] [0, b]

‘or’ used in condition (ii) is mutually exclusive [15-1

We see all the interval semirings built using Z+

� {{0} or R + � {0} have no S- idempotents. On simil

can define Smarandache units in an interval semiring

DEFINITION 2.10: Let S be an interval semiring with

1]. We say x � S \ {[0, 1]} to be a Smarandache un

there exists a [0, y] � S such that (i) [0, x]. [0, y] = [0, 1]

(ii) There exists [0, a], [0, b] � S \ {[0, x]1]} such that

[0 ] [0 ] [0 ] [0 ] [0 ]



Chapter Three

NEW CLASSES OF INTERVAL SEM

In this chapter we introduce new classes of intand describe them. These new classes of int

satisfy several interesting properties which are

interval semirings introduced in Chapter two of

chapter has two sections. Section one introduces

semirings. Interval polynomial semirings are

section two.

3.1 Matrix Interval Semirings



We illustrate this situation by some examples.

Example 3.1.1: Let V = {([0, a1], [0, a2], [0, a3], [0, a

� {0}; 1 d i d 4}; V is a row interval matrix semiring

Example 3.1.2: Let W = {([0, a1], [0, a2], …, [0, a7])

{0}; 1 d i d 7} be a row interval matrix semiring.

Example 3.1.3: Let T = {([0, a1], [0, a2], [0, a3]) | ai �1 d i d 3} be a row interval matrix semiring.

We can define row interval matrix subsemiring and r

matrix ideal.

We will illustrate these substructures by example

Example 3.1.4: Let V = {([0, a1], [0, a2], …, [0, a12])

{0}; 1 d i d 12} be an interval row matrix semiring

a1], [0, a2], …, [0, a12]) / ai � 3Z+ � {0}; 1 d i d 12}

easily verified to be an interval row matrix subsemir

W is not a interval row matrix ideal of V.

Example 3.1.5: Let V = {([0, a1], [0, a2], …, [0, a6])

{0}; 1 d i d 6} be an interval row matrix semiring.

{([0, a1], [0, a2], …, [0, a6]) | ai � 3Z+ � {0}; 1 d i dis a interval row matrix subsemiring as well as in

matrix ideal of V.

Example 3.1.6: Let V = {([0, a1], [0, a2], …, [0, a10])

{0}; 1 d i d 10} be an interval matrix semiring. C



Example 3.1.8: Let V = {([0, a1], [0, a2], …, [0,

{0}; 1 d i d 25} be an interval row matrix semi

a1], 0, 0, …, 0) | a1 � Z+ � {0}} � V; W is

matrix subsemigroup and W is not an interval rof V.

Example 3.1.9: Let V = {([0, a1], [0, a2], …, [0, {0}; 1 d i d 19} be an interval row matrix semir

= {([0, a1], …, 0, 0, 0, [0, a9]) | a9 , a1 � Q+ � {0

interval row matrix subsemiring of V. Howev

interval row matrix ideal of V.

Example 3.1.10: Let V = {([0, a1], [0, a2], [0, {0}; 1 d i d 3} be an interval row matrix semirin

{([0, a1], [0, a2], [0, a3]) | ai � 5Z+ � {0}; 1 dsubset of V. W is an interval row matrix subse

well as an interval row matrix ideal of V.

Infact V has infinitely many interval ideals.

We see all these row matrix interval seinfinite cardinality further all them are interv

commutative semirings.

Now we call an interval row matrix sem

Smarandache row matrix interval semiring of S i

subset P � S such that P is a row matrix interval


Example 3.1.11 : Let T = {([0, a1], [0, a2], …, [0



Example 3.1.12: Let V = {([0, a1], [0, a2], …, [0, a32

� {0}} be a row matrix interval semiring. Clearly

contain a proper subset A such that A is a row mat

semifield. So V is not a S-row matrix interval semirin

Infact we have an infinite class of row mat

semirings which are not S-row matrix interval sem

see it is not possible to have a 1 u n row intesemifield where n > 1; with interval entries different

The reader is expected to prove this.

We can define Smarandache pseudo row mat

subsemiring, S-ideal, S-subsemiring, S-pseudo id

pseudo dual ideal in case of row matrix interval sem

analogous way.

We reserve this work to the reader, however give e

these concepts.

Example 3.1.13: Let V = {([0, a1], [0, a2], [0, a3], [0

Q

+

� {0}; 1 d i d 4} be a row interval matrix semir= {([0, a1], 0, 0, 0) | a1 � Q+ � {0}} � V; P is a r

matrix semifield; hence V is a S-row interval matri

Take W = {([0, a1], 0, [0, a2], 0) | a1, a2 � Q+ � {0}}

a row interval matrix subsemiring of V. Take G = {(

0) / a � Q+ � {0}} � W; G is a row interval matrix s

W. Hence W is a S-row matrix interval semiring.

In view of this we have the following theorem th

which is simple.



Interested reader can give examples of this struct

Now we proceed onto give example of Smaranda

ideals of a row matrix interval semiring.

Example 3.1.18: Let V = {([0, a1], [0, a2], …, [0, a12

� {0}; 1 d i d 12} be row matrix interval semiring.

Consider P = {([0, a1], [0, a2], [0, a3], 0, 0, …, 0

� Z

+

� {0}} be a S-row matrix interval subsemiring{(0, 0, [0, a], 0, 0, …, 0) / a � Z+ � {0}} � P, is a

interval semifield in P.

For every p � P and for every a � A we have pa,

Thus P is a S-pseudo row matrix interval subsem

Now we proceed onto give examples of Smarandaelements in a row matrix interval semirings.

Example 3.1.19 : Let V = {([0, a1], [0, a2], …, [0, a1

� {0}; 1 d i d 10} be a row matrix interval semirin

{([0, a1], [0, a2], 0, …, 0) and b = {(0, 0, 0, [0, a1]

a3], , 0, 0, 0, 0) be in V. We see a. b = (0, 0, 0, …, 0). Now take x = (0, 0, [0, a], 0, …, 0) and y = (0, [0

0, 0, …, 0) in V. We see ax = (0, 0, …, 0) and by = (

further xy = (0, 0, [0, a], 0, …, 0) u (0, [0, a], [0, b], 0

= (0, 0, [0, ab], 0, …, 0)

z (0, 0, 0, …, 0).

Thus a, b � V is a S-zero divisor of the row mat

semiring. Interested reader can construct more exam

zero divisors.

N d i l f



Let a = ([0, 1

1a ], 0, …, 0) and b = (0, 0, 0, [0

S. We see ax = ([0, a1 .1

1a ], 0, …, 0) z (0, 0, …,0, 0, [0, a3a], 0, 0, …, 0) z (0, 0, …, 0).

But ab = (0, 0, …, 0).

Thus x, y � V is a S-anti zero divisor of

interval semiring V.

We see in row matrix interval semiring a S-a

in general is not a zero divisor. However if a rowsemiring S has S-anti zero divisors then S has zer

The reader is given the task of defining S

matrix interval semiring and illustrate it with exa

Now having studied some properties ab

interval semirings, we now proceed onto define

interval semirings. Clearly we cannot define mmatrix interval semirings where m z 1.

DEFINITION 3.1.2: Let V = {Set of all n u n in

with intervals of the form [0, a] where a � Z +

{0} or R+ � {0}}; V is a semiring under interval

and interval matrix multiplication. We define V square matrix interval semiring.

We will illustrate this situation by some exam

Example 3.1.21: Let

V =1 2

i

3 4

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

[0, a ] [0, a ]

ª º° � � d® « »

¬ ¼° ¯ be a 2 u 2 square matrix interval semiring



Example 3.1.24: Let G = {All 8 u 8 lower triangu

matrices with intervals of the form [0, xi] where xi � be a 8 u 8 square matrix interval semiring.

Now as in case of semirings we can defin

substructure in these semirings which is left as an exreader.

However we will illustrate this situation by some exa

Example 3.1.25: Let P = {All 3 u 3 interval ma

intervals of the form [0, a] where a � Z+ � {0}}

square matrix interval semiring. Choose G = {All 3

matrices with intervals of the form [0, a] where a � 5

� P; G is a square matrix interval subsemiring of P.

Example 3.1.26: Let

V =[0, a] [0,b]

a,b,c,d Z {0}

[0,c] [0,d]

½§ ·° °

� �® ¾¨ ¸

© ¹° °¯ ¿

be a 2 u 2 interval matrix semiring.

T =[0, a] [0, a]

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

½§ ·° °

� �® ¾¨ ¸© ¹° °¯ ¿

� V,

T is a 2 u 2 interval matrix subsemiring of V.



Example 3.1.28: Let

T =[0, a] [0, a]

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

§ ·° � �® ̈ ¸

© ¹° ¯ be a square interval semigroup.

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

a 15Z {0}[0, a] [0, a]

½§ ·° °� �® ¾¨ ¸© ¹° °¯ ¿

P is a square interval subsemiring of T.

We now proceed onto give examples of

matrix ideal of the semiring.

Example 3.1.29: Let V = {all 10 u 10 square in

with intervals of the form [0, a] with a � Z+ � {

interval semiring. Take I = {all 10 u 10 square in

with intervals of the form [0, a] where a � 5Z+

an ideal of V.

Example 3.1.30: Let V = {all 7 u 7 square in

with intervals of the form [0, a]; a � Q+ � {0

matrix interval semiring. W = {all 7 u 7 dimatrices with intervals of the form [0, a] where

� V; W is a square matrix interval ideal of V.



interval semiring. Consider W = {all 3 u 3 uppe

interval matrices with intervals of the form [0, a]

15Z+ � {0}} � V, it is easily verified W is only

square matrix subsemiring and is not an ideal of V.

Now having seen ideals and subsemirings

intervals matrix semirings now we proceed o

Smarandache square matrix interval semiring.

DEFINITION 3.1.3: Let V = {set of all n u n squa

matrices with intervals of the form [0, a], a � Q+ �

{0}} be a square interval matrix semiring.

Let P � V, if P is a square interval matrix se

define V to be a Smarandache square matrix interv(S-square matrix interval semiring).

Example 3.1.33: Let V = {2 u 2 interval matrices w

of the form [0, a]; a � Q+ � {0}} be a square inte

semiring. Take

W =a 0

a Q {0}0 0

½ª º° °

� �® ¾« »¬ ¼° °¯ ¿

� V,

W is a square matrix interval semifield, hence W is

matrix interval semifield.

Example 3.1.34 : Let V = {All 10 u 10 triangu

matrices with intervals of the form [0, a], a � Q+ �

square matrix interval semiring.



Consider

W =

[0,a] 0 0 00 [0,a] 0 0

a Q0 0 [0,a] 0

0 0 0 [0,a]

ª º° « »

° « » � �® « »° « »° ¬ ¼¯ W is a square matrix interval semifield containe

is a S-square matrix interval semiring.It is interesting to note that all square

semirings are not in general Smarandache square

semirings. We will illustrate this situation by som

Example 3.1.36: Let V = {all 2 u 2 upper triamatrices with intervals of the form [0, a]; a � 3

matrix interval semiring. We see V has no prope

such that T is a matrix interval semifield. Thus V

interval semiring.

Example 3.1.37: Let

V =[0,a] 0

a,b,c,d Z {00 0

½ª º°

� �® « »¬ ¼° ¯

be a matrix interval semiring.

W =[0,a] 0

a Z {0}0 0

½ª º° °

� �® ¾« »¬ ¼° °¯ ¿

�



a interval matrix subsemiring of S. If P contains a

� T � P such that T is a interval matrix semifie

define P to a Smarandache interval matrix subs

interval matrix subsemiring).


Example 3.1.38: Let S = {all 5 u 5 interval ma

intervals of the form [0, a] where a � Q+ � {0}} be

matrix semiring. Consider P = {all 5 u 5 diago

matrices with intervals of the form [0, a] where a �� S, P is a S-interval matrix subsemiring as P contain

T =

[0,a] 0 0 0 0

0 0 0 0 0

a Q {0}0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

½ª º° °« »° °« »° °« » � �® ¾

« »° °

« »° °« »° °¬ ¼¯ ¿

�

so that T is a semifield.

Example 3.1.39: Let V = {All 6 u 6 upper triangu

matrices with intervals of the form [0, a] where a � be the interval matrix semiring. Take W = {all 6 uinterval matrices with intervals of the form [0, a] wh

� {0}} � V; W is an interval matrix subsemiring of



However V is a S-interval matrix semiring.

claim.

Now we proceed onto define S-ideals of a inte

semiring.

DEFINITION 3.1.5: Let V be a S-matrix interval semi

empty subset P of V is said to be a Smarandache inteideal (S-interval matrix ideal) of V if the following

are satisfied.

1. P is an S-interval matrix subsemiring.

2. For every p � P and A � P where A is the

P we have for all a � A and p � P, pa and ap

We will illustrate this situation by some examples.

Example 3.1.41: Let V = {all 4 u 4 interval ma

intervals of the form [0, a] where a � Q+ � {0}} be

matrix semiring.

Choose

W =

[0,a] 0 0 0

0 [0,b] 0 0a,b,c Q {0}

0 0 [0,c] 0

0 0 0 0

½ª º° « »° « » � �® « »° « »° ¬ ¼¯

W is a S-matrix interval ideal of V.



The following result is left as an exercise f

prove.

THEOREM 3.1.3: Let V be a S-matrix interval s

S-matrix interval ideal of V is a S-matrix interval

V but every S-matrix interval subsemiring of V

not be a S-ideal of V.

We now proceed onto define the notion of S-

matrix subsemiring.

DEFINITION 3.1.6: Let S be a interval matrix s

empty proper subset A of S is said to be Smara

interval matrix subsemiring (S-pseudo in

subsemiring) if the following condition is true.

If there exists a subset P of S such that A �

S-interval matrix subsemiring, that is P has a su

B is a semifield under the operations of S

semifield under the operations of S.


Example 3.1.43: Let S = {All 4 u 4 interva

intervals of the form [0, a] where a � Q+

� {0interval semiring.

Let

[0,a] [0,b] 0 0ª º°



P =

[0, a] [0, b] [0,c] [0,d]

0 [0,e] [0,f ] [0,h] a,b,c,d,e,f ,h,m,0 0 [0,m] [0,n] n, t Q {0}

0 0 0 [0, t]

½ª º° « »° « »® « » � �° « »° ¬ ¼¯ ¿

is a S-matrix interval subsemiring of S. Thus A is

interval matrix subsemiring of S.

Example 3.1.44: Let V = {set of all 5 u 5 interval m

intervals of the form [0, a] where a � Z+ � {0}} be

matrix semiring.

Let

A =

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

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

a,b,c0 0 [0,b] 0 [0,a]

0 0 0 [0,a] 0

0 0 0 0 [0,b]

ª º° « »° « »° « » � ®

« »° « »

° « »° ¬ ¼¯

� V be a subset of interval 5 u 5 matrices of V. No

collection of 5 u 5 upper triangular interval ma

entries of the form [0, a], a � Z+ � {0}} = P � V

matrix interval subsemiring of V. Now

0 0 0 0 0ª º°« »



Now we proceed onto define the notion o

pseudo matrix interval ideal of an interval matrix

DEFINITION 3.1.7: Let V = {collection of all

matrices with intervals of the form [0, a] where a

Q+ � {0} or R+ � {0}} be an interval matrix se

V be a S-pseudo subsemiring of V and A �

interval semifield or a S-interval matrix subsemi

every p � P and for every a � A ap and pa � P t

to be a Smarandache pseudo matrix interval

pseudo matrix interval ideal of V).

We will illustrate this situation by an exampl

Example 3.1.45: Let V = {4 u 4 interval matrice

of the form [0, a] with a � R + � {0}} be an

semiring. Choose

P =

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

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

0 0 [0,c] 0

0 0 0 [0,d]

ª º°« »°« »®« »°« »°¬ ¼¯

| a, b, c, d � R

P is a S-pseudo matrix interval subsemiring of V

[0 ] [0 ] [0 ] [0 ]ª º



T =

[0,a] 0 0 0

0 0 0 0 a R {0}0 0 0 0

0 0 0 0

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

� A �

is a interval matrix semifield in V. Thus P is a S-pseu

matrix ideal of V. Now we can as in case of other interval semir

Smarandache zero divisors and Smarandache anto z

[ ]. The task of defining these notions are left as an

the reader.

However we will illustrate this situation by an ex

Example 3.1.46: Let

V =[0, a] [0, c]

a,b,c,d Z {0}

[0,b] [0,d]

½ª º° °

� �® ¾« »

¬ ¼° °¯ ¿

be an interval matrix semiring. It is easily verified

divisors and right or left zero divisor.

For instance

x = 0 0[0,a] 0

ª º« »¬ ¼

and y = 0 00 [0,b]

ª º« »¬ ¼

in V is such that



Consider p =[0,a] 0

0 0

ª º« »

¬ ¼

and q =0 0

0 [0,b]

ª º« »

¬ ¼

in V

p. q =[0,a] 0

0 0

ª º« »¬ ¼

0 0

0 [0,b]

ª º« »¬ ¼

=0

0

ª « ¬

Now

q.p = 0 0

0 [0,b]ª º« »¬ ¼

[0,a] 0

0 0ª º« »¬ ¼

= 0

0ª « ¬

Thus p.q = (0) is a zero divisor in V.

Now choose

x =0 0

[0, t] 0ª º« »¬ ¼

� V

is such that

x.p =0 0

[0, t] 0

ª º« »

¬ ¼

[0,a] 0

0 0

ª º« »

¬ ¼=

0 0

[0, ta] 0

ª º« »¬ ¼

z (0)

and

px =[0,a] 0

0 0

ª º

« »¬ ¼

0 0

[0, t] 0

ª º

« »¬ ¼

=0 0

0 0

ª º« »¬ ¼

.



and

yq =0 [0,m]

0 0

ª º« »¬ ¼

0 0

0 [0,b]

ª º« »¬ ¼

=0 [0,mb]

0 0

ª º« »¬ ¼

Now

xy =0 0

[0, t] 0

ª º« »¬ ¼

0 [0,m]

0 0

ª º« »¬ ¼

=0 0

0 [0, tm]

ª º« »¬ ¼

z

yx =0 [0,m]

0 0

ª º« »¬ ¼

0 0

[0, t] 0

ª º« »¬ ¼

=[0,mt] 0

0 0

ª º« »¬ ¼

z

Thus p, q � V is a Smarandache zero divisor of V.

The following observation is important.

If an interval matrix semiring V has Smaranda

matrix zero divisors then V has zero divisors, but h

question is, will every zero divisor in an inter

semiring will always be a S-zero divisor; is open.

Now we proceed onto give an example of a Smara

zero divisors of an interval matrix semiring V.

Example 3.1.47: Let

V =[0, a] [0, c]

a,b,c,d Q {0}[0,b] [0,d]

½ª º° °� �® ¾« »

¬ ¼° °¯ ¿be an interval matrix semiring.



It is still interesting to note that in general if x in V,

interval semiring is a Smarandache matrix interva

divisor in V then in general x need not be a zero divi

Further it is important to note these inter

semirings can have right zero divisors or left zero div

The author is left with the task of defining

examples of Smarandache idempotents in intersemirings.

The study of Smarandache units in interval matri

can be carried out as a matter of routine [ ]. N

following section we proceed onto define and descr

polynomial semirings.

3.2 Interval Polynomial Semirings

In this section also we only concentrate on intervals

[0, a] where a is from Z+

� {0}, Q+

� {0} or R +

� {0If x is any variable or an indeterminate we def

polynomial semiring in an analogous way.

DEFINITION 3.2.1: Let S =f

®¯¦

i

i

i 0

[0,a ]x ai � Z + �

{0} or Q+ � {0}} be the collection of all interval pol

under addition is an abelian semigroup with 0 = [

l i l hi h h th t ti f th



Before we proceed to give examples of this stru

how the addition and multiplication of interval p

carried out.

Let p(x) = [0, 5] + [0, 3] x2 + [0, 4] x3 + [0, 9] x

3] + [0, 7] x + [0, 12] x2 + [0, 40] x3 + [0, 18

interval polynomials in the variable x an

coefficients from Z+

� {0}.

p (x) + q (x)

= ([0, 5] + [0, 3] x2 + [0, 4] x3 + [0, 9] x7) +

x + [0, 12] x2 + [0, 40] x3 + [0, 18] x5)

= ([0, 5] + [0, 3]) + ([0, 3] + [0, 0]) x2 + (

x3 + ([0, 0] + [0, 7]) x + ([0, 0] + [0, 18

[0, 0]) x7

= [0, 8] + [0, 3] x2 + [0, 44] x3 + [0, 7] x +

9] x7.

Now we will show how the product is define= ([0, 5] + [0, 3] x2 + [0, 4]x3 + [0, 9] x7) (

+ [0, 12] x2 + [0, 40] x3 + [0, 18] x5)

= [0, 5] [0, 3] + [0, 3] [0, 3] x2 + [0, 4] [0,

3] x7 + [0, 5] [0, 7] x + [0, 3] [0, 7] x2.x

x3.x + [0, 9] [0, 7] x7.x + [0, 5] [0, 12]x2

x2.x2 + [0, 4] [0, 12] x3. x2 + [0, 9] [0, 12

[0, 40] x3 + [0, 3] [0, 40] x2. x3 + [0, 4]7 3 5



36]) x4 + ([0, 48] + [0, 120] + [0, 90]) x5 + [

([0, 27] + [0, 54]) x7 + ([0, 63] + [0, 72]) x

x9 + [0, 360]x10 + [0, 162]x12

= [0, 15] + [0, 35] x + [0, 69]x2 + [0, 233]x3 +

[0, 258]x5 + [0, 160]x6 + [0, 81]x7 + [0, 13

108] x9 + [0, 360] x10 + [0, 162] x12.

We see the interval polynomial multiplication a polynomial multiplication is commutative and [0,

[0,0]x + [0,0] x2 + …+ [0,0] xn, acts as the unit eleme

We will illustrate the distributive law.

Let

a(x) = [0, 5] + [0, 8] x + [0, 1] x8

b(x) = [0, 1] x + [0, 3] x2 and

c(x) = [0, 7] + [0, 3] x4 + [0, 5] x7

be three interval polynomials of an interval

semiring.

To show

a(x) [b(x) + c(x)] = a (x) b (x) + a (x) c (x)

Consider

a (x) [b (x) + c (x)]

= ([0, 5] + [0, 8] x + [0, 1] x8) [([0, 1]x + [0, 3

7] + [0, 3] x4 + [0, 5] x7)]

= ([0, 5] + [0, 8] x + [0, 1] x8) + ([0, 1] x + [0

7] + [0, 3] x4 + [0, 5] x7)

= [0, 5] [0, 1] x + [0, 8] [0, 1]x.x + [0, 1] [0, 12 2



= [0, 35] + [0, 61] x + [0, 23] x2 + [0, 24] x

[0, 24] x5 + [0, 25] x7 + [0, 47] x8 + [0, 1

+ [0, 3] x12 + [0, 5] x15

Consider

a(x) b(x) + a(x) c(x) =

= ([0, 5] + [0, 8]x + [0, 1] x8

) ([0, 1]x + [0+ [0, 8]x + [0, 1]x8) ([0, 7] + [0, 3] x4 + [

= [0, 5] [0, 1] x + [0, 8] [0, 1] x.x + [0, 1]

5] [0, 3] x2 + [0, 8] [0, 3] x.x2 + [0, 1] [0

5] [0, 7] + [0, 8] [0, 7] x + [0, 1] [0, 7] x

x4 + [0, 8] [0, 3] x.x4 + [0, 1] [0, 3] x8.x

x7 + [0, 8] [0, 5] x.x7 + [0, 1] [0, 5] x8.x7

= [0, 35] + ([0, 5] [0, 1] + [0, 8] [0, 7]) x

15]) x2 + ([0, 24] x3 + [0, 15]) x4 + [0,

x7

+ ([0, 7] + [0, 40]) x8

+ [0, 1] x9

+ [0x12 + [0, 15] x15

= [0, 35] + [0, 61] x + [0, 23] x2 + [0, 24] x

[0, 24] x5 + [0, 25] x7 + [0, 47] x8 + [0, 1

+ [0, 3] x12 + [0, 5] x15

It is easily verified I and II are equal, thus S

polynomial semiring.



Now having seen examples of interval

semirings we now proceed onto define Smarandac

polynomial semirings.

DEFINITION 3.2.3: Let V be any interval polynomial

proper subset T of V, is such that T is an interval sem

we define V to be a Smarandache polynomial interv

(S – polynomial interval semiring).


Example 3.2.11: Let

P = ii i

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

f

½� �® ¾¯ ¿¦

be a polylnomial interval semiring. Take T = {[0, ai]

{0}} � P, T is an interval semifield, so P is a S-

interval semiring.

Example 3.2.12: Let

W = i

i i

i 0

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

½� �® ¾

¯ ¿¦

be an interval polynomial semiring. Choose G = {[

Q+ � {0}} � W, G is an S-interval polynomial semir

THEOREM 3.2.1: Let

W d fi S l i l i l b



We can define S-polynomial interval sub

analogous way. We leave this task to the reade

some examples of S-interval polynomial subsem

Example 3.2.13: Let

T = i

i ii 0

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

½� �

® ¾¯ ¿¦ be an interval polynomial semiring.

Choose

W =

i

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

f

½

� �® ¾¯ ¿¦ �

W is a S-interval polynomial subsemiring as G

Z+ � {0}} � W is an interval semifield of W.

Example 3.2.14: Let V = {6 [0, ai] xi / ai � Z

interval polynomial semiring. G = {6 [0, ai] x2i /

� V, G is a S-interval polynomial subsemiring o

[0, ai] / ai � Z+ � {0}} � G is an interval semifie

Now we have the following theorem.

THEOREM 3.2.2: Let V be an interval polynomia

has a S interval polynomial subsemiring then V



be an interval polynomial subsemiring of G. C

ideal of the interval polynomial semiring G.Infact this interval polynomial semiring G

many ideals.

We will give examples of interval polynomial s

has interval polynomial subsemirings which areknow every ideal of an interval polynomial

interval polynomial subsemiring.

Example 3.2.17: Let

P = i

i i[0,a ]x a Q {0} ½� �® ¾

¯ ¿¦

be an interval polynomial semiring.

Consider

G = i

i i

i 0

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

½� �® ¾

¯ ¿¦ �

G is only an interval polynomial subsemiring w

ideal of P.

Now we will give examples of S-subsemi

not S-ideals of an interval polynomial semiring.

½



P = 2i

i i

i 0

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

½� �® ¾

¯ ¿

¦ � V

be a S-interval polynomial subsemiring of V. P is no

of V.

Concept of S-pseudo ideal and S-pseudo sub

case of interval polynomial semirings can be studi

and analysed by the reader with appropriate example

However the concept of S-zero divisors, S-anti z

or S-idempotents have no relevance in case

polynomial semirings built using Z+ � {0} or Z p (p

Q+ � {0} or R + � {0}. However when Zn iscompositive number these concepts have relevance.



Chapter Four

GROUP INTERVAL SEMIRINGS

In this chapter we for the first time introduce the

group interval semirings, semigroup interval

interval semiring and groupoid interval semirin

few of the properties associated with them. This sections. In section one we introduce the co

interval semirings and semigroup interval semiri

a few of the properties associated with them In

DEFINITION 4 1 1: Let S = {[0 ai] / ai � Z or Z+ �



DEFINITION 4.1.1: Let S = {[0, ai ] / ai � Z n or Z �

� {0} or R+ � {0}} be the commutative semiring w

1]. G be any group. The group semiring of the group

semiring S consists of all finite formal sums

[0, ]¦ i i

i

D (i runs over finite number) where [0, D i ]

� G satisfying the following conditions.

(i)1

[0, ]

¦n

i i

i

D =1

[0, ]

¦n

i i

i

g E if and only if D i

1,2,…,n.

(ii)1

[0, ]¦n

i i

i

g D +1

[0, ]¦n

i i

i

g E =1

[0, ]

¦n

i i

i

g D E

(iii)1

[0, ]

§ ·¨ ¸© ¹¦

n

i i

i

g D 1

[0, ]

§ ·¨ ¸© ¹¦

n

i i

i

g E =1

[0, ]

¦t

i j

j

mJ

= m j and J j = 6 [0, D i E k ]

(iv) For [0, D i ] � S and g i � G; g i [0, D i ] = [0, D

(v) [0, a]1

[0, ]

¦n

i i

i

b g =1

[0, ][0, ]

¦n

i i

i

a b g =1

[0

¦n

i

[0, a] � S and g i � G.

SG is an associative interval semiring with 0

Example 4.1.1: Let S = {[0 a] | a � Z+ � {0}



Example 4.1.1: Let S {[0, a] | a � Z � {0}

semiring. G = ¢g | g2 = 1² be a cyclic group. SG

b]g | [0, a], [0, b] � S; g � G} is the group interv

Clearly SG is of infinite order and is a comm

semiring with unit [0, 1].

Example 4.1.2: Let S = {[0, a] | a � Q+ � {0}

semiring with unit [0, 1]. G = D2.7 = {1, a, b / a2

a} be the group.

SG = i i i i

i

[0,a ]g a Q {0},g D� � �® ¯ ¦

the group interval semiring is of infinite ordcommutative.

Example 4.1.3: Let S = {[0, a] / a � Z7} be an in

and G = ¢g | g5

= 1² be the group, SG group inte

of finite order and is commutative.

SG = {[0, a0] + [0, a1] g + [0, a2] g2 + [0, a3]

g5 = 1, [0, ai] � S, 0 d i d 4}. [0, 1] acts as the ide

Example 4.1.4: Let S = {[0, a] / a � Z6 = {0, 1,

finite commutative semiring with identity [0, 1].

p2, p3, p4, p5} be the permutation group where

SG = {[0, a0] + [0, a1] p1 + [0, a2] p2 + [0, a3] p3 + [0,



{[ , 0] [ , 1] p1 [ , 2] p2 [ , 3] p3 [ ,

a5] p5 | ai � Z6; 0 d i d 5}.

Example 4.1.5: Let S = {[0, a] / a � Q+ �commutative interval semiring with unit. Choose G =

1} be a cyclic group of finite order.

SG = i

ii

[0,a]g a Q {0} ½� �

® ¾¯ ¿¦; gi � G, 1 d i

is the group interval semiring of the group G over

semiring S. Clearly SG is a commutative interval

infinite order with the unit [0, 1].

It is important to mention here that the

constructing group interval semirings leads to getclass of both finite and infinite and commutativ

commutative interval semirings.

We can define substructures and special elemen

interval semirings.

We see S = {[0, a] / a � Z+ � {0} or Q+ � {0}

� {0}} (p a prime) are interval semirings which

semifields.

We will describe some zero divisors, units and idem

these group interval semirings.

Example 4.1.6: Let S = {[0, a] / a � Z4} be an interv

G = ¢g / g4 = 1² be any group. The group interval s

= [0, 1] + [0, 1]g2 + [0, 1]g4 + [0, 1]g6 + 2



[ , ] [ , ]g [ , ]g [ , ]g

1] g2

+ 2[0, 1] g3

+ 2 [0, 1] g3

+ 2 [0, 1] g

= [0, 1] + [0, 1]g2 + [0, 1] + [0, 1]g2 + 2

1]g2

+ 2 [0, 1]g3

+ 2 [0, 1] g3

+ 2 [0, 1] +

= 0.

Thus D

2

= 0 in SG is nilpotent. Also if b = [0, 2] b

2= ([0, 2])

2+ 2 ([0, 2] [0, 2]) g + ([0,

= 0 is again an nilpotent element of order two. Ta

g + [0, 2]) ([0, 2]g2 + [0, 2]g3) = 0 where x = [0,

y = [0, 2]g2 + [0, 2]g3 are in SG. Thus x, y

divisor in SG.

Example 4.1.7: Let S = {[0, a] / a � Z2} and G =

the interval semiring and group respectively.

interval semiring of G over the interval semiring

SG = {0, [0, 1], [0, 1]g, [0, 1]g2

, [0, 1]g3

, [0, 1] 1]g2 + [0, 1]g3, [0, 1] + [0, 1]g, [0, 1] + [0, 1]g2,

[0, 1]g + [0, 1]g2, [0, 1]g + [0, 1]g3, [0, 1]g2 + [0

[0, 1]g + [0, 1]g2, [0, 1] + [0, 1]g + [0, 1]g3, [0,

[0, 1]g3 + [0, 1]g + [0, 1]g2 + [0, 1]g

Now

D = [0, 1] + [0, 1]g + [0, 1]g2 + [0

in SG is such that

D2 = [0, 1] + [0, 1]g2 + [0, 1]g4 + [0, 1]g6 + 2

Take D = [0, 1] + [0, 1]g + [0, 1]g2 + [0, 1]g3 and E =



[ , ] [ , ]g [ , ]g [ , ]g E1]g in SG. ClearlyDE = 0.

Example 4.1.8: Let S = {[0, a] / a � Z3} be an interv

and G = ¢g | g2 = 1² be the cyclic group of orde

interval group semiring or group interval semiring SG

0, [0, 2], [0, 1]g, [0, 2]g, [0, 1] + [0, 1]g, [0, 1] + [0, 2

[0, 1]g, [0, 2] + [0, 2]g}. Now ([0, 2]g)2 = [0, 1] is a unit in SG. Consider D =

2]g in SG.

D2 = [0, 2]2 + [0, 2]2 g2 + 2 [0, 2] [0, 2] g

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

= [0, 2] + [0, 2] g

= D.

Thus D is an idempotent in SG. Consider E = [0, 2]

SG.

E2 = ([0, 2] + [0, 1]g)2

= [0, 2]2 + [0, 1]g2 + 2 [0, 2] [0, 1]g

= [0, 1] + [0, 1] + [0, 1]g

= [0, 2] + [0, 1]g = E

Thus E is an idempotent in SG.

Example 4.1.9: Let S = {[0, a] / a � Z3} be an intervand G = {g / g3 = 1} be a cyclic group of order thre

[0, 1], [0, 2], [0, 1] g, [0, 2]g, [0, 1]g2, [0, 2]g2, [0, 1

[0 1] + [0 2]g [0 2] + [0 1]g [0 2] + [0 2]g +

Now we work with infinite group interval semiri



Example 4.1.10 : Let S = {[0, a] / a � Z+ � {0}semiring. G = ¢g / g2 = 1² be a cyclic group of or

group interval semiring. SG = {[0, a] + [0, b] g /

S}.

Take D = [0, 1]g � SG; D2 = [0, 1] is a self in

of SG. We see if a = [0, a] + [0, b] g then an

= [0all n and [0, a] < [0, x] and [0, b] < [0, y] thus

[0, t] + [0, u] g where t and u are very large.

Clearly SG has no zero divisors.

Example 4.1.11: Let S = {[0, a] / a � Z+ � {0}

semiring. G = ¢g / g3 = 1² be a group of order th

interval semiring SG has no zero divisors.

Now we proceed onto define substructures in

semirings SG.

Let us illustrate them by examples and thsubstructures can be defined as a matter of routin

Example 4.1.12: Let S = {[0, a] / a � Z+ � {0}

semiring. G = S3 the symmetric group of degree

[0, a]g / g � G a � Z+

� {0}} is the group interthe group G over the interval semiring S.

Consider

1 2 3§ · 1 2 3 ½§ ·°

SP is an interval subsemiring of SG.



Now we can define interval subsemiring also as f

TG = {6 [0, a] g / g � G, a � 2Z+ � {0}} � SGinterval subsemiring. (This subsemiring has no unit)

Example 4.1.13: Let S = {[0, a] / a � Z+ � {0}} be

semiring. G = ¢g | g p = 1² be a cyclic group of order p

SG = i i

i

[0,a]g g G;a Z {0} ½� � �® ¾

¯ ¿¦

is the group interval semiring of the group G over

semiring S. G is simple, as G has no proper However

PG = i

i

[0,a]g a 5Z {0} ½� �® ¾

¯ ¿¦ � SG.

Now P = {[0, a] / a � 5Z+ � {0}} � S = {[0, a] / a �

so PG � SG is an group interval subsemiring of SG

no unit.

Example 4.1.14: Let S = {[0, a] / a � Z4} be

semiring. Consider G = ¢g / g5 = 1² be a cyclic grou

5. SG be the group interval semiring of the group

semiring S. SG = {[0, 1], [0, 2], [0, 3], 0,4

i 0

[0,a]g¦

2 3

It is interesting to note that none of the +



semirings SG when S = {[0, a] | a � Z+ � {0} or

� {0}} have zero divisors for any group G. Infastrict interval semirings and S-interval semiring

commutative group SG turns out to be an interva

But when SG, where S = {[0, a] | a � Zn, n <

can have zero divisors, nilpotents and idempote

not an interval semifield as [0, a] + [0, b] = 0 (min which a z 0 and b z 0.

Thus question of SG becoming an interval se

out for S built using Zn, n < f.

Infact all interval semirings S = {[0, a] | a �� {0} or R + � {0}} happen to be a complete p

set with least element [0, 0] = 0 such that the su

operations are continuous. Interested author c

interval semirings under * operation and cha

interval * semirings.

A study of group interval semirings SG whe

of order n (n a non prime) and S = {[0, a] | a �such that (1) where p / n. (2) when p \ n.

Also the question when G is a cyclic group

prime and S = {[0, a] / a � Zn}, n a composite n

and p \ n can be studied by interested reader.

Now we proceed onto study semigroup int

SP where S is an interval semiring and P is a

semigroup.

u 0 1 2 3



0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Now SG = {[0, a] + [0, b] 2 + [0, c] 3 / a, b

{0}} is an interval semigroup semiring. SG has zero

For take [0, b] 2 . [0, c] 2 = 0

For all a, b � Z+. [0, 1] 3 . [0, 1] 3 = [0, 1] is

be noted that SG is an infinite semigroup interval se

has nilpotent element of order two. For all ([0, a] 2 )

a � Z+.

Now we proceed onto give another example

semigroup interval semiring.

Example 4.1.16: Let S = {[0, a]| a � Z6} be

semiring. Let G = (Z3, u) be the semigroup, under mu

modulo three given by the table.

u 0 1 2

0 0 0 01 0 1 2

2 0 2 1

[0, 3] . [0, 2] 2 = 0 and so on.



Also [0, 5] [0, 5] = [0, 1], [0, 5] 2 / [0, 5] 2

on. [0, 3]. [0, 3] = [0, 3] is an interval idempoten

Example 4.1.17 : Let S = {[0, a]| a � Z2}

semiring. G = (Z6, u) be the semigroup under mu

SG = {[0, 1] + [0, 1] 2 + [0, 1] 3 + [0, 1] 4

[0, 1], [0, 1] + [0, 1] 2 , …, [0, 1] 3 + [0, 1] 4

be the semigroup interval semiring.

Clearly SG is of finite order and |SG| = 32.

Consider x = [0, 1] +[0, 1] 2 , + [0, 1] 4 in

= [0, 1], so x is a unit in SG.

Consider y = [0, 1] + [0, 1] 3 � SG; y2 = [= y, so y is an idempotent of SG.

Take p = [0, 1] + [0, 1] 5 � SG.

p2 = (0). Several other properties related w

obtained by the interested reader.

Example 4.1.18: Let S = {[0, a] / a � Z12}

semiring. G = {Z6, u} be the semigroup under

SG be the semigroup interval semiring of G over

SG has zero divisors, units and idempotents

Now we give examples of semigroup interval sub

Example 4.1.19: Let S = {[0, a] | a � Z2}

Example 4.1.20: Let S = {[0, a] | a � Z24} be

i i G {Z } b th i d



semiring. G = {Z5, u} be the semigroup under mu

modulo 5. Let

SG = 24

g

[0,a]g a Z½° °

�® ¾° °¯ ¿¦

be the semigroup interval semiring of the semigroup

interval semiring S.

Now consider W = {6 [0, a] g / g � G and a � {

…, 22} � Z24} � SG; W is a semigroup interval sub

SG.

Example 4.1.21 : Let S = {[0, a] | a � Q+ � {0} } be

semiring. Let G = {Z19, u} be a semigroup under mu

modulo 19. SG be the semigroup interval semir

semigroup G over the interval semiring S.

SG =g

[0,a]g a Q {0} and g G½° °

� � �® ¾° °¯ ¿

¦

Now let M = {6 [0, a] g / a � Z+ � {0} and g �

clearly M is an interval semiring and is the semigro

subsemiring of SG. Infact SG has infinite number of

interval subsemirings. The order of SG is infinite an

of every semigroup interval subsemiring is also infin

Example 4.1.22: Let S = {[0, a] | a � Z+ � {0}} be

semiring. G = {Z71, u} be the semigroup under mu

S. Now let J = {6 [0, a] g / g � G and a � {0, 2



SG, J is an ideal of SG. Take D = [0, a] + [0, b]

[0, c] 6 in J. It is easily verified J is an ideal of S

Interested reader can study more about ideals an

in a semigroup interval semiring. We now cons

of semigroups and study them.

Example 4.1.24 : Let S(3) be the set of all mapp

itself S(3) is the semigroup under the operation

of maps. Let S = {[0, a] / a �Z3} be the interval s

is the semigroup interval semiring of the semig

the interval semiring S. Clearly order of SS (SS(3) is a non commutative interval semiring.

Further it is interesting to note that SS (3) c

interval semiring SS3 which is a semi

subsemiring. We can also call SS (3) as a S-sem

semiring as S (3) is a S-semigroup for S3 � S (3)

Example 4.1.25: Let S (3) be the symmetric sem

= {[0, a] | a � Z+ � {0}} be an interval semiring

a]g | g � S(3) and a � Z+ � {0}} is a sem

semiring of the semigroup S(3) over the interv

Clearly SS(3) is of infinite cardinality and is a ninterval semiring. Infact SS(3) is a S-interval sem

E l 4 1 26 Let S(7) be a s mmetric semigr

However SS (7) is not a semigroup interval S-sem



Example 4.1.27 : Let S = {[0, a] | a � Z7} be semiring and S(m) be a symmetric semigroup (m

SS(m) the semigroup interval semiring of the semi

over the interval semiring S.S is not a S-semiring a

non commutative and not a strict semiring.

Now we give examples of interval semidivision ring.

Example 4.1.28: Let S = {[0, a] | a � Z+ � {0} or Q

R + � {0}} be an interval semiring. Take S (n)

symmetric semigroup. The semigroup interval semi

is non commutative but is a strict semiring and SS

zero divisors. Hence SS (n) is a division interval sem

SS (n) is a S-semiring.

Now having defined and seen the concept o

interval semigroup and group interval semiring we nonto define the notion of interval semigroup se

interval group semiring which we choose to call

interval semirings.

DEFINITION 4.1.2: Let S be any commutative semirin

which is not an interval semiring. P an interval sem

= { 6 ai [0, pi ] | ai � S and [0, pi ] � P} under usual a

multiplication is a semiring defined as the interval

interval semigroup P over the semiring S. SP = {

Z ai � Z+ � {0}} The following observations



Z12 , ai � Z � {0}}. The following observations

(i) SP is not a semifield for x = 5 [0, 6 ] a

in SP are such that xy = 0. So SP has zer

(ii) SP has idempotents, for take a = [0, 4

that a2 = a.

(iii) SP is a S-interval semiring as S � SP fothe identity element of SP; so S = S [0,1

S is a semifield.

(iv) S is strict interval semiring as s + p = 0 i

0 and p = 0.

(v) SP has nilpotents for consider t = n [0, [0, 0] = 0.

Example 4.1.30 : Let S = {Q+ � {0}} be a sem

P = {[0, a] | a � Z+ � {0}} an interval semiring

interval semigroup semiring is of infinite orde

formal sums of the form 6 ai [0, xi] where ai �P. ai [0, 0] = 0 for all ai � Q+ � {0} and ai [0, 1]

is assumed to be the multiplicative identity of SP

It is easily verified SP is an infinite s

semiring. Further SP is commutative. Also SP is

special semiring.Thus SP itself is a semifield.

Thus we by this method construct semifi

di lit

Now by this method we can use distributive

construct interval semigroup semirings of finite orde



construct interval semigroup semirings of finite orde

all distributive lattices are semirings. We will ulattices to construct interval semigroup rings.

Example 4.1.32: Let

S = { be the distributive lattice}

which is a semiring. P = {[0, a] / a � Z7} be t

semigroup under multiplication. SP is an interval

semiring where SP = {0, 1 [0, a], 6 x [0, a] where x =

a � Z7}. Clearly SP is of finite order. SP has no zero

idempotents but SP has units for 1 [0, 6]. 1 [0, 6] =

[0, 4]. 1 [0, 2] = 1 [0, 8] = 1 [0, 1] and so on.

Example 4.1.33 : Let S = { the chain lattice given b

a3, a4, 1, 2

ia = ai, ai 1 = ai, ai 0 = 0, 0 < a1 < a2 < a3 < a

d 4} is a semiring. P = {[0, a] / a � Z6} is t

semigroup under multiplication. SP = {6 x [0, a] f

sums where x � S and a � Z6} is the interval

semiring of finite order. This special interval semiri

divisors units and idempotents. However SP is a S-se

Example 4.1.34 : Let S = {0, a, b, 1, a lattice gi

diagram

1

0

1



Example 4.1.35 : Let S = {0, 1, a, b, c, d, e, f |

be the semiring (which is a distributive lattice) a

a � R + � {0}} be an interval semigroup. Consid

a], finite formal sums with x � S and a � R + �P} is the interval semigroup semiring of infinite

SP has zero divisors and units.

Thus using the special type of semirings w

lattices we can construct interval semigroup semorder as well as infinite order.

Now we proceed onto define interval group und

multiplication. However we will be using only

under multiplication.

DEFINITION 4.1.3: Let G = {[0, a] | a � Z n;

group under addition of finite order defined as

under addition.

Example 4.1.36: Let G = {[0, a] | a � Z8}, G

group under addition.

d

1

b

0

a

c

e

f

R + � {0}} is not an interval group under addition. Th

proceed onto define interval group under multiplicati



p g p p

DEFINITION 4.1.3: Let S = {[0, a] | a � Q+ } be an

S is an interval group under multiplication of infinite

If we replace Q+ by R+ then also S is an inte

under multiplication of infinite order.

However if we replace Q

+

by Z

+

, S is not an intunder multiplication only an interval semigr

multiplication.

Now if we replace Q+ by Z p \ {0} p any prime p t

an interval group under multiplication of finite o

using Z p \ {0} we get a collection of interval gr

multiplication of finite order which is commutative.


Example 4.1.38: Let S = {[0, a] | a � Z5 \ {0}} = {[0

[0, 3], [0, 4]} is an interval group under multiplicatfive. We see [0, 4] [0, 4] = [0, 1] [0, 1] is the identity

S.

We will illustrate this by the following table.

u [0, 1] [0, 2] [0, 3] [0, 4]

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

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

[0 3] [0 3] [0 1] [0 4] [0 2]

and is commutative. [0, 2] [0, 4] = [0, 1], i.e

inverse of [0, 4].



[0, 3] [0, 5] = [0, 1] and [0, 6] [0, 6] = [0, 1given by the following table.

u [0, 1] [0, 2] [0, 3] [0, 4] [0,

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

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

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

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

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

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

Clearly S is cyclic and generated by <[0,

commutative.

Now we give yet another example of interval gro

Example 4.1.40: Let G = {[0, 1], [0, 2], [0, 3], [

6], [0, 7], [0, 8], [0, 9], [0, 10], u} where u is mo

interval group of order 10 and <[0, 6]> generate

cyclic interval group of order 10.

Now having seen examples of interval groupand infinite order now we proceed onto defi

group semirings.

be the interval group semiring of the interval group

semiring G. Clearly SG is a semiring. We see SG



but SG is commutative. Further SG is a strict semirin

Example 4.1.42: Let S = {R + � {0}} be a semifield.

= {[0, a] / a � Q+} be an interval group. Let

SG = i i

i

[0,a ]D®¯¦

we consider only finite formal sums of the interv

from the interval group and coefficients from the sem

{0}}, SG is a semiring which is the interval group

the interval group G over the semiring S. SG is com

infinite order infact SG is a semifield. SG is also knspecial interval semiring. Suppose D = 8 [0, 4] + 19

[0, 11/2] and E = [0, 3] + 3 [0, 5/9] + 8 [0, 3/7] be in

D + E = (8 [0, 4] + 19 [0, 3/7] + 2 [0, 11/2] ) + [

5/9] + 8 [0, 3/7]

= (8 [0, 4] + 27 [0, 3/7] + 2 [0, 11/2] ) + 1[0, 3

Now

DE = (8 [0, 4] + 19 [0, 3/7] + 2 [0, 11/2] ) u (1[0

5/9] + 8 [0, 3/7] )

= (8.1 [0, 4] [0, 3] + 19.1 [0, 3/7] [0, 5/9] + 8

5/9] + 19.1 [0, 3/7] [0, 3] ++ 8.8 [0, 4] [0, 5/9] + 19.1 [0, 3/7] [0, 3] + 2

[0, 3] + 2.1 [0, 11/2] [0, 5/9] + 2.8 [0, 11/2] [

3] + 2 [0, 9] + 4 [0, 6] + 3/7 [0, 7] and E = 3 [0

3/2 [0, 6] be in SG. Now



D + E = (9 [0, 3] + 2 [0, 9] + 4 [0, 6] + 3/7 [0

+ [0, 10] + 3/2 [0, 6])

= 9 [0, 3] + 2 [0, 9] + 3/7 [0, 7] + (4 + 3/

8] + [0, 10].

DE = (9 [0, 3] + 2 [0, 9] + 4 [0, 6] + 3/7 [0, 7

[0, 10] + 3/2 [0, 6])

= 9.3 [0, 3] [0, 8] + 9.1 [0, 3] [0, 10] + 9.3/

2.3 [0, 9] [0, 8] + 2.1 [0, 9] [0, 10] + 2.3/

4.3 [0, 6] [0, 8] + 4.1 [0, 6] [0, 10] + 4.3/

3/7.3 [0, 7] [0, 8] + 3/7.1 [0, 7] [0, 10] +[0, 6]

= 27 [0, 2] + 9 [0, 8] + 27/2 [0, 7] + 6 [0,

[0, 10] + 12 [0, 4] + 4 [0, 5] + 6 [0, 3]

= (27 [0, 2] + 2 [0, 2]) + 9 [0, 8] + 27/2 [0

3 [0, 10] + 12 [0, 4] + 4 [0, 5] + 6 [0, 3]= 29 [0, 2] + 9 [0, 8] + 27/2 [0, 7] + 6 [0,

12 [0, 4] + 4 [0, 5] + 6 [0, 3].

SG is a special interval semiring of infinite order

Example 4.1.44: Let S = {R + � {0}} be a semir

/ a � R +} be an interval group under multiplica

interval group semiring of the interval grou

D . E = 2 [0, 5] + 3 [0, 19 ] + 2 [0, 4/3] ( 41

3 [0, 1] + 17 [0, 8 ]



3 [0, 1] + 17 [0, 8 ]

= 2. 41 [0, 5], [0, 5 ] + 2. 3 [0, 5

2. 17 [0, 5], [0, 8 ] + 3 41 [0, 19 ]

3. 3 [0, 19 ] [0, 1] + 3. 17 [0, 19 ] [

41 [0, 4/3], [0, 5 ] + 2. 3 [0, 4/3] [0, 1

[0, 4/3] [0, 8 ]

= 82 [0, 5 5 ] + 6 [0,5] + 34 [0, 5 8

[0, 95 ] + 3 3 [0, 19 ] + 3 17 [0,

414 5

0,3

ª º« »« »¬ ¼

+ 2 3 [0, 4/3] + 2 7 [0, 4

Clearly SG is an infinite commutative interval grou

Clearly SG has no zero divisors. Further SG is a stsemiring hence is an interval semifield.

Now we proceed onto give examples of int

semirings of finite order using distributive finite latti

Example 4.1.45: Let S = {0, 1, a, b, c, d, e, f, g}

chain lattice given by the following diagram. That is

semiring

Choose G = {[0, a] | a � Z3 \ {0}} be an int

= {a1 [0, 1] + a2 [0, 2] / ai � S 1d i d 2} be the



semiring of the interval group G over the semSG is of finite order. SG has no zero divisor.

Example 4.1.46: Let S = {0, a, b, 1} be a sem

example 4.1.34. G = {[0, a] / a � Z5 \ {0}} be an

of order four. SG be the interval group semiring

group G over the semiring S. Clearly SG is of

has zero divisors and units.

Take

a [0, 4] . b [0, 2] = a.b [0,4] [0, 2]

= a.b [0, 3]

= 0 as a.b = 0.1 [0, 4] 1 [0, 4] = 1. [0, 1]

and

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

But SG is a strict commutative interval group has zero divisor.

Now having seen special types of interval sem

proceed onto define interval groupoid sem

interval semirings, loop interval semirings an

semirings.

SG =½

� �® ¯ ¿¦ i i

i

[0,a ]g [0,a ] S and g G



¯ ¿

where the sum ¦ i

i

[0,a]g is a finite formal

coefficients from S. The product and sum are defined

of semigroup interval semiring. SG is a non associat

semiring known as the groupoid interval semiring.

the groupoid interval semiring of the groupoid interval semiring S.

We will illustrate this situation by some simple exam

Example 4.2.1: Let S = {[0, a] / a � Z+ � {0}} be

semiring with unit. G = {Z5, *, a * b = ta + ub (mod

u = 2} be a groupoid. SG = {6 [0, a] g / [0, a] � S a

be the groupoid interval semiring of the groupoid

interval semiring S. Consider D = [0, 3] 4 + [0, 8] 3

E = [0, 5] 2 + [0, 7] 3 in SG.

D + E = [0, 3] 4 + ([0, 8] + [0, 7]) 3 + [0, 4] 1 +

= [0, 3] 4 + ([0, 15]) 3 + [0, 4] 1 + [0, 5]

Now

D u E = ([0, 3] 4 + [0, 8] 3 + [0, 4] 1 ) u [0, 5] 2

= [0, 3] [0, 5] 4 * 2 + [0, 8] [0, 5] 3 * 2 + 1 * 2 + [0, 3] [0, 7] 4 * 3 + [0, 8] [0, 7] 3

4] [0, 7] 1 * 3

We show SG is not associative under the pr



in the groupoid interval semiring.Consider D = [0, 2] 3 , E = [0, 5] 2 and J =

Now

(DE) J = ([0, 2] 3 . [0, 5] 2 ) u ([0, 7] 4 )

= ([0, 2]. [0, 5] (9 + 4 (mod 5)) [0, 7]

= [0, 10] 3 u [0, 7] 4= [0, 70] (9 + 8 (mod 5))

= [0, 70] 2 .

Consider

DEJ) = [0, 2] 3 ( [0, 5] 2 . ([0, 7] 4 )

= [0, 2] 3 ( [0, 5] [0, 7] (6 + 8 (mod 5= [0, 2] 3 ([0, 35] 4 )

= ([0, 2] .[0, 35) (9 + 8 (mod 5))

= [0, 70] 2 .

I and II are equal hence this set of DEJ is assgeneral DEJ) z (DEJ. For take D = [0, 7] 4 , E =

= [0, 10] 3 .

Consider

DEJ = ([0, 7] 4 . [0, 12] 2 ) ([0, 10] 3 )

= [0, 84] (12 + 4 (mod 5)) ([0, 10] 3 )

= [0, 84] 1 . [0, 10] 3

= [0, 840] (3 + 6 (mod 5))

Clearly I and II are not equal. Thus the operation

of two elements in the groupoid interval semiring i



non associative. We have shown for certain triple D(DEJ = DEJ but there exists triples a, b, c in SG w

a(bc). Thus SG is a non associative interval semiring

order.

Example 4.2.2: Let S = {[0, a] / a � R + � {0}} be

semiring. G = {Z8, * (3, 4)} be a groupoid. SG be th

interval semiring of the groupoid G over the interval

SG = {6 [0, ai] gi | gi � Z8 and ai � R + � {0}}. For D

+ [0, 7] 5 and E = [0, 2 ] 6 + [0, 5 ] 2 + [0, 3] 4

D + E = [0, 5] 3 + [0, 7] 5 + [0, 2 ] 6 + [0

[0, 3] 4 .

D E = ([0, 5] 3 + [0, 7] 5 ) ([0, 2 ] 6 + [0

[0, 3] 4 )

= ([0, 5] ([0, 2 ] 3 * 6 + [0, 5] [0, 5 ] 3 *

[0, 3] 3 * 4 + [0, 7] [0, 5 ] 5 * 2

= [0, 5 2 ] [9 + 24 (mod 8)) + [0, 5 5 ] (9 +

+ [0, 15] ( 9 + 16 (mod 8)) + ([0, 7 2 ] (15

8)) + ([0, 21] (15 + 16 (mod 8)) + ([0, 7(mod 8))

= [0, 5 2 ] 1 + [0, 5 5 ] 1 + [0, 15] 1 + [

Example 4.2.3: Let S = {[0, a] / a � Q+ � {0}+



semiring. Let G = {Z � {0}, *, (3, 5)} be a grouorder (for any a, b � Z+ � {0}, a * b = 3a + 5b;

Z+ � {0}).

Consider

SG = i i i i

i

[0,a ]g a Q {0} and g Z � � �® ¯ ¦

where i run over only finite index; that is all the

as finite formal sums. SG is the groupoid inter

the groupoid G over the interval semiring S.

It is easily verified SG is a non associative interinfinite order.

We study a few properties enjoyed by

interval semirings like ideals and subsemiring. W

these concepts the interested reader is requested t

We only give a few examples related with these

Example 4.2.4: Let S = {[0, a] | a � Z+ � {0}

semiring. P = {Z7, *, (3, 4)} be a groupoid

interval semiring SG is a non commutative gr

semiring.

Example 4.2.5: Let S = {[0, a] | a � Q+ � {0}

Example 4.2.6: Let S = {[0, a] / a � Q+ � {0}} be

semiring. G = {Z4, *, (3, 2)} be a groupoid. SG be th



interval semiring of the groupoid G over the interval Now P = {0, 2} � Z4, SP is a groupoid interv

and SP � SG so SP is a groupoid interval subsemi

SP is a right ideal and is not a left ideal of SG.

T = {1, 3} � Z4, ST is a groupoid interval semir

� SG so ST is a groupoid interval subsemiring of right ideal, both ST and SP are only right ideals of

not left ideal of SG.

Example 4.2.7: Let G = {Z12, *, (2, 10)} be a groupo

a] / a � Q+ � {0}} be an interval semiring. SG is th

interval semiring of the groupoid G over the interval

SP = {6 [0, a] gi / gi � P {0, 2, 4, 6, 8, 10} � Z1

{0}} � SG; SP is a groupoid interval subsemiring.

Now having seen examples of group

subsemiring and ideal of groupoid interval semirin

define S-groupoid interval semiring.

DEFINITION 4.2.2: Let G be any groupoid. S

semiring. If G is a S-groupoid then G has a subset P

P is a semigroup. SG is a groupoid interval sem

groupoid G over the interval semiring S. SP is a

interval semiring and SP is a proper subset of SG. W

S-groupoid interval semiring and G is a S-groupoid.

It is interesting and important to note th

interval semirings are not S-groupoid interval sem



Interested reader is expected to give exastructures. Now we get different classes of gr

semirings like P-groupoid interval semiring

alternative groupoid interval semiring, Mou

interval semiring, idempotent groupoid interval

on.

We call a groupoid interval semiring SG o

over an interval semiring S to be a P-groupoid in

if G is a P-groupoid likewise all other structure

only examples of them.

Example 4.2.9: Let S = {[0, a] / a � Z+

� {0}semiring. G = {Z12, *, (7, 7)} be a P-groupo

groupoid interval semiring is a P-groupoid interv

Example 4.2.10: Let S = {[0, a] / a � Q+ � {0}

semiring G = {Z6, *, (3, 3)} be a groupoid. C

alternative groupoid. SG be the groupoid inter

the groupoid G over the semiring S. SG is

groupoid interval semiring.

All groupoid interval semirings are not alter

interval semirings. For take G = {Z p, *, (3, 3)

groupoid. S = {[0, a] / a � Z+ � {0}} be an inSG is a groupoid interval semiring but is no

groupoid interval semiring as G is not an alternat

Example 4.2.12 : Let G = {Q+ � {0}, *, (4, 7)} be a

= {[0, a] / a � R + � {0}} be an interval semiri

id i l i i i i fi i



groupoid interval semiring is an infinite grouposemiring.

Example 4.2.13 : Let G = {R + � {0}, *, (3, 11)} be

S = {[0, a] / a � R + � {0}} be an interval semiring

groupoid interval semiring of infinite order.

Now having seen infinite groupoid interval se

now proceed onto give examples of interval groupoi

of an interval groupoid over a semiring.


However the reader is requested to referinformation about interval groupoid.

Example 4.2.14: Let G = {[0, a], *, (2, 3), a �interval groupoid. For any [0, a] and [0, b] in G we h

[0, b] = [0, 2a+3b (mod 11)] S = {Z+

� {0}} be a se= {6 x [0, a] where x � S and [0, a] � G denotes th

of all finite formal sums}; SG under addition and mu

is a non associative semiring known as the interv

semiring.

For D = 9 [0, 8] + 3 [0, 7] + 2 [0, 3] and E = 5 [0

3] + [0, 5] in SG define DE = 9 [0, 8] + 8 [0, 7] +

[0, 5] in SG.

DE = (9 [0 8] + 3 [0 7] + 2 [0 3]) u (5 [0 7]

(mod 11)] + 3 [0, 14 + 15 (mod 11)]

(mod 11)]

45 [0 4] + 15[0 2] + 10 [0 5] + 108 [0



= 45 [0, 4] + 15[0, 2] + 10 [0, 5] + 108 [03 [0, 7] + 36 [0, 1] + 24 [0, 4]

= 69 [0, 4] + 36 [0, 1] + 15 [0, 2] + 10 [0,

[0, 7] + 108 [0, 3] � SG.

Example 4.2.15 : Let G = {[0, a] / a � Z10, *, [0,

(1a + 5b) mod 10], (1, 5)} be an interval grou

interval groupoid. Take S = Q+ � {0} to be the

{finite formal sums of the form 6 a [0, x] / a �G} is an interval groupoid semiring of infinite o

3] + 2 [0, 7] and E = 3 [0, 7] + 4 [0, 9] + 2 [0, 5]

DE = 8 [0, 3] + 2 [0, 7] + 3 [0, 7] + 4 [0, 9

= 8 + 5 [0, 7] + 4 [0, 9] + 2 [0, 5] and

DE = (8 [0, 3] + 2 [0, 7]) (3 [0, 7] + 4 [0, 9

= 8.3 [0, 3] [0, 7] + 2.3 [0, 7] [0, 7] + 8.4

2.4 [0, 7] [0, 9] + 2.2 [0, 7] [0, 5] + 8.2 [= 24 [0, 3 + 35 (mod 10)] + 6 [0, 7 + 35

[0, 3 + 45 (mod 10)] + 8 [0, 7 + 45 (mo

+ 25 (mod 10] + 16 [0, 3+25 (mod 10)]

= 24 [0, 8] + 6 [0, 2] + 32 [0, 8] + 8 [0, 2]

[0, 8]

= 72 [0, 8] + 18 [0, 2].

We call SG the S-interval groupoid interval

Example 4.2.16: Let S = {Z+ � {0}} be a semiring.

a � Z12, *, (3, 9)} be an interval groupoid. G is a Sm

strong Mo fang inter al gro poid Consider SG t



strong Moufang interval groupoid. Consider SG tgroupoid semiring of the interval groupoid G over t

S. Clearly SG is a special Smarandache interval se

SG in this case is a Smarandache strong interva

groupoid semiring. Clearly SG is an infinite non c

non associative interval semiring.

It is pertinent at this stage to mention that by the

we can construct infinite non associative interva

which are also commutative as well as they ha

subsemirings which are associative. Now we procee

ways to construct groupoid interval semirings of finit

To this end we will be using mainly distributivsemirings. So we will be making use of these sem

construct finite interval groupoid semirings.

Example 4.2.17 : Let S = {be the chain lattice of ord

S = {0, a1, a2, …., a18, 1} {0 < a1 < a2 < … < a18

semiring of finite order. Take G = {[0, a], a � Z8, *

an interval groupoid of finite order.

Now SG = {a1 [0, 1 ] + a2 [0, 2 ] + … + a7 [0, 7 ]

| ai � S and [0, a] � G} be a interval groupoid sem

interval groupoid G over the semiring S.

Clearly SG is non commutative non associativ

finite order. SG is a special Smarandache groupoi

Infact SG has two sided ideal given by I = {6 ai [0,



SG is a non associative interval semiring with 0

additive identity and [0, 1] � S and e � L (e identity

[0 1] L � SL and S e = S � SL



[0, 1] L � SL and S.e = S � SL.

We will first illustrate this situation by some exam

we proceed onto discuss other related properties with


semiring. Let L be a loop given by the following tabl

‘o’ e a1 a2 a3 a4 a5

e e a1 a2 a3 a4 a5

a1 a1 e a3 a5 a2 a4

a2 a2 a5 e a4 a1 a3

a3 a3 a4 a1 e a5 a2

a4 a4 a3 a5 a2 e a1

a5 a5 a2 a4 a1 a3 e

SL = {¦ [0, x] ai / x � Z+

� {0}; ai � L, 1 < iloop interval semiring of the loop L over the interv

S. Clearly SL is of infinite order and SL is a non asso

commutative interval semiring.

Example 4.2.21: Let S = {[0, a] / a � Q+ � {0}} be

semiring. L = {e, a, b, c, d, g} be the loop giv

following table .

SL = {¦ [0, x] y / y � L and x � Q+ � {0

interval semiring of the loop L over the interv

Clearly SL is an infinite non associative interval



Clearly SL is an infinite non associative intervalcommutative interval semiring.

Example 4.2.22: Let S = {[0, a] / a � R + � {0}

semiring. Let L = {e, g1, g2, …, g7} be a loo

following table.

0 e g1 g2 g3 g4 g5 g6

e e g1 g2 g3 g4 g5 g6

g1 g1 e g5 g2 g6 g3 g7

g2 g2 g5 e g6 g3 g7 g4

g3 g3 g2 g6 e g7 g4 g1

g4 g4 g6 g3 g7 e g1 g5

g5 g5 g3 g7 g4 g1 e g2

g6 g6 g7 g4 g1 g5 g2 e

g7 g7 g4 g1 g5 g2 g6 g3

Consider

SL =7

i

i 0

[0,a]g

®¯¦ / a � R + � {0} and gi � L, 1 <

be the loop interval semiring of loop L ovsemiring S. Clealry SL is a non associative

infinite interval semiring.

‘o’ e a b c d

e e a b c d



e e a b c da a e c d b

b b d a e c

c c b d a e

d d c e b a

L is clearly non commutative. Consider SL = {¦� Q+ � {0} and gi � L} be the loop interval sem

loop L over the interval semiring S. Take P = {¦[0

Z+ � {0} and gi � L} � SL be a subset of SL. Cle

interval subsemiring of SL of infinite order.


semiring. L7(3) = {e, 1, 2, 3, 4, 5, 6, 7} be a loop g

following table.

‘o’ e 1 2 3 4 5 6 7e e 1 2 3 4 5 6 7

1 1 e 4 7 3 6 2 5

2 2 6 e 5 1 4 7 3

3 3 4 7 e 6 2 5 1

4 4 2 5 1 e 7 3 65 5 7 3 6 2 e 1 4

6 6 5 1 4 7 3 e 2

7 7 3 6 2 5 1 4

The product is performed using the table given fo

x.y = [0, 5] [0, 3] 2.6 + [0, 7] [0, 3] 3.6 + [0,[0 10] [0 3] 6 + [0 5] [0 4] 2 5 + [0



x.y [0, 5] [0, 3] 2.6 [0, 7] [0, 3] 3.6 [0, [0, 10] [0, 3] e.6 + [0, 5] [0, 4] 2.5 + [0

[0, 20] [0, 4] 5.5 + [0, 10] [0, 4] e.5

= [0, 15] 7 + [0, 21] 5 + [0, 60]1 + [0, 30]

[0, 28] 2 + [0, 80]e + [0, 40] 5

= [0, 15] 7 + [0, 61] 5 + [0, 60] 1 + [0, 30]

[0, 28] 2 + [0, 80]e.

Thus having seen how multiplication in SL7 (3

now we proceed onto define substructures.

Consider P = {¦ [0, x] ti / x � Z+ � {0} and

(3)} � SL7 (3); it is easily verified P is asubsemiring of SL7 (3).

Also P is a strict interval subsemiring. Thus

semifield contained in SL7 (3), so SL7 (3) is a Sm

interval semiring. SL7 (3) has atleast 7 d

semifields.

THEOREM 4.2.1: Let S = {[0, a] / a � Q+ � {0}

Z + � {0}} be an interval semiring. Ln (m) = {e,

set where n > 3, n odd and m is a positive integ

n) = 1 and (m-1, n) =1 with m < n.

Define on Ln (m) a binary operation * as foll1. e * i = i * e = i for all i � Ln (m)

2. i2 = i * i = e for all i � Ln (m)

from Q+ � {0} or Z+ � {0} or R + � {0} but of in

containing any one of these subsemifields.

T O 4 2 2 L t L ( +1/2) b l S {[0



THEOREM 4.2.2: Let Ln (n+1/2) be a loop. S = {[0,

� {0} or Q+ � {0} or R+ � {0}} be an interval se

loop interval semiring SLn (n+1/2) is a commutat

semiring which is an interval semifield of infinite ord

The proof is direct and follows from the fact

commutative when m = n+1/2 for more refer [14].

For the first time we proceed onto define int

using Zn (n > 3; n odd).

DEFINITION 4.2.4: Let G = {[0, a] | a � Ln(m); Ln(built using Z n , n > 3, n odd, with m < n (m, n) = 1 (

1} G under the operations of Ln(m) is an interval loop

For if [0, a], [0, b] � G then

[0, a] * [0, b] = [0, a * b]= [0, (mb – (m – 1) a) (

(if a z e, b z a and b z e); if a = b then [0, a] [0, a] [0, a]. [0, e] = [0, e] [0, a] = [0, a] for all [0, a] � G

It is left as an exercise for the reader to prove G is a

called an interval loop.

We will proceed onto give examples of them.

Example 4.1.25 : Let G = {[0, a] | a � L5(2)} G is

loop of order six given by the following table

We see any [0, 3] * [0, 4] = [0, 5] got from the ta

We can have infinite number of interval lo

in this way where n > 3 and n odd is the only crinterval loops are of finite order and their order i



y yinterval loops are of finite order and their order i

Further the reader is given the task of provin

theorem.

THEOREM 4.2.3: Let

G = {[0, a] | a � Ln§ ·

¨ ¸© ¹

n 1

2 }

be an interval loop, G is a commutative interval

For more information about loops please refer [1

Example 4.1.26: Let G = {[0, a] | a � L7(4),

L7(4)} be an interval loop. It can be easily veri

commutative loop of order 8.

We see several nice properties enjoyed by

loops. These interval loops constructed by thMoufang interval loops. They are also not inter

interval Bruck loop. However we have interv

satisfy the weak inverse property. That is we hav

loops. The following theorem the proof of w

gurantees the existence of WIP interval loops.

THEOREM 4.2.4: Let (G, *) = {[0, a] | a � L

interval loop Ln(m) is a weak inverse property l

However these class of interval loops conta

alternative interval loop and right alternative interva

is evident from the following theorem, the proof of was an exercise to the reader For more information ref



g pas an exercise to the reader. For more information ref

THEOREM 4.2.5: Let G = {[0, a] / a � (Ln (2),

interval loop. G is a left alternative interval loop.

THEOREM 4.2.6: Let G = {[0, a] / a � (Ln (n – 1)

interval loop. G is a right alternative interval loop.

THEOREM 4.2.7: No interval loop G constructed usi

alternative.

Hint: Both the left alternative interval loop G o

alternative interval loop G are built using Ln(2) an

respectively and both of them are non commutative.

We can define associator of an interval loop ex

case of the loops. The following theorem is also

hence is left as an exercise for the reader.

THEOREM 4.2.8: Let G = {[0, a]| a � (Ln(m), *)} be

loop built using Ln (m). The associator A (G) = G.

The proof directly follows from the fact A(Ln(m) = these interval loops built using Ln (m) are G-interval

We can define Smarandache Lagrange inte

Lagrange interval loop) as in case of general lo

the notion of Smarandache weakly Lagrange in be defined [14]. This task is left as an exercise to



We will however give examples of the

connected with them.

Example 4.2.28: Let G = {[0, a] | a � L15 (2)}loop. G is a S-Lagrange interval loop as G ha

subgroups of order 2 and 4.

We give the following theorems and the proof

direct and the reader is expected to prove.

THEOREM 4.2.11: Every S Lagrange interval loo Lagrange interval loop.

THEOREM 4.2.12: Let G = {[0, a] | a � Ln (m prime} be an interval loop. Then G is a S-Laloop for all m such that (m-1, n) = 1 and (n, m) =

THEOREM 4.2.13: Let G = {[0, a] | a � Ln (m)}loop. Every interval loop G is a S-weakly Lagran

We say an interval loop L of finite order is

pseudo Lagrange interval loop if order of

subloop divides order of L. If the order of atleas

subloop of L divides the order of the interval call L to be a S-weakly pseudo Lagrange interval

We can as in the case of finite loops define

G = {[0, a] | a � Ln (m)} gives an interval loop< m < n, as n is given to be a prime. Now every inconstructed in this manner are Smarandache stroloop as every element in G is of order two and 2/p+1



n is a prime.

We will illustrate this by examples.

Example 4.2.29 : Let G = {[0, a] | a � L11 (m); 1 <

an interval loop. We can construct 9 distinct interv

varying m, m = 2, 3, …, 10. It is easily verified ever

in G is such that x2

= [0,e] hence G is a S-stroninterval loop.

Infact we have infinite class of S-strong 2-Syl

loops using Ln (n) where n is a prime; as the numbe

is infinite. We can define as in case of S loops fo

loops the notion of Smarandache interval loop hom[14].

Now with these concepts of interval loops n

define the new notion of interval loop semirings a

loop interval semirings.

DEFINITION 4.2.5: Let G = {[0, a] | a � Ln (m)} be

loop. S = Z + � {0} be a semiring.

SG = ®¯¦ i i

i

a [0,x ] i ,

runs over finite formal sums and ai � S, xi � Ln

interval semiring under usual addition and mu

SL = i i i 13 i

i

a [0,x ] x L (5) and a� �®

¯ ¦



is of infinite order. Clearly this interval se

associative and non commutative.

Example 4.2.32: Let G = {[0, a] | a � L15 (8)}

loop and S = R + � {0} be a semiring.

SG = i i i i 15

i

a [0,x ] a S and x L� �® ¯

¦

be the interval loop semiring of the intervasemiring S. SG is of infinite order. SG is a comm

loop semiring, which is a non associative interva

Example 4.2.33 : Let G = {[0, a] | a � L17 (8)}

loop and S = Z+ � {0} be a semiring.

SG = i i i i

i

a [0,x ] a S and x G½

� �® ¯ ¦

is the interval loop semiring of the interval lo

semiring S. SG is non commutative and is of1.[0, e] is the identity element of SG.

We can construct 15 such interval loop sem

i t l l L (2) L (3) L (4) L

P = i i i 9

i

a [0,x ] a Z {0} and x L (7) ½� � �® ¾

¯ ¿¦

P is a loop interval subsemiring of the interval loo



P is a loop interval subsemiring of the interval loo

SG. Clearly P is not an ideal of SG.

Consider

T = i i i

ia [0,x ] x 8,a Q {0}

½

� �® ¾¯ ¿¦ � SG

T is an interval group subsemiring which is also a sem

Example 4.2.35 : Let G = {[0, a] | a � L19 (10)} b

loop and S = {Z+

� {0}} be a semiring;

SG = i i i 19

i

a [0,x ] a Z {0} and x L (1� � �® ¯ ¦

be the interval loop semiring of the interval loop semiring S.

Clearly SG is a commutative interval semiring. C

P = i i 19x [0,a] x 5Z {0} and a L (10) ½� � �® ¾

¯ ¿¦

P is an interval loop subsemiring of SG as well as

SG

P = i i i i 19

i

a [0,x ] a 5Z {0} and x L (� � �®

¯

¦



P is an interval loop subsemiring of SG as wel

SG.

Example 4.2.37 : Let G = {[0, a] | a � L23 (12)

loop. S = Z+ � {0} be the semiring.

SG = i i i i 23x [0,a ] x S and a L (� �® ¯ ¦

be the interval loop semiring of the interval lsemiring S.

Consider

P = i i i i 23x [0,a ] x 9Z {0}, a L (12� � �® ¯ ¦

P is an ideal of SG for SG is a commutative inter

We can as in case of loops build in case o

also the notion of first normalizer and second nor

First normalizer of G = {[0, a] | a � Ln (m)}

N1 (H1 (t)) = {[0, a] � Ln (m)} | [0, a] Hi (t)

and second normalizer of G is N2(Hi (t)) = {[0, xx] (Hi (t)) [0, x] = Hi (t)}.

W ill hi i i b l

H1(15). So the two normalizers of the interval sublo

always equal. Now using interval subloops of G a

construct in interval loop subsemirings and ideals. T

left as an exercise to the reader.



Example 4.2.40: Let G = {[0,a] / a �L21 (20)} be S

a1 < a2 < … < a7 < 1} be a chain lattice which is a sem

SG = i i i i 21x [0,a ] a S, x L (20)½

� �® ¾¯ ¿¦

be the interval loop semiring. Clearly SG is of finit

has non trivial idempotents but no zero divisors.

Example 4.2.41 : Let G = {[0, a] | a � L25 (8)} beloop.

S = {

a.b = 0, a+b = 1, a.0 = 0 = b0, 1+a = 1+b = 1, 1.a = a be a semiring.

SG = i i i i 25x [0,a ] a S, x L (8)½

� �® ¾¯ ¿¦

be an interval loop semiring. SG has zero divisors ide

For take x = a [0, 9] and y = b [0, 12] ;

b ([0 9] [0 12])

a

1

b

0



THEOREM 4.2.15: Let G = {[0, a] | a � § ·

¨ ¸© ¹

p

p 1 L

2

be the interval loop. S = {C n , 0 < a1 < … < an-2

semiring. SG the interval loop semiring is a semifield



The proof is left as an exercise for the reader.

Example 4.2.46: Let G = {[0, a] | a � Ln (m)} be

loop m z n+1/2. S = {Cn; 0 < a1 < … < an-1 < 1} be thSG be the interval loop semiring. SG is a division se

is SG has no zero divisors and is strict.


THEOREM 4.2.16: Let G = {[0, a] | a � Ln (m); m zan interval loop. S = {C d ; 0 < a1 < … < ad-1 < 1} beThe interval loop semiring SG is an interval division

The proof is direct and is left as an exercise for t

prove.

It is important to mention here that all interva{[0, a] | a � Ln (m)} are S-simple. Hence to find ide

are not very easy that too when the semiring is Q+ �� {0}. Now we proceed on to define interval subse

ideals of finite interval semirings.

Example 4.2.47: Let G = {[0, a] | a � L17 (15)} beloop. S = {C5, 0 < a1 < a2 < a3 < 1} be the semir

interval loop semiring of the interval loop G over th

P is an interval loop subsemiring of SG but clear

ideal of SG.

Example 4.2.48: Let G = {[0, a] | a � L21 (11)} bS = {0 < a1 < a2 < a3 < a4 < a5 < a6 < 1} be the s



S {0 < a1 < a2 < a3 < a4 < a5 < a6 < 1} be the s

interval loop semiring. SG has several

subsemirings.

Now we will proceed onto define interv

semiring.

DEFINITION 4.2.17: Let G = {[0, a] | a � Ln(m)

loop. S = {[0, x] | x � Z + � {0} or Q+ � {0} or Rinterval semiring. SG the interval loop interval s

SG = �®¯¦ [0,x ][0,a ] [0,x ] S

and [0, a] � G where the sums are finite foraddition and product respecting S and G} is a se

For instance if x = [0, a1 ] [0, g 1 ] + [0, a2 ] [

[0, g 3 ] and y = [0, b1 ] [0, h1 ] + [0, b2 ] [0, h2 ]) ar

x.y = ([0, a1 ] [0, g 1 ] + [0, a2 ] [0, g 2 ] +([0, b1 ] [0, h1 ] + [0, b2 ] [0, h2 ])

= [0, a1 ] [0, b1 ] [0, g 1 ] [0, h1 ] + [0, a1 ] [0h2 ] + [0, a2 ] [0, b1 ] [0, g 1 ] [0, h1 ] + …

[0, g 3 ] [0, h2 ]= [0, a1b1 ] [0, b1h1 ] + [0, a1b2 ] [0, g 1 h2 ] +

[0, g 3 h2 ].

Example 4.2.50 : Let G = {[0, a] | a � L11(3)} be

loop and S = {[0, x] | x � Q+ � {0}} be an interva

SG be the interval loop interval semiring.

Example 4 2 51: Let G = {[0 a] | a � L13(8)} and S



Example 4.2.51: Let G {[0, a] | a � L13(8)} and S

� R + � {0}} be the interval loop and interva

respectively. SG is the interval loop interval semiring

We will now give examples of substructure

interval loop interval semirings which are of infinite

Example 4.2.52: Let G = {[0, a] | a � L19(7)} and S

� Z+ � {0}} be interval loop and interval semiring r

SG = {6 [0, x] [0, a] | x � Z+ � {0} and a � L19

interval loop interval semiring of the interval loop

interval semiring S. Take P = {6 [0, x] [0, a] | x �and a � L19 (7)} � SG; P is an interval lo

subsemiring; infact P is also an ideal of SG.

Example 4.2.53: Let G = {[0, a] | a � L27 (13)} be

loop and S = {[0, x] | x � Q+ � {0}} be the interva

SG = {finite formal sums of the form 6 [0, x] [0, (13) and x � Q+ � {0}} be the interval loop interv

Let H = {6 [0, x] [0, a] | a � L27 (13) and x � 4Z+ � be the interval loop interval subsemiring of SG. Clea

an interval loop interval ideal of SG.

Example 4.2.54: Let G = {[0, a] | a � L5 (4) = ^ e , 1 ,

{e, x0, x1, x2, …, x5} that is xn = n ; 0 d n d 5 }} be

l S {[0 ] | Q+ {0}} b h i l i

Now we proceed onto illustrate interval

semiring of finite order; the definition is left as

the reader.

Example 4.2.55: Let G = [0, a] | a � L7 (4)} be a



p [ , ] | 7 ( )}

S = {[0, x] | x � Z12} be the interval semiring. SG

a] | a � G and x � Z12} be the interval loop in

Clearly SG is of finite order. SG has zero divisor

and ideals.

Consider P = {6 [0, x] [0, a] | x � {0, 2, 4, 6

� L7 (4)} � SG; P is an interval loop interval se

also an ideal.

Take x = [0, 2] [0, 3] + [0, 6] [0, 5] + [0, 4]

[0, 6] [0, 2] in SG.

x. y = ([0, 2] [0, 3] + [0, 6] [0, 5] + [0, 4] [0, 6]

= [0, 2] [0, 6] [0, 3] [0, 2] + [0, 6] [0, 6] [0

4] [0, 6] [0, 6] [0, 2]

= 0. (0, 3.2]) + 0 ([0, 5.2]) + 0 ([0, 6.2])= 0.

Thus SG has non trivial zero divisors. Furt

strict interval loop interval semiring.

For [0, 3] [0, 5] + [0, 9] [0, 6]= 0 but none ois zero. [0, 1] [0, e] + [0, 1] [0, 1] + [0, 1] [0, 2] +

7] = x � SG is such that x2 = 0.

Example 4.2.56: Let S = {[0, a] | a � Z30}

semiring. G = {[0, x] | x � L5 (2)} be an interv

i t l l i t l i i i f fi it d

Example 4.2.57: Let G = {[0, a], *, (3, 4), a �interval groupoid. S = {[0, x] | x � Z+ � {0}} besemiring. SG is the interval groupoid interval s

infinite order. This has subsemirings given by P = {

g] where x � 3Z+ � {0} and [0, g] � G} � SG.



g] where x � 3Z � {0} and [0, g] � G} � SG.

Example 4.2.58: Let G = {[0, a], *, (3, 11), a �interval groupoid. Let S = {[0, a] | a � Q+ � {0}} be

semiring. SG the interval groupoid interval seminfinite order. P = {6 [0, x] [0, g] | x � Z+ � {0} and

SG is an interval groupoid interval subsemiring. Ho

not an ideal of SG.

Example 4.2.59: Let G = {[0, a], *, (8, 12), a �

interval groupoid. S = {[0, x] | x � Z+

� {0}} besemiring. SG the interval groupoid interval semiring

x] [0, a] | x � 3Z+ � {0} and a � Z35} � SG is an i

Infact SG has infinite number of ideals.

Now having seen examples of interval group

semirings of infinite order we now proceed onto givof interval groupoid interval semirings of finite order

Example 4.2.60: Let G = {[0, a], *, (13, 7), Z20} be

groupoid. S = {[0, x] | x � Z10} be an interval sem

the interval groupoid interval semiring of finite orde

P = {6[0, x] [0, g] | x � {0, 2, 4, 5, 6, 8} � Z10 and SG, P is not only a subsemiring but also an ideal of S

Example 4.2.62: Let G = {[0, g] | g � Z32}

semiring under multiplication modulo 32.

Let S = {[0, a] | a � Z+ � {0}} be an interv

= {6 [0, a] [0, g] | a � Z+ � {0} and g � Z32}



{ [ , ] [ , g] | { } g 32}semigroup interval semiring.

SG is of infinite order. SG has zero divisors.

P = {6 [0, a] [0, g] | a � 3Z+ � {0} and g �20, 24, 28} � Z

32} � SG is a interval sem

subsemiring of SG. It is easily verified P is an id

Example 4.2.63: Let G = {[0, g] | g � Z45}

semigroup. S = {[0, a] | a � Q+ � {0}} be an in

SG be the interval semigroup interval semiring. P

g] | g � Z45, a � Z+

� {0}} � SG; is an inteinterval subsemiring of SG, but P is not an

However SG has zero divisors.

Example 4.2.64: Let G = {[0, g] | g � Q+ � {0}

semigroup. S = {[0, x] | x � Z+ � {0}} be an in

SG be the interval semigroup interval semiring. order. Take P = {6 [0, x] [0, g] | x � 3Z+ �{0}

{0}} � SG, P is an interval semigroup interv

infact an ideal of SG.

If we take T = {6 [0, x] [0, h] | x � 5Z+ � {

� {0}} � SG, P is only a interval sem

subsemiring and not an ideal of SG.

Example 4.2.65: Let G = {[0, g] | g � Z40}

{6 [0, a] [0, g] | a � {0, 10, 20, 30, …, 110} � Z120,

� SG, P is an interval semigroup interval subsemalso an ideal of SG. This interval semigroup interv

has non trivial zero divisors and it is not a strict semi Now we proceed onto give examples of int



interval semirings.

Example 4.2.67: Let G = {[0, g] | g � Z7 \ {0}} be

group under multiplication modulo 7. S = {[0, x]

{0}} be an interval semiring. SG = {6 [0, x] [0, g]

{0}, g � Z7 \ {0}} is the interval group interva

Clearly SG is of infinite order.

Example 4.2.68: Let G = {[0, x] | x � Z23 \ {0}} be

group. S = {[0, a] | a � Z

+

� {0}} be the interval seis the interval group interval semiring. SG is of inf

SG is a strict semiring. P = {6 [0, a] [0, x] | a � 3Z+

Z23 \ {0}} � SG is the interval group interval s

which is an ideal of SG.

Example 4.2.69: Let G = {[0, g] | g � Z23 \ {0}} begroup. S = {[0, a] | a � R + � {0} } be the interval se

= {6[0, a] [0, g] | a � R + � {0} and g � Z23 \

interval group interval semiring. SG is commutativ

strict interval group interval semiring.

SG has no zero divisors. SG has idempotents.

Consider P = {6 [0, a] [0, g] | a � Q+ � {0} an{0}} � SG, P is an interval subsemiring and is not

SG. Infact SG has infinite number of subsemirings.

THEOREM 4.2.17: Let G = {[0, g] | g � Z p \ {0}

an interval group. S = {[0, x] | x � Z + � {0} or

� {0}} be the interval semiring. SG be the

interval semiring.The following conditions are satisfied by SG.



1. SG is commutative, strict infinite interval sem2. SG is a S-semiring.

3. SG has no zero divisors.

4. SG has idempotents if and only if S is built o

R+ � {0}.

5. SG has infinite number of ideals if S is built o

6. SG has only one ideal if S is built over R+

{0}.

The proof is direct and hence is left as an ereader.

Example 4.2.70: Let G = {[0, a] | a � Q+} be an

under multiplication. S = {[0, x] | x � Z+ � {0

semiring.

SG be the interval group interval semiring. Sorder and commutative. SG has ideals given by

a] | x � 17Z+ � {0} and a � Q+} � SG; P is easian ideal of SG. Infact SG has infinite numbe

subsemirings in it.

Example 4.2.71: Let G = {[0, a] | a � R +

} be anS = {[0, x] | x � R + � {0}} be an interval semi

interval group interval semiring.

We will now see some examples of finite int

interval semirings.

Example 4.2.72: Let G = {[0, a] | a � Z13 \ {0}} begroup. T = {[0, x] | x � Z20} be an interval semirin



[0, x] [0, a] | a � Z13 \ {0} and x � Z20} be the intinterval semiring. TG is of finite order. TG has ze

TG is not a strict semiring but TG is commutative.

THEOREM 4.2.18: Let G = {[0, a] | a � Z p \ {0} and

be a finite interval group. S = {[0, x] | x � Z n } be

semiring. SG = { 6 [0, x] [0, a] | x � Z n and a � Z p \interval group interval semiring. The following are tr

1. SG is a commutative finite semiring.2. SG is not strict.3. SG has ideals provided n is not a prime.

The proof is direct and hence is left as an exercise to

We can construct interval semirings which

commutative.

Example 4.2.73: Let G = {SX the interval symm

where X = {[0, a1], [0, a2], [0, a3]} and S = {[0, x]

{0}} be an interval semiring.

SG is the interval group interval semiring of in

and SG is a non commutative semiring.

Thus using the interval symmetric group SX, whea1], …, [0, an] } we can get the class of non cointerval semirings of infinite order. It is pertinent

h th t S(¢X²) ill i th i l i t l i



Chapter Five

INTERVAL NEUTROSOPHIC SEMI

In this chapter we define different types of neutr

semirings. We use neutrosophic intervals in thintervals. The two types of neutrosophic intervalbe using are pure neutrosophic intervals an

DEFINITION 5.1: Let S = {[0, aI] | a � Z n or Z + � {

{0} or R+ � {0}; S under interval addition and multan interval semiring known as the pure neutrosop

semiring.

W ill i l f th



We will give examples of them.

Example 5.1: Let S = {[0, aI] | a � Z9, I2

= I}. S is a

neutrosophic interval semiring.

Example 5.2: Let S = {[0, aI] | a � Z+I � {0}} be pure neutrosophic interval semiring of infinite order

interval semirings are commutative.

If in the definition 5.1 we replace the interval [0

a+bI] ; a, b � Zn or Z+ � {0} or Q

+ � {0} or R + �

neutrosophic interval semiring and is not a pure n

interval semiring.

We will illustrate this situation also by some exampl

Example 5.3: Let S = {[0, a+bI] | a, b � R + � {

neutrosophic interval semiring. Now P = {[0, bI] |

{0}} � S as a neutrosophic interval subsemirin

clearly pure neutrosophic.

Example 5.4: Let S = {[0, a+bI] | a, b � Z20} be a n

interval semiring. Clearly S is of finite order; wneutrosophic interval semiring given in example

infinite order. Both of them are commutative and bo

b i i hi h i t hi i t l

Example 5.6: Let S = {[0, aI] | a � Z7} be the pu

interval semiring, S has no subsemirings. Hence


THEOREM 5 1: Let S = {[0 aI] | aI � Z ; p a p



THEOREM 5.1: Let S = {[0, aI] | aI � Z p; p a pneutrosophic interval semiring. S has no subsemhas no proper ideals.

The proof is straight forward hence left as an reader.

THEOREM 5.2: Let S = {[0, aI] | aI � Q+ I � neutrosophic interval semiring. S has subsemirideals.

Proof is left as an exercise for the reader.

Example 5.7: Let S = {[0, a + bI] | a, b � Q

neutrosophic interval semiring. P = {[0, bI] | b �is a pure neutrosophic interval subsemiring of S

a] | a � Q+ � {0}} � S is an interval subsemirin

neutrosophic interval subsemiring of S.This interval subsemiring will be de

subsemiring.

We will give more examples.

Example 5.8: Let S = {[0, a + bI] | a, b � Zneutrosophic interval semiring S has both pur

THEOREM 5.4: Let S = {[0, a + bI] | a,b � Z n or Z

Q+ � {0} or R+ � {0}} be an interval neutrosophic has both pure neutrosophic interval subsemiring a

neutrosophic interval subsemiring.

THEOREM 5 5: Let S = {[0 a+bI] | a b � Z+ �



THEOREM 5.5: Let S = {[0, a+bI] | a,b � Z �neutrosophic interval semiring. S has ideals.

THEOREM 5.6: Let S = {[0, D I] | D � Z + � {0}}

neutrosophic interval semiring. S has ideals.

Now we proceed onto define and give examples of

of neutrosophic semirings.

DEFINITION 5.2: Let

S =

f

�®¯¦ i

n

i 0

[0,aI ]x a Z

or Z + � {0} or Q+ � {0} or R+ � {0}}; S under addition and multiplication is an interval semiringthe pure neutrosophic interval polynomial semiring.

We will illustrate this by one or two examples.

Example 5.9: Let

S =i

12

i 0

[0,aI]x a Zf

½�® ¾

¯ ¿¦

be an infinite pure neutrosophic polynomial interval

be an infinite pure neutrosophic polynomial inter

Example 5.12: Let

S =5

5 6

12

i 0

[0,aI]x x 1, a Z

½ �® ¾¯ ¿¦



be a pure neutrosophic polynomial interval se

order.

Example 5.13: Let

S =i

i 0

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

½ � �®

¯ ¦

be a neutrosophic interval polynomial semiring o

Example 5.14: Let

S =i

i 0

[0,a bI]x a,b R {0}f

½ � �®

¯ ¦

be the neutrosophic interval polynomial sem

order.

Example 5.15: Let

S =i

16

i 0

[0,a bI]x a,b Zf

½ �® ¾

¯ ¿¦

be a neutrosophic interval polynomial semiring o

Example 5.16: Let

Si[0 a bI]x a b Z {0}

f ½

� �®¦



P =i

1

i 0

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

½� �®

¯ ¦

P is both a subsemiring as well as ideal of S.Take

2i[0 I] {0 3 6 9} Zf ½

® ¾¦



T =2i

12

i 0

[0,aI]x a {0,3,6,9} Z

½� �® ¾

¯ ¿¦

T is only a subsemiring and not an ideal of S.

Thus S has many subsemirings which are no

Example 5.21: Let

E =

9i 10

70

i 0

[0,aI]x x 1,a Z

½ �® ¾

¯ ¿¦

be a pure neutrosophic interval polynomial semirConsider

F =

9i

7

i 0

[0, aI]x a {0,10,20,..., 60} Z

½� �®

¯ ¦

F is a subsemiring as well as an ideal of E.

Having defined substructures in pure neutro

polynomial semirings we now proceed onto gi

neutrosophic interval polynomial semiring

substructures.

Example 5.22: Let

S =i

i 0

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

½ � �®

¯ ¦be a neutrosophic polynomial interval semiring

is a pure neutrosophic interval polynomial subsem

Clearly W is also an ideal of S, however T is not an

Infact S has several subrings and ideals.

Example 5.23: Let9

i 10 ½® ¾¦



S =i 10

42

i 0

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

½ � ® ¾

¯ ¿¦

be a neutrosophic interval polynomial semiring.

P =9

i 10

42

i 0

[0,bI]x b Z ;x 1

½� ® ¾¯ ¿¦ � S

is a pure neutrosophic interval polynomial subsemir

as ideal of S.

T =

9i 10

42i 0 [0,a bI]x a,b Z ;x 1

½

� ® ¾¯ ¿¦ �is only pseudo neutrosophic interval polynomial subS and is not an ideal of S.

W =9

i 10

i 0

[0,a bI]x x 1,a,b {0,7,14,21,28,3

� ® ¯ ¦

� S is both a subsemiring as well as an ideal of S.

Now we can as in case of interval matrix semiring

notion of neutrosophic interval matrix semirings. H

only examples of them and their substructures.

Example 5.24: Let S = {([0, a1I], …, [0, a9I]) where

d i d 9} be a pure neutrosophic row interval matrix

{([0 I] [0 bI] 0 0 0 0 0 [0 I] [0 dI]) | b

interval matrix semiring. S has T = {([0, a1], [0

a4]} | ai � Z7, 1 d i d 4} � S to be a pseudo ne

interval matrix subsemiring. W = {([0, b1I], [0, b

b4I]) | bi � Z7, 1 d i d 4} � S is a pure neutrosopinterval subsemiring.



In view of this we have the following theorem

which are direct.

THEOREM 5.7: Let S = {([0, a1 + b1 I], [0, a2 + bbn I]) | ai , bi � Z p , p a prime, 1 d i d n} be a nematrix interval semiring. S has pseudo neuinterval matrix subsemiring, pure neutrosophmatrix semiring and neutrosophic row m

subsemiring.

THEOREM 5.8: Let S = {([0, a1 I], [0, a2 I], …, [0

p a prime, 1 d i d n} be a pure neutrosophic semiring. S has both pure neutrosophic intervand subsemirings.

THEOREM 5.9: Let S = {([0, aI], [0, aI], …, [0,

a prime} be a pure neutrosophic interval matrix no subsemirings.

COROLLARY 5.1: S given in theorem 5.9 has no

Example 5.27: Let S = {([0, aI], [0, aI], [0, aI],

[0, aI]) | a � Z11} be a pure neutrosophic semiring S has no subsemirings and ideals

Consider

A = 13

[0,aI] [0,aI] [0,aI]

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

[0,aI] [0,aI] [0,aI]

½ª º° °« » �® ¾« »° °« »¬ ¼¯ ¿

�

is a pure neutrosophic 3 u 3 interval matrix subsemi



is a pure neutrosophic 3 3 interval matrix subsemi

as ideal of S.

Take W = {All 3 u 3 pure neutrosophic uppe

interval matrices with intervals of the form [0, aI] |

S, is only a subsemiring and not an ideal of S.

Example 5.29: Let S = {5 u 5 pure neutrosoph

matrices with intervals of the form [0, aI] where a � be a pure neutrosophic square matrix interval sem

subsemirings and ideals given by P = {5 u 5 pure nintervals matrices with intervals of the form [0, aI]

7Z+ � {0}} � S; P is an ideal of S.

Example 5.30: Let

S =

[0, a] [0,b]

[0,c] [0,d]

ª º°

®« »°¬ ¼¯ |

a = a1 + a2I, b = b1 + b2I, c = c1 + c2I and d = d1 + d2I

� Z12, 1 d i d 2} be a neutrosophic interval matri

Clearly

P =

1 1

1 1

[0,a ] [0,b ]

[0,c ] [0,d ]

ª º°

®« »°¬ ¼¯ | a1, b1, c1, d1 � Z12} �is a pseudo neutrosophic interval matrix subsemiring



over the interval neutrosophic semiring S. Clearly

associative and is of finite order.

Example 5.34: Let L = L9 (7) be a loop. S = {[0, aI]{0}} be a pure neutrosophic interval semiring. SL

interval neutrosophic semiring of the loop L ove

t hi i t l i i S SL i



neutrosophic interval semiring S. SL is non

neutrosophic interval semiring of infinite order.

Example 5.35: Let L = L13 (7) be a loop of order 14+ bI] | a, b � Z3} be a neutrosophic interval semiring

loop neutrosophic interval semiring. SL is non asso

commutative, SL has non associative pseudo n

interval subsemiring of finite order given by

P = i 3 i

g[0,a]g a Z and g L

½° °� �® ¾° °¯ ¿¦ � SL

and pure neutrosophic interval subsemiring given by

T = i 3 i

i

[0,aI]g a Z and g L½

� �® ¾

¯ ¿

¦ .

Further

V = 1 i 3 i[0 ,a aI]g a Z and g L½

� �® ¾¯ ¿¦ �

is also a neutrosophic interval subsemiring of SL

commutative.

Example 5.36: Let G = S6 be the symmetric group o+

Take

T = 6 6

h

[0,aI]h a Z {0}; h A S ½� � �®

¯

¦

T is an ideal of SG.

Example 5.37: Let G = <g | g5 = 1 > be the



Example 5.37: Let G <g | g 1 > be the

degree 5. S = {[0, a + bI] | a, b � Z10} be a neutrsemiring. SG the group neutrosophic interval sem

finite order and commutative. SG has both pseudinterval subsemiring and pure neutroso

subsemiring.

THEOREM 5.10: Let S n be the symmetric group o

{[0, a+bI] | a, b � Q+ � {0}} be the neutro

semiring. SS n is the symmetric group neutro semiring. SS n has ideals.

Proof: We only give an hint to the proof of the

Sn; An is the normal subgroup of Sn SAn � SSn

neutrosophic interval subsemiring of SSn and is a

the normal subgroup of Sn.It is interesting to note that in the above th

pseudo neutrosophic interval subsemirneutrosophic interval subsemirings also.

Now having seen groupoid neutrosophic int

loop neutrosophic interval semirings and grouinterval semirings we now proceed onto

semigroup neutrosophic interval semirings.

and idempotents. Further SP is a commutative sem

of finite order.

Example 5.39: Let P = {Z25, under multiplication m be a semigroup of order 25. S = {[0, a+bI] | a, b � Q

an interval neutrosophic semiring. SP the

neutrosophic interval semiring is of infinite



neutrosophic interval semiring is of infinite

subsemirings ideals and pseudo interval n

subsemirings and pure neutrosophic interval subsemi

Example 5.40: Let S(9) be the symmetric semigrou

aI] | a � Q+ � {0}} be the pure neutrosophic interv

SS (9) is the semigroup pure neutrosophic interval

infinite order and is non commutative SS(9) has

subsemirings and ideals in it.

Example 5.41: Let S (3) be the symmetric semigrou

aI] | a � Z14} be the pure interval neutrosophic sem

the semigroup interval neutrosophic semiring is of

non commutative and has ideals and subsemirings

also zero divisors.

Example 5.42: Let S (5) be the symmetric semigroup

+ bI] | a, b � R + � {0}} be the neutrosophic interv

SS(5) is the semigroup neutrosophic interval semirin

order non commutative but has subsemirings.

Now having seen these new structures we ca

neutrosophic interval semirings using neutrosophsemigroups, groupoids, groups and loops.

Example 5.44: Let G = {[0, aI] | a � Q+}

neutrosophic group under multiplication. S = {R

semiring. SG the group neutrosophic interval

infinite order.

Example 5.45: Let P = {[0, a+bI] | a, b � Z+

interval neutrosophic semigroup of infinite



interval neutrosophic semigroup of infinite

multiplication. S = {Q+ � {0}} be the semi

semigroup interval neutrosophic semiring o

neutrosophic semigroup P over the semiring S.

Example 5.46: Let S = {[0, aI] | a � Z12}

neutrosophic semigroup of finite order under mu

{ Z+ � {0}} be the semiring. FS is the inte

neutrosophic semiring. FS is commutative an

order has ideals and subsemirings.

Example 5.47: Let G = {[0, aI] | a � Z12I, *

interval neutrosophic groupoid of finite order. S

be the semiring. SG is the groupoid interv

semiring of the neutrosophic interval grouposemiring S. Clearly SG is a non commut

associative interval semiring of infinite order.

Example 5.48: Let S = {[0, aI] | a � Z+ � {0}, *

interval neutrosophic groupoid of infinite order.

be the semiring. FS is the neutrosophic insemiring of infinite order which is non associati

finite formal sums of the form

¦

Example 5.50: Let L = {LnI (m)} = {The interval lo

neutrosophic having interval elements [0, eI], [0, I],

[0, nI], n-odd n > 3, with operation * such that [0, aI

mbI + (m – 1) aI (mod n) where m d n, (m, n) = 1 aI,(m – 1, n) = 1} be the interval neutrosophic loop of

S = {Z+ � {0}} be the semiring. SL = {6 si [0, aI]

formal sums where s � S and [0 aI] � L} is the lo



formal sums where si � S and [0, aI] � L} is the lo

neutrosophic semiring of infinite order, non commut

n+1/2 and non associative.

We can construct any number of such non associativ

using this class of loops.

Now having seen several types of interval se

reader is left with the task of defining properties

related with them.

Now we proceed onto give the applications ostructures.



Chapter Six

APPLICATION OF INTERVAL SEM

These new structures will find applications in

analysis. Since these structures do not in g

negative numbers one can use them in real valu

solutions. Since both associative and non assosemirings are defined; in situations were the ope

associatives these structures can be used.



Chapter Seven



SUGGESTED PROBLEMS

In this chapter we suggest around 118 proble

make the interested reader get better grasp of the

1. Obtain some interesting properties enjoy

semirings built using Z+ � {0}.

2. Is every interval semiring a S-interval semiri

3. Give an example of an interval semiring wh

interval semiring.

4 Does there exist an interval semiring fo

7. What are the special properties enjoyed by t

semiring constructed using Q+ � {0}?

8. Determine a method of solving interval equations built using Q+ � {0}.

9. Let S = {[0, a] | a � 3Z+ � {0}} be an interval se



a. Does S have ideals and interval subsemiring

b. Does S have interval subsemirings which are

c. Can S be a S-semiring?

Justify your claims.

10. Let P = {[0, a] | a � Q+ � {0}} be a semiring

properties (a), (b) and (c) given in problem (9) fo

11. Let T = {[0, a] | a � R + � {0}} be an interval seman interval semifield?

12. Obtain some interesting properties about intervan interval semiring.

13. How is an S-interval ideal different from an intean interval semiring?

14. Obtain some interesting properties about S-d

ideals of an interval semiring.

15. Describe some nice properties enjoyed by S p

interval ideals of an interval semiring.

16 Define an interval semiring which has zero div

20. Does there exist an interval polynomial sem

every interval polynomial subsemiring i

polynomial subsemiring?

21. Find for the interval group ring SG where S

Z7} and G = S7.

a. What is the order of SG?

b Fi d id l i SG



b. Find ideals in SG.

c. Find interval subsemirings in SG.

d. Does SG have zero divisors?e. Can SG have units?

22. Let G be any group of finite order. S = {[0

{0}} be an interval semiring, SG the

semiring of the group G over S, the interval s

a. Can SG have zero divisors?

b. Can SG have S-zero divisors?

c. Find ideals of SG and characterize them

d. Can SG have idempotents or S-idempote

23. Let G = D27 = {a, b / a2 = b7 = 1, bab = a}

group. S = [0, a] / a � Q+ � {0}} be the in

Enumerate all the special properties enjoy

group interval semiring.

24. Let G = {g / g23 = 1} be a cyclic group of or

a] / a � R + � {0}} be an interval semiring

interval semiring of G over S. Can SG have

subsemirings? Can SG have ideals? Is SGJustify your claim.

f. Can SG have S-anti zero divisors? Justify yo

26. Let Sn be the symmetric group of degree n.S = {

Q

+

� {0}} be an interval semiring. SSn be thegroup interval semiring of the symmetric group

interval semiring S.a. Find subsemirings of SSn

b C SS h id t t ?



b. Can SS n have idempotents?

c. Can SSn have ideals?

27. Let G be a group of finite order say n, an

symmetric group n < m. S = {[0, a] / a � Q+ �

interval semiring. SG and SSn be group intervand symmetric group interval semiring respectiv

a. Prove there exists an embedding of SG in Sn b. SSn is always non commutative, prove.

c. SSn have commutative interval subsemirings

d. Find a homomorphism K : SG o SSn so tinjective homomorphism.

28. Determine interesting properties enjoyed by SS

{[0, a] / a � R + � {0}}.

29. Distinguish between SSn and TSn where S = {[0,

� {0}} and T = {[0, a] / a � Z+ � {0}}.

30. Let S = {[0, a] / a � Z20} be an interval semirin

g10 = 1² be a cyclic group. Find the properties

SG. Can SG have zero divisors and S-zero diviso

b. If p and q are not primes but (p, q)

additional properties satisfied by SG.

c. If p and q are non primes with (p, q) =

properties enjoyed by SG?Which of the three group semirings SG given

(c) are a rich algebraic structure? We sa

structure is rich if it has several properties

l t b t t



elements or substructures.

33. Let G be a group of order n, n, a composi{[0, a] | a � Zm}, m a composite number su

d where d is again a composite number.

group interval semiring of the group G over

Enumerate all the properties enjoyed by SG.

34. Let S = {[0, a] / a � Z+ � {0}} be an interva

{(D2.3 u S4 u P) where P is a cyclic group of

group. SG is the group interval semiring

over the interval semiring S.

a. Is SG a S-semiring?

b. Does SG contain ideals?

c. Does SG contain subsemirings which ared. Find some special properties enjoyed by

35. Let S = {Z+ � {0}} be a semiring. G = {[0,

an interval group. SG be the interval g

Determine some distinct properties enjoyed b

36. Let S be the chain lattice {0 < a1 < a2 < …. <is a semiring. G = {[0, a] / a � Z19 \ {0}}

d lti li ti d l 19 SG

37. Let S = {

0 1 b d f } b fi it i i G {

a

1

f

0

d

c

e

b



; 0, 1, a, b, c, d, e, f } be a finite semiring. G = {Z37 \ {0}} be an interval group under multiplicat

37. SG be the interval group semiring.

a. Find ideals in SG.

b. Find zero divisors in SG

c. Is SG a S-interval group semiring?

d. Can SG have idempotents?

e. Find subsemirings which are not ideals in SG

38. Let S = {

| 0, a, b, 1} be a semiring of order four. G = {[0,

\ {0}} be an interval group. SG be the interval gr

the interval group G over the semiring S.

a. Find zero divisors in SG.

b. Find the cardinality of SG.

c. Find substructures of SG.

39 Let S (6) be the symmetric semigroup F = {[0

a

1

b

0

Find zero divisors, S-zero divisors, S-an

idempotents if any in the semigroup interval

41. Let S = {[0, a] / a � Z12} be an interval semu} be a semigroup under multiplication mod

a. Find the order of the semigroup interval

b. Is SP a S-semiring?



b. Is SP a S semiring?

c. Find ideals and subsemirings of SP.

42. Let G = D2.7 be the dihedral group of order

a � Z21} be an interval semiring. SD2.7 = S

interval semiring of the group G over the in

S.a. Find the order of SG.

b. Can SG have ideals?

c. Can SG have zero divisors?

d. If SG has zero divisors are they S-zero

antizero divisors?

e. Is SG a S-semiring?

f. Can SG have S-idempotents?

g. Give reasons for SG to be only an interv

not an interval semifield. Justify all your

43. Let G = {[0, a] | a � Z15} be an interval se

multiplication modulo 15. S = {Z+ � {0}} b

SG the interval semigroup semiring of th

over the semiring S.

a. Find zero divisors in SG.

b. Can SG have S-zero divisors?

45. Let S = {[0, a] | a � Z12} be an interval semigro

� {0}} be a semiring. FS the interval semigrou

of the interval semigroup S over the semiring F.

a. Can FS have zero divisors? b. Can FS be a S-semiring?

c. Can FS have ideal? Justify your claim.

46 Let G = {[0 a] | a � Z11 \ {0}} be an interval gro



46. Let G = {[0, a] | a � Z11 \ {0}} be an interval gro

� {0}} be a semiring. SG the interval group sem

a. Is SG a S-semiring? b. Is SG a semifield?

c. Obtain proper subsemirings of SG.

47. Let G = {[0, a] | a � Z18} be an interval semigro

Z+ � {0} be a semiring. SG the interval semigro

of the interval semigroup G over the semiring S problems (a), (b) and (c) given in problem 46.

48. Let S = {[0, a] | a � Z20} be an interval semirin

g12 = 1² be a group. SG the group interval semirsome interesting properties enjoyed by SG.

49. Let S = {[0, a] / a � Z20} be an interval semigr

� {0} be a semiring. FS be the interval semigro

of the interval semigroup S over the semiring S.

a. Find zero divisors in FS. b. Can FS be a S-semiring?

c. Does FS have S-ideals?d. Does FS have subsemirings which a

subsemirings?

SP be the interval semigroup interval semiri

special properties enjoyed by this new structu

52. What are groupoid interval semirings? Giv

finite and infinite groupoid interval semiring

53. Let G = {[0, a] / a � Z12, *, (3, 7)} be an int

S = {Z+ � {0}} be a semiring. SG be the in



{ { }} g

semiring. Find the special properties enj

structures.

54. Let G = {[0, a] / a � Z11, *, (3, 4)} be an int

S = {Q+ � {0}} be a semiring, SG be the insemiring of G over S.

a. Find zero divisors of SG if any.

b. Find S-ideal of SG if any.

c. Does SG contain S-subsemiring?

d. Show SG is non associative.

e. Does SG satisfy any special identity?

55. Let G = {[0, a], *, (3, 11), a � Z+ � {0}}

groupoid. S = {Z+ � {0}} be a semiring; S

groupoid semiring of infinite order whassociative semiring.

a. SG has no zero divisors, prove.

b. Can SG have units and idempotents?

c. Is SG a S-semiring?

d. Find subsemirings in SG.

e. Can SG have S-subsemirings?f. Can SG have ideals and S-ideals?

D SG h S b i i hi h

semiring. Can SG be a S-semiring? Can SG h

divisors?

58. Let G = {[0, a] / a � Z12, (3, 6), *} be an intervaS = Z+ � {0} be a semiring. SG be the interv

semiring. Describe all the distinct properties enjo

59. Let S = {[0, a] / a � Z+ � {0}} be an interval se



{ { }}

{a � Z12, *, (3, 7)} be a groupoid. SG the group

semiring. Find S-zero divisors and zero divisor

SG.

60. Let G = {Z19, *, (2, 3)} be a groupoid. S = {[0,

� {0}} be an interval semiring. SG be the group

semiring.

a. Find S-subsemirings in SG.

b. Can SG have idempotents?

c. Does SG have S-ideals?

d. Can SG have S-pseudo ideals?

e. Can SG have S-anti zero divisors?

61. Let G = {Z214, *, (0, 3)} be a groupoid. S = {[0,

be an interval semiring. SG the groupoid intervof G over S.a. Find the order of SG.

b. Is SG a S-ring?

c. Can SG have an S-ideal?

d. Can SG have S-subsemirings which are not S

e. Can SG have S-zero divisors?

62 Let G ¢Z * (0 5)² be the gro poid S {[0 a

65. Can there be a groupoid interval semiring SG

prime? Justify your claim.

66. Can there be a groupoid interval semiring SG

67. Let G = {Z12, *, (3, 6)} be a groupoid. S =

the interval semiring. SG the groupoid inter

G over S. Does SG satisfy any of the classic



groupoids?

68. Let G = {[0, a] | a � Z12, *, (3, 4); [0, a] * [0(mod 12)]} be an interval groupoid. S = {0

… < a10 < 1 } be a finite semiring. Stu

groupoid semiring SG.

69. Let G = {[0, a] | a � Z15, (0, 7)} be an interva

Cn be a chain lattice where n = 30. SG groupoid semiring.a. Can SG have S-subsemiring?

b. Can SG have ideals?

c. Can SG have S-ideals?

d. Can SG have zero divisors?

e. Obtain some interesting properties enjoyf. Find the order of SG.

g. Does the order of subsemiring divide the

70. Let S = {[0, a] / a � Q+ � {0}} be an interva

L17 (5) be the loop of order 18. SG be th

semiring of the loop G over the interval semia. Can SG have ideals?

b Is SG a S-semiring?



interval semiring. Find the order of LS

properties enjoyed by LS.

79. Enumerate some properties enjoyed by n

interval semiring.

80. Construct non associative interval semirithose obtained by groupoid interval sem

groupoid semirings loop interval semirin



groupoid semirings, loop interval semirin

loop interval semirings.

81. Give an example of a groupoid interval sem

not a S-semiring.

82. Give an example of a loop interval semiring

S-semiring.

83. Prove the class of interval semirings SLn (

where a is in Z+ � {0}} and Ln (m) defined

interval semirings.

84. Let G = {Z41, *, (3, 12)} be a groupoid. S =

semiring. SG be the groupoid semiring. Con

a] / a � Z41, *, (3, 12)} be the interval group

interval groupoid semiring of G1 over S.

a. Is SG # SG1? b. Is SG a S-semiring?

c. Can SG1 be a S-semiring?

85. Let G = {[0, aI] / a � Z20, *, (3, 7)} be

interval groupoid. S = {Z+ � {0}} be the s

86. Study all the properties enjoyed by S = {[0, a +

Q+ � {0}} the neutrosophic interval semiring.

87. What are the probable applications of this nesemirings?

88. Let L = {[0, aI] / a � {eI, I, 2I, …, 15I}, *, m =

[0, bI] = [0, 8bI + 7aI (mod 15)]} be the pure n

i t l l D i ll th ti i t d



interval loop. Derive all the properties associated

Suppose T = {[0, a +bI] / a, b � {e, 1, 2, …,

8} be the neutrosophic interval loop. Compare th

L and T. What is the cardinality of T? Does T h

subloops?

89. Let G = {[0, a+bI] / a, b � Z32} be the neutrosop

semigroup. Discuss all the properties related with

a. What is the cardinality of G? b. Does G have zero divisors?

c. Can G have idempotents?

d. Is G a S-semigroup?

90. Let H = {[0, a + bI] / a, b � 31} be the neutrosop

semiring. Study the properties (a) to (d) me problem 89 for H.

91. Let P = {[0, a + bI] / a, b � Z+ � {0}} be a n

interval semigroup.

a. Is P a S-semigroup?

b. Can P have ideals?c. Can P have S-subsemigroups which are not S

d Can P have zero divisors or idempotents?

c. Can F have S-ideals?

94. Let K = {[0, a + bI] | a, b � R + � {0}} be

interval semiring. Study properties (a),

mentioned in problem (93) for K.

95. Let S = {[0, a + bI] | a, b � Z28} be a neutro

semigroup. F = {C10; 0 < a1 < … < a8 < 1} be

the neutrosophic interval semigroup semiring



the neutrosophic interval semigroup semiring

a. What is the cardinality of FS?

b. Find S-ideals in any in FS.

c. Can FS have S-zero divisors?

d. Can FS have idempotents?

e. Is FS a S-semiring?

96. Let P = {[0, aI] | a � Ln (m), *) be an interv

loop. Study and characterize the propertiesSuppose S = {C7, 0 < a1 < … < a5 < 1} be t

the interval neutrosophic loop semiring.a. Prove order of SP is finite.

b. Is SP a S-semiring?

c. Is SP a semifield?

d. Can SP have S-zero divisors?e. What are the special properties enjoyed b

97. Let L = {[0, a + bI] | a, b � L21(11)} be

interval loop.

a. Find the order of L?

b. Is L a S-loop?

98 Let L = {[0 a + bI] | a b � L23(12)} be

99. Let G = {[0, aI] | a � Z17 \ {0}} be the pure n

interval group. S = {0 < a1 < a2 < … < a10 <

semiring; SG the group interval neutrosophic sem

a. Find the order of SG? b. Is SG a S-semiring?

c. Can SG have S-ideals?

d. Does SG have S-zero divisors?e. Can SG have S-subsemirings which are not S



100. Let G = {[0, aI + b] | b, a � Z23

\ {0}} be a n

interval group. S = Z+ � {0} be the semiring.

group interval neutrosophic semiring. Study (b)

in problem (99) for this SG.

101. Let S = {[0, a + bI] | a, b � Z240} be a neutrosop

semigroup. K = {0 < a1 < … < a100 < 1} be a se

the neutrosophic interval semigroup semiring.

properties (a) to (e) given in problem 99.

102. Let S = {[0, a + bI] | a, b � Z24} be a neutrosop

semigroup. K = {[0, a + bI] / a, b � Z+ � {

interval neutrosophic semiring. KS the

neutrosophic interval semiring. Obtain the

enjoyed by KS.

103. Let W = {[0, a+bI] | a, b � Z26, *, (3, 11)} be a n

interval groupoid. S = {0 < a1 < a2 < a3 < 1} be

SW be the neutrosophic interval groupoid semiri

a. Find the order of SW. b. Is SW a S-semiring?

D SW h S di i ?

105. Let G = {[0, a + bI] | a, b � Z46} be a neutro

semigroup. S = {[0, a + bI] | a, b � Q+ � {0}

neutrosophic semiring. SG be the neutro

semigroup interval semiring. Enumerate alenjoyed by SG.

106. Let S = i

i 0

[0,a bI]x a,b Z {0}f

½ � �® ¾

¯ ¿¦



i 0¯ ¿neutrosophic polynomial semiring. Des

properties enjoyed by S.

107. Let P = i

i 0

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

½� �® ¾

¯ ¿¦ b

neutrosophic polynomial semiring.

a. Can P have S-ideals?

b. Is P a strict semiring?

c. Can P be a semifield?

Justify all your claims.

108. Let P = {all 10 u 10 neutrosophic intervaintervals of the form [0, a + bI] | a, b

neutrosophic interval matrix semiring.

a. Find the order of P.

b. Is P a semifield?

c. Find S-zero divisors in P.

d. Find S-ideals if any in P.e. Can P have S-subsemirings which ar

Justify your claim.



FURTHER R EADING



1. Birkhoff. G., and Maclane, S.S., A Brief Survey Algebra, New York, The Macmillan and Co. (19

2. Bruck, R.H., A Survey of binary systems, Berl

Verlag, (1958).

3. Golan, JS., Semirings and their ApplicationAcademic Pub. (1999)

4. Padilla, R., Smarandache Algebraic Structures,Pure and Applied Sciences, Delhi, Vol. 17 E, N

121,

http://www.gallup.unm.edu/~smarandache/ALG

TXT.TXT

5. Smarandache, Florentin (editor), Proceedings

International Conference on Neutrosophy, N Logic, Neutrosophic set, Neutrosophic probSt ti ti D b 1 3 2001 h ld t th U

9. Vasantha Kandasamy, W.B., New c groupoids using Z n, Varahmihir Journal o

Sciences, Vol 1, 135-143, (2001).10. Vasantha Kandasamy, W.B., On Ordered Gr

Groupoid Rings, Jour. of Maths and Com

145-147, (1996).

11. Vasantha Kandasamy, W.B., Smarandachttp://www.gallup.unm.edu/~smarandache/G

12 Vasantha Kandasamy W B Groupids and



12. Vasantha Kandasamy, W.B., Groupids andGroupoids, American Research Papers, Rebo

13. Vasantha Kandasamy, W.B., Smaranda semifields and semivector spaces, Ame

Papers, Reboth, (2002).

14. Vasantha Kandasamy, W.B., Smarandache lResearch Papers, Reboth, (2002).

15. Vasantha Kandasamy, W.B., and Smarand

Fuzzy Interval Matrices, Neutrosophic Inand their application, Hexis, Reboth, (2002)

16. Vasantha Kandasamy, W.B., and Smarand

Interval Semigroups, Kappa and Omega, Gle

17. Vasantha Kandasamy, W.B., Smarandache

Moon Kumar Chetry, Interval Groupoids,Glendale, (2002).

INDEX



C

Commutative interval semiring, 9-10

D

Direct product of interval semirings, 16

G

Group interval semirings, 57-8,65Groupoid interval neutrosophic semiring, 120-125

Groupoid interval semiring, 79-81

I

Interval semigroup interval semiring, 109

Interval semigroup semiring, 109

Interval semiring homomorphism, 19Interval semirings, 9-10

Interval subsemiring, 10

L

Left (right) ideal of interval semiring, 17-8



Loop interval semiring, 89

M

Moufang interval loop, 95

N

Neutrosophic interval semiring, 113-6

Neutrosophic loop semiring, 123-5

Neutrosophic matrix interval semiring, 121

Non- commutative interval semiring, 9-10

n u n square matrix interval semiring, 31

P

Pseudo neutrosophic interval subsemiring, 114-6

Pure neutrosophic interval matrix semiring, 121

Pure neutrosophic interval polynomial semiring,

Pure neutrosophic interval semiring, 113-6Pure neutrosophic interval subsemiring, 120-12

S-groupoid interval semiring, 84-7

S-ideal of a row matrix interval semiring, 27-8

S-idempotent, 24S-interval anti zerodivisors, 23-4

S-interval matrix anti zerodivisor, 44-5

S-interval matrix ideal, 38

S-interval matrix subsemiring, 35-6

S-interval zerodivisors, 22-4

S-Lagrange interval loop, 95-7



g g pS-loop interval semiring, 89-93

Smarandache interval semiring, 19-20

Smarandache interval subsemiring, 19-20

Smarandache pseudo interval subsemiring, 20-2

Special interval semiring, 70

S-polynomial interval semiring, 52

S-pseudo left (right) ideal of the interval semiring, 20

S-pseudo matrix interval ideal, 41

S-pseudo matrix interval subsemiring, 39

S-pseudo row matrix interval subsemiring, 27-8

S-row matrix interval semiring, 25-7

S-simple interval loop, 96-7

S-square matrix interval semiring, 34

S-unit, 24

S-weakly Lagrange interval loop, 95-9

Symmetric group neutrosophic interval semiring, 125

W

WIP interval loop, 95

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.S



projects. 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 53rd 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@gm

Web Site: http://mat.iitm.ac.in/home/wbv/public_or http://www.vasantha.in

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 numbeEuclidean geometry, synthetic geometry, algebstatistics neutrosophic logic and set (generaliz

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

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