# Role of operator semirings in characterizing $\Gamma$-semirings in terms of fuzzy subsets

**ABSTRACT** The operator semirings of a $\Gamma$-semiring have been brought into use to

study $\Gamma$-semiring in terms of fuzzy subsets. This is accomplished by

obtaining various relationships between the set of all fuzzy ideals of a

$\Gamma$-semiring and the set of all fuzzy ideals of its left operator semiring

such as lattice isomorphism between the sets of fuzzy ideals of a

$\Gamma$-semiring and its operator semirings.

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**ABSTRACT:**In this paper a correspondence between the set of all intuitionistic fuzzy ideals of a Γ-semiring and the set of all intuitionistic fuzzy ideals of its operator semirings is established and used them to study some properties of the semiring S_{2}.ANNALS OF FUZZY MATHEMATICS AND INFORMATICS. 03/2014; 7(3):529-542.

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arXiv:1101.4807v1 [math.RA] 25 Jan 2011

Role of operator semirings in characterizing

Γ−semirings in terms of fuzzy subsets

Sujit Kumar Sardar

Department of Mathematics,

Jadavpur University, Kolkata

E-mail: sksardarjumath@gmail.com

Y. B. Jun

Department of Mathematics Education

Gyeongsang National University

Chinju 660-701, Korea

skywine@gmail.com

Sarbani Goswami

Lady Brabourne College, Kolkata, W.B., India

E-mail: sarbani7−goswami@yahoo.co.in

Abstract

The operator semirings of a Γ−semiring have been brought into use

to study Γ−semiring in terms of fuzzy subsets. This is accomplished

by obtaining various relationships between the set of all fuzzy ideals of

a Γ−semiring and the set of all fuzzy ideals of its left operator semir-

ing such as lattice isomorphism between the sets of fuzzy ideals of a

Γ−semiring and its operator semirings.

Mathematics Subject Classification[2000]:16Y60, 16Y99, 03E72

Keywards: Γ-semiring, operator semiring, fuzzy left(right) ideal, fuzzy

k-ideal, Γ-semifield.

1 Introduction

The notion of Γ−semiring was introduced by M.M.K Rao[9]. This generalizes

not only the notions of semiring and Γ−ring but also the notion of ternary

semiring. This also provides an algebraic home to the nonpositive cones of the

totally ordered rings (It may be recalled here that the nonnegative cones of

the totally ordered rings form semirings but the nonpositive cones do not as

the induced multiplication is no longer closed). Γ−semiring theory has been

enriched with the help of operator semirings of a Γ−semiring by Dutta and

Sardar[3]. To make operator semirings effective in the study of Γ−semirings

Dutta et al[3] established correspondence between the ideals of a Γ−semiring

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2

Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

S and the ideals of the operator semirings of S. As it was done for Γ−rings in

[4] we also establish here various relationships between the fuzzy ideals of a

Γ−semiring S and the fuzzy ideals of the operator semirings of S. We obtain

a lattice isomorphism between the sets of fuzzy ideals of a Γ−semiring S and

its left operator semiring L. This has been used to give a new proof of the

lattice isomorphism between the sets of ideals of S and L which was originally

due to Dutta and Sardar[3]. This is also used to obtain a characterization of a

Γ−semifield and relationship between the fuzzy subsets of Γ−semiring S and

matrix Γn−semiring Sn.

2 Preliminaries

We recall the following definitions and results for their use in the sequel.

Definition 2.1 [11] Let S be a non empty set. A mapping µ : S → [0,1] is

called a fuzzy subset of S.

Definition 2.2 [9]Let S and Γ be two additive commutative semigroups.

Then S is called a Γ−semiring if there exists a mapping

S×Γ×S → S (images to be denoted by aαb for a,b ∈ S and α ∈ Γ) satisfying

the following conditions:

(i) (a + b)αc = aαc + bαc,

(ii) aα(b + c) = aαb + aαc,

(iii) a(α + β)b = aαb + aβb,

(iv) aα(bβc) = (aαb)βc for all a,b,c ∈ S and for all α,β ∈ Γ.

Further, if in a Γ−semiring, (S,+) and (Γ,+) are both monoids and

(i) 0Sαx = 0S= xα0S

(ii)x0Γy = 0S= y0Γx for all x,y ∈ S and for all α ∈ Γ then we say that S is

a Γ−semiring with zero.

Throughout this paper we consider Γ−semiring with zero. For simplification

we write 0 instead of 0S.

Example 2.3 [3]Let S be the additive commutative semigroup of all m × n

matrices over the set of all non-negative integers and Γ be the additive com-

mutative semigroup of all n × m matrices over the same set. Then S forms

a Γ−semiring if xαy denotes the usual matrix multiplication of x,α,y where

x,y ∈ S and α ∈ Γ.

Definition 2.4 [3]Let S be a Γ−semiring and F be the free additive com-

mutative semigroup generated by S × Γ. Then the relation ρ on F, defined by

Page 3

Role of operator semirings in characterizing Γ−semiring in terms of fuzzy subsets3

m

?

i=1

for all a ∈ S(m,n ∈ Z+), is a congruence on F. The congruence class con-

m

?

i=1

i=1

classes, is an additive commutative semigroup and this also forms a semiring

with the multiplication defined by

(xi,αi)ρ

n

?

j=1

(yj,βj) if and only if

m

?

i=1

xiαia =

n

?

j=1

yjβja

taining

(xi,αi) is defined by

m

?

[xi,αi]. Then F/ρ, the set of congruence

(

m

?

i=1

[xi,αi])(

n

?

j=1

[yj,βj]) =

?

i,j

[xiαiyj,βj]

This semiring is denoted by L and called the left operator semiring of the

Γ−semiring S.

Dually the right operator semiring R of the Γ−semiring S is defined.

Definition 2.5 [3]Let S be a Γ−semiring and L be the left operator semiring

and R be the right one. If there exists an element

m

?

i=1

[ei,δi] ∈ L(

n

?

j=1

[γj,fj] ∈ R)

such that

m

?

i=1

m

?

i=1

eiδia = a (

n

?

j=1

aγjfj= a) for all a ∈ S then S is said to have the

left unity

[ei,δi] (resp. the right unity

n

?

j=1

[γj,fj]).

Definition 2.6 [3]Let S be a Γ−semiring and L be the left operator semiring

and R be the right one. For P ⊆ L (⊆ R), P+:= {a ∈ S : [a,Γ] ⊆ P}

(respectively P∗:= {a ∈ S : [Γ,a] ⊆ P}). For Q ⊆ S,

m

?

i=1

i=1

set of all finite sums

?

i,k

m

?

i=1

i=1

set of all finite sums

?

i,k

Q+′:= {[xi,αi] ∈ L : (

m

?

([xi,αi])S ⊆ Q} where (

m

?

i=1

[xi,αi])S denotes the

xiαisk, sk∈ S and

Q∗′:= {[αi,xi] ∈ R : (

m

?

S([αi,xi]) ⊆ Q} where S(

m

?

i=1

[xi,αi]) denotes the

skαixi, sk∈ S.

Throughout this paper unless otherwise mentioned S denotes a Γ-semiring with

left unity and right unity and FLI(S), FRI(S) and FI(S) denote respectively

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4

Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

the set of all fuzzy left ideals, the set of all fuzzy right ideals and the set of

all fuzzy ideals of the Γ-semiring S. Similar is the meaning of FLI(L), FRI(L),

FI(L) where L is the left operator semiring of the Γ−semiring S. Also through-

out we assume that µ(0) = 1 for a fuzzy left ideal (fuzzy right ideal, fuzzy

ideal) µ of a Γ−semiring S. Similarly we assume that µ(0) = 1 for a fuzzy

left ideal (fuzzy right ideal, fuzzy ideal) µ of the left operator semring of a

Γ−semiring S.

Definition 2.7 [3] A Γ−semiring S is said to be zero-divisor free (ZDF) if

aαb = 0 implies that either a = 0 or α = 0 or b = 0 for a,b ∈ S, α ∈ Γ.

Definition 2.8 [3]A commutative Γ−semiring S is said to be a Γ−semifield

if for any a(?= 0) ∈ S and for any α(?= 0) ∈ Γ there exists b ∈ S,β ∈ Γ such

that aαbβd = d for all d ∈ S.

Example 2.9 [10]Let S = {rω : r ∈ Q+∪ {0}} and Γ = {rω2: r ∈ Q+∪

{0}}, where Q+is the set of all positive rational numbers. Then S forms a

Γ−semifield.

Definition 2.10 [1]Let µ be a non empty fuzzy subset of a semiring S (i.e.

µ(x) ?= 0 for some x ∈ S). Then µ is called a fuzzy left ideal [ fuzzy right ideal]

of S if

(i) µ(x + y) ≥ min[µ(x),µ(y)] and

(ii) µ(xy) ≥ µ(y) [resp. µ(xy) ≥ µ(x)] for all x,y ∈ S.

Definition 2.11 [6]Let µ be a non empty fuzzy subset of a Γ−semiring S

(i.e. µ(x) ?= 0 for some x ∈ S). Then µ is called a fuzzy left ideal [ fuzzy right

ideal] of S if

(i) µ(x + y) ≥ min[µ(x),µ(y)] and

(ii) µ(xγy) ≥ µ(y) [resp. µ(xγy) ≥ µ(x)] for all x,y ∈ S,γ ∈ Γ.

Definition 2.12 [7]Let S be a Γ-semiring and µ1,µ2∈ FLI(S) [FRI(S), FI(S)].

Then the sum µ1⊕ µ2of µ1and µ2is defined as follows:

(µ1⊕ µ2)(x) = sup

x=u+v[min[µ1(u),µ2(v)] : u,v ∈ S]

= 0 if for any u,v ∈ S,u + v ?= x.

Note. Since S contains 0, in the above definition the case x ?= u + v for

any u,v ∈ S does not arise.

Definition 2.13 [5]Let S be a Γ-semiring and n be a positive integer. The

sets of n × n matrices with entries from S and n × n matrices with entries

from Γ are denoted by Snand Γn respectively. Let A,B ∈ Sn and ∆ ∈ Γn.

Then A∆B ∈ Sn and A + B ∈ Sn. Clearly, Sn forms a Γn-semiring with

these operations. This is called the matrix Γ−semiring over S or the matrix

Γn−semiring Snor simply the Γn−semiring Sn.

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Role of operator semirings in characterizing Γ−semiring in terms of fuzzy subsets5

The right operator semiring of the matrix Γn−semiring Snis denoted by

[Γn,Sn] and the left one by [Sn,Γn]. If x ∈ S, the notation xEijwill be used

to denote a matrix in Sn with x in the (i,j)-th entry and zeros elsewhere.

αEij∈ Γn, where α ∈ Γ, will have a similar meaning. If P ⊆ S, Pnwill denote

the set of all n×n matrices with entries from P. Similar is the meaning of ∆n

where ∆ ⊆ Γ.

Proposition 2.14 [5]Let S be a Γ−semiring and R (resp. L) be its right (resp.

left) operator semiring. Let Rn(resp. Ln) denote the semiring of all n × n

matrices over R (resp. L). Then

(i) the right operator semiring [Γn,Sn] of the Γn−semiring Snis isomorphic

p

?

i=1

and (ii) the left operator semiring [Sn,Γn] of the Γn−semiring Snis isomorphic

p

?

i=1

with Rnvia the mapping [[γi

jk],[xi

uv]] ?→

p

?

i=1

(

n

?

t=1

[γi

jt,xi

tv])1≤j,v≤n,

with Lnvia the mapping [[xi

uv],[γi

jk]] ?→

p

?

i=1

(

n

?

t=1

[xi

ut,γi

tk])1≤u,k≤n.

In view of the above proposition we henceforth identify [Sn,Γn] and Rn;

[Sn,Γn] and Ln.

3 Main results.

Throughout this section S denotes a Γ−semiring and L denotes the left oper-

ator semiring of the Γ−semiring S.

Definition 3.1 Let µ be a fuzzy subset of L, we define a fuzzy subset µ+of

S by µ+(x) = inf

γ∈Γµ([x,γ]) where x ∈ S.

If σ is a fuzzy subset of S, we define a fuzzy subset σ+′of L by

σ+′(

?

ii

[xi,αi]) = inf

s∈Sσ(

?

xiαis) where

?

i

[xi,αi] ∈ L.

By routine verification we obtain the following lemma.

Lemma 3.2 If {µi : i ∈ I} is a collection of fuzzy subsets of L then

µ+

?

i∈I

?

i∈I

i= (µi)+.

Proposition 3.3 Suppose σ,σ1,σ2∈ FI(S)and µ ∈ FI(L). Then

(i) σ+′∈ FI(L). Moreover, if σ is non constant then σ+′is non constant.

(ii) (σ+′)+= σ,

(iii) σ1?= σ2implies that σ+′

(iv) (σ1⊕ σ2)+′= σ+′

1 ?= σ+′

2,

2,

1⊕ σ+′

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6

Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

(v) (σ1∩ σ2)+′= σ+′

(vi) σ1⊆ σ2implies that σ+′

(vii) µ+∈ FI(S). Moreover, if µ is non constant then µ+is non constant.

(viii) (µ+)+′= µ,

(ix) µ1⊆ µ2implies that µ+

1∩ σ+′

2,

1 ⊆ σ+′

2,

1⊆ µ+

2.

Proof. (i) Let σ ∈ FI(S). Then σ(0) = 1. Let γ ∈ Γ. Then

σ+′([0,γ]) = inf

s∈S[σ(0γs)] = σ(0) = 1,γ ∈ Γ. Thus we see that σ+′is non

empty and σ+′(0) = 1 as for all γ ∈ Γ, [0,γ] is the zero element of L.

Let

?

ij

Then σ+′(

?

i

j

≥ inf

?

i

= min[inf

?

i

= min[σ+′(

?

i

Again σ+′(

?

ij

≥ [min[σ(

?

ii

≥ inf

?

i

Similarly we can show that σ+′(

?

i

Consequently, σ+′∈ FI(L).

Further, let σ be a non constant fuzzy ideal of S.

If σ+′is constant then σ+′(

?

i

σ(x) = (σ+′)+(x) = inf

[xi,αi],

?

[yj,βj] ∈ L.

[xi,αi] +

?

[yj,βj]) = inf

s∈S[σ(

?

i

xiαis +

?

j

yjβjs)]

s∈S[min[σ(

xiαis),σ(

?

j

yjβjs)]]

s∈S[σ(

xiαis)], inf

s∈S[σ(

?

j

[yj,βj])].

yjβjs)]]

[xi,αi]),σ+′(

?

j

[xi,αi]

?

[yj,βj]) = σ+′(

?

i,j

xiαiy2),σ(

[xiαiyj,βj]) = inf

s∈Sσ(

?

i,j

xiαiyjβjs)

xiαiy1),σ(

?

?

i

xiαiy3),..........]]

s∈S[σ(

(xiαis))] = σ+′(

?

i

[xi,αi]

[xi,αi]).

?

j

[yj,βj]) ≥ σ+′(

?

j

[yj,βj]).

[xi,αi]) = 1 for all

?

i

[xi,αi] ∈ L. Then

γ∈Γσ+′([x,γ]) = 1 for all x ∈ S where σ is constant.

This contradicts that σ is non constant . Consequently, σ+′is non constant.

(ii) Let x ∈ S. Then

((σ+′)+)(x) = inf

γ∈Γ[σ+′([x,γ])] = inf

[γi,fi] be the right unity of S. Then

γ∈Γ[inf

s∈S[σ(xγs)]] ≥ σ(x). Hence σ ⊆ (σ+′)+.

?

i

xγifi) ≥ min[σ(xγ1f1),σ(xγ2f2),.........]

Let

?

i

xγifi= x for all x ∈ S.

Now σ(x) = σ(

?

i

≥ inf

γ∈Γ[inf

s∈S[σ(xγs)]] = (σ+′)+(x).

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Role of operator semirings in characterizing Γ−semiring in terms of fuzzy subsets7

Therefore (σ+′)+⊆ σ and hence (σ+′)+= σ.

(iii) Let σ1 ?= σ2. If possible, let σ+′

σ1= σ2, (by (ii)) – which contradicts our assumption. Hence σ+′

(iv) Let

?

i

((σ1⊕ σ2)+′)(

?

i

= inf

?

k

= sup[min[inf

?

k

= sup[min[σ+′

?

k

= (σ+′

?

i

Thus (σ1⊕ σ2)+′= σ+′

(v) (σ1∩ σ2)+′(

?

i

= inf

?

i

= min[inf

?

i

= min[σ+′

?

i

= (σ+′

?

i

Hence (σ1∩ σ2)+′= σ+′

(vi) Let σ1,σ2∈ FI(S) be such that σ1⊆ σ2. Then

σ+′

?

ii

= σ+′

?

i

Thus σ+′

(vii) Let µ ∈ FI(L). Then µ(0) = 1.

Now µ+(0) = inf

1

= σ+′

2. Then (σ+′

1)+= (σ+′

2)+. i.e.,

1 ?= σ+′

2

[ai,αi] ∈ L. Then

[ai,αi])=inf

s∈S(σ1⊕ σ2)(

?

i

aiαis)

s∈S[sup[min[σ1(

xkδks),σ2(

?

j

yjβjs)] :

?

i

yjβjs)]]

aiαis =

?

k

xkδks +

?

j

yjβjs]]

s∈Sσ1(xkδks), inf

s∈Sσ2(

?

j

[yj,βj])]]

1([xk,δk]),σ+′

2(

?

j

1⊕ σ+′

2)(

[ai,αi])

1⊕ σ+′

[ai,αi]) = inf

2.

s∈S[(σ1∩ σ2)(

?

i

aiαis)]]

aiαis)]

s∈S[min[σ1(

aiαis),σ2(

?

i

s∈Sσ1(

aiαis), inf

s∈Sσ2(

?

i

[ai,αi])]

aiαis)]

1(

[ai,αi]),σ+′

2(

?

i

1∩ σ+′

2)(

[ai,αi])

1∩ σ+′

2.

1([xi,αi]) = inf

s∈S[σ1(

?

xiαis)] ≤ inf

s∈S[σ2(

?

i

xiαis)]

2([xi,αi]) for all

?

i

[xi,αi] ∈ L.

1 ⊆ σ+′

2.

γ∈Γ[µ([0,γ])] = 1 [Since for all γ ∈ Γ, [0,γ] is the zero element

of L].

This also shows that µ+is non empty.

Let x,y ∈ S and α ∈ Γ. Then

µ+(x+y) = inf

γ∈Γ[µ([x + y,γ])] = inf[µ([x,γ]+[y,γ])] ≥ inf[min[µ([x,γ]),µ([y,γ])]]

γ∈Γ[µ([x,γ])], inf

Therefore µ+(x + y) ≥ min[µ+(x),µ+(y)].

= min[inf

γ∈Γ[µ([y,γ])] = min[µ+(x),µ+(y)]

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8

Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

Again µ+(xαy) = inf

γ∈Γ[µ([xαy,γ])] = inf

γ∈Γ[µ([xαy,γ])] = inf

= µ+(x).

γ∈Γ[µ([x,α][y,γ])] ≥ inf

γ∈Γ[µ([x,α][y,γ])] ≥ µ([x,α]) ≥ inf

γ∈Γµ[y,γ] = µ+(y)

δ∈Γ[µ([x,δ])] and µ+(xαy) = inf

Consequently µ+∈ FI(S).

Suppose µ is a non constant fuzzy ideal of L. If µ+is constant then µ+(x) = 1

for all x ∈ S as µ+(0S) = 1. Suppose

?

ii

= 1 for all

?

i

our assumption. This completes the proof.

(viii) Let µ ∈ FI(L). Then for

?

i

((µ+)+′)(

?

ii

= inf

?

i

Hence µ ⊆ (µ+)+′.

Let

?

i

µ(

?

jj

i

≥ min[µ(

?

j

≥ inf

?

j

Therefore (µ+)+′⊆ µ and so (µ+)+′= µ.

(ix) Proof is similar to that of (vi).

Note. All the results of the above proposition also hold for FRI(S) and

FRI(R).

In view of the above Proposition and the fact that the set of fuzzy ideals of

a Γ−semiring S and that of its operator semirings form a lattice under the

operations ⊕ and ∩ we obtain the following result.

[xi,αi] ∈ L. Then µ(

?

[xi,αi]) = ((µ+)+′)(

?

i

[xi,αi]) = inf

s∈Sµ+(

?

i

xiαis)

[xi,αi] ∈ L. This implies that µ is constant, which contradicts

[xi,αi] ∈ L,

[xi,αi]) = inf

s∈S[µ+(

?

xiαis)] = inf

s∈S[inf

γ∈Γ[µ(

?

i

?

[xiαis,γ])]]

s∈S[inf

γ∈Γ[µ(

[xi,αi][s,γ])]] ≥ µ(

i

[xi,αi]).

[ei,δi] be the left unity of S.Then

[xj,αj]) = µ(

?

[xj,αj]

?

[ei,δi])

[xj,αj][e1,δ1]),µ(

?

j

[xj,αj][e2,δ2]),............]

s∈S[inf

γ∈Γ[µ(

[xj,αj][s,γ])]] = (µ+)+′(

?

j

[xj,αj]).

Theorem 3.4 The lattices of all fuzzy ideals [ fuzzy right ideals ] of S and L

are isomorphic via the inclusion preserving bijection σ ?→ σ+′where σ ∈ FI(S)

[resp. FRI(S)] and σ+′∈ FI(L) [resp. FRI(L)].

Corollary 3.5 FLI(S) [resp. FRI(S), FI(S)]] is a complete lattice.

Proof. The corollary follows from the above theorem and the fact that

FLI(L) [resp. FRI(L), FI(L)] is a complete lattice[2].

Page 9

Role of operator semirings in characterizing Γ−semiring in terms of fuzzy subsets9

Lemma 3.6 Let I be an ideal (left ideal, right ideal) of a Γ−semiring S and

λIbe the characteristic function of I. Then (λI)+′= λI+′. Moreover, I+′is an

ideal of L.

Proof. Let

?

i

[xi,αi] ∈ L. Then either

?

i

[xi,αi] ∈ I+′or

?

i

[xi,αi] / ∈ I+′.

If

?

i

xiαis ∈ I for all s ∈ S. Hence (λI)+′(

[xi,αi] ∈ I+′then λI+′(

?

i

[xi,αi]) = 1 and

?

i

Again, if

?

i

[xi,αi]) = 0 and

[xi,αi]) = inf

s∈SλI(

?

i

?

i

xiαis) = 1.

?

i

[xi,αi] / ∈ I+′then λI+′(

?

i

xiαis / ∈ I for

some s ∈ S. Hence (λI)+′(

?

i

[xi,αi]) = inf

s∈SλI(

?

i

xiαis) = 0. Thus we obtain

(λI)+′= λI+′. Now by Proposition 3.3(i), (λI)+′is a fuzzy ideal of the left

operator semiring L. Hence λI+′ is a fuzzy ideal of L. Consequently, I+′is an

ideal of L[1]

Lemma 3.7 Let I be an ideal (left ideal, right ideal) of the left operator

semiring L of a Γ−semiring S and λIbe the characteristic finction of I. Then

(λI)+= λI+. Moreover, I+is an ideal of S.

Proof. Let x ∈ S. Then either x ∈ I+or x / ∈ I+. If x ∈ I+then

λI+(x) = 1. Again x ∈ I+implies that [x,γ] ∈ I for all γ ∈ Γ. Hence

(λI)+(x) = inf

γ∈ΓλI([x,γ]) = 1.

Again, if x / ∈ I+then λI+(x) = 0 and [x,γ] / ∈ I for some γ ∈ Γ. Hence

(λI)+(x) = inf

γ∈ΓλI([x,γ]) = 0. Consequently, (λI)+= λI+. Now by Proposition

3.3(ii), (λI)+is a fuzzy ideal of S. Hence I+is an ideal of S[6]

Remark. The last parts of Lemma 3.6 and Lemma 3.7 are originally due

to Dutta and Sardar. They are established here via fuzzy subsets.

The following theorem is also due to Dutta and Sardar[3]. We give an

alternative proof of it by using the lattice isomorphism of fuzzy ideals obtained

in Theorem 3.4 and by using Lemma 3.6 and Lemma 3.7.

Theorem 3.8 The lattices of all ideals [ right ideals ] of S and L are iso-

morphic via the mapping I ?→ I+′where I denotes an ideal ( right ideal) of

S.

Proof. That I ?→ I+′is a mapping follows from Lemma 3.6. Let I1and

I2be two ideals of S such that I1?= I2. Then λI1and λI2are fuzzy ideals of

S where λI1and λI2are characteristic functions of I1and I2respectively[6].

Evidently, λI1?= λI2. Then by Theorem 3.4, λ+′

I1?= λ+′

I2. Hence by Lemma 3.6,

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10

Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

λI+′

Next let J be an ideal of L. Then λJis a fuzzy ideal of L[2]. By Proposition

3.3(vii), (λJ)+is a fuzzy ideal of S and by Theorem 3.4 we obtain ((λJ)+)+′=

λJ. Now by successive use of Lemma 3.7 and Lemma 3.6 we obtain λ(J+)+′ =

λJand consequently, (J+)+′= J. Hence the mapping is onto.

Let I1, I2be two ideals of S such that I1⊆ I2. Then λI1⊆ λI2. Hence by

Theorem 3.4, λ+′

I+′

1

⊆ I+′

proof.

1

?= λI+′

2

whence I+′

1 ?= I+′

2

Consequently, the mapping I ?→ I+′is one-one.

I1⊆ λ+′

I2whence by Lemma 3.6, λI+′

2. Thus the mapping is inclusion preserving. This completes the

1

⊆ λI+′

2. Consequently

Theorem 3.9 A commutative semiring M is a semifield if and only if for

every non constant fuzzy ideal µ of M, µ(x) = µ(y) < µ(0) for all x,y ∈

M \ {0}.

Proof. Let M be a semifield and let x,y ∈ M \ {0} and µ a non constant

fuzzy ideal of M. Then µ(y) = µ(yxx−1) ≥ µ(xx−1) ≥ µ(x) where x−1is the

inverse of x. Similarly µ(x) ≥ µ(y). Therefore µ(x) = µ(y). It is known that

µ(0) ≥ µ(x) for all x ∈ M. Now we claim that µ(0) > µ(x) for all x ∈ M \{0}.

Otherwise, if µ(0) = µ(x) for some x ∈ M\{0} then by what we have obtained,

µ(0) = µ(x) for all x ∈ M \ {0} –a contradiction to the supposition that µ is

non constant.

Thus µ(x) = µ(y) < µ(0) for all x,y ∈ M \ {0}.

Conversely, let M be a commutative semiring and for every non constant

fuzzy ideal µ of M, µ(x) = µ(y) < µ(0) for all x,y ∈ M \ {0}. Now let I be a

non zero ideal of M. If possible, suppose I ?= M. Then there exists an element

x ∈ M \ I.

Let λI be the characteristic function of I. Then λI(x) = 0 ?= 1 = λI(0).

This implies that λIis a non constant fuzzy ideal of M. Suppose y(?= 0) ∈ I.

Then by hypothesis λI(x) = λI(y) but λI(x) = 0 and λI(y) = 1. Thus we

get a contradiction. Hence I = M. Thus M has no non-zero proper ideals.

Consequently, M is a semifield[8].

Theorem 3.10 A ZDF commutative Γ−semiring S is a Γ−semifield if and

only if for every non constant fuzzy ideal µ of S, µ(x) = µ(y) < µ(0) for all

x,y ∈ S \ {0}.

Proof. Let S be a ZDF commutative Γ−semifield and µ a non constant

fuzzy ideal of S. Let x,y ∈ S \ {0} and α(?= 0) ∈ Γ. Then there exists z ∈

S,β ∈ Γ such that xαzβs = s for all s ∈ S. In particular, xαzβy = y. Then

µ(y) = µ(xαzβy) ≥ µ(x). Similarly µ(x) ≥ µ(y). Therefore µ(x) = µ(y).

Now we claim that µ(0) > µ(x) for all x ∈ S \ {0}. Otherwise, if µ(0) = µ(x)

for some x ∈ S \{0} then µ(0) = µ(x) for all x ∈ S \{0} –a contradiction that

Page 11

Role of operator semirings in characterizing Γ−semiring in terms of fuzzy subsets11

µ is non constant.

Thus µ(x) = µ(y) < µ(0) for all x,y ∈ S \ {0}.

Converse follows by applying the similar argument as applied in the converse

part of Theorem 3.9 and by using Theorem 9.7 of [3].

Using the above theorem and Proposition 3.3 (hence Theorem 3.4) we give

a new proof of the following result of Dutta and Sardar[3].

Theorem 3.11 Let S is a ZDF commutative Γ−semiring. Then S is a

Γ−semifield if and only if its left operator semiring L is a semifield.

Proof. Let S is a Γ−semifield. Let µ be a non constant fuzzy ideal of L.

Then by Proposition 3.3(vii), µ+is a non constant fuzzy ideal of S. Hence by

Theorem 3.10, µ+(x) = µ+(y) < µ+(0S) for all x,y ∈ S \ {0S}. Let

?

ij

= inf

?

ij

Hence by Theorem 3.9, L is a semifield.

Conversely, let L be a semifield and let µ be a non constant fuzzy ideal of S.

Then by Proposition 3.3(i), µ+′is a non constant fuzzy ideal of L. Hence by

Theorem 3.9, µ+′(

?

i

j

?

ij

µ(x) = (µ+′)+(x) = inf

[xi,αi],

?

[xj,βj] ∈ L \ {0L}. Then µ(

?

i

[xi,αi]) = ((µ+)+′)(

?

i

[xi,αi])

s∈Sµ+(

xiαis) = inf

s∈Sµ+(

?

xjβjs) = ((µ+)+′)(

?

j

[xj,βj]) < µ(0L).

[xi,αi]) = µ+′(

?

[yj,αj]) < µ+′(0L) for all

[xi,αi],

?

[yj,αj] ∈ L \ {0L}. Now let x,y ∈ S \ {0S}. Then

γ∈Γµ+′([x,γ]) = inf

γ∈Γµ+′([y,γ]) = µ(y) < µ(0S). Hence S is

a Γ−semifield.

The following result is also a consequence of Theorem 3.4.

Theorem 3.12 Suppose S is a Γ−semiring with unities and n is a positive

integer. Then there exists an inclusion preserving bijection between the set of

all fuzzy ideals of S and the set of all fuzzy ideals of the matrix Γn−semiring

Sn.

Proof. Let L be the left operator semiring of S. Then by Theorem 5.2 of [2]

there is an inclusion preserving bijection between FI(L) and FI(Ln). In view

of Theorem 3.4, there is an inclusion preserving bijection between FI(Sn)

and FI([Sn,Γn]). Again by Proposition 2.14, FI([Sn,Γn]) and FI(Ln) are

isomorphic. Further, there is an inclusion preserving bijection between FI(S)

and FI(L)(cf. Theorem 3.4). Combining all these we obtain the theorem.

Remark. The above theorem can also be obtained directly via the map-

ping µ ?→ µnwhere µ ∈ FI(S) and µn∈ FI(Sn) and µnis defined by

µn([aij]) = min[µ(aij) : aij∈ [aij];1 ≤ i,j ≤ n].

Page 12

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Sujit Kumar Sardar, Y. B. Jun and Sarbani Goswami

Remark 3.13 Throughout the paper we defined the concepts and obtained

results for left operator semiring. Similar concepts and results can be obtained

for right operator semiring R of S by using analogy between L and R.

Remark 3.14 From Theorem 3.4 and its dual (cf. Remark 3.13) for right

operator semiring R we easily obtain lattice isomorphism of all fuzzy ideals of

R and that of L.

Concluding Remark. It is well known that operator semirings of a Γ−semiring

are very effective in the study of Γ−semirings. Lemma 3.6, Lemma 3.7, Theo-

rems 3.8 and 3.12 illustrate that operator semirings of a Γ−semiring can also

be made effective in the study of Γ−semiring in terms of fuzzy subsets.

References

[1] Dutta, T. K. and Biswas, B. K.: Fuzzy Prime Ideals Of A Semiring; Bull.

Malaysian Math. Soc.(Second Series) 17 (1994), 9-16.

[2] Dutta, T. K. and Biswas, B. K.: Structure of fuzzy ideals of semirings;

Bull. Calcutta Math. Soc. 89(4) (1997), 271-284.

[3] Dutta, T.K. and Sardar, S.K.:

Γ−semiring; Southeast Asian Bull. of Math. 26(2002), 203-213.

On the Operator Semirings of a

[4] Dutta, T.K. and Chanda, T.: Structures of Fuzzy Ideals of Γ−Ring; Bull.

Malays. Math. Sci. Soc. (2) 28(1) (2005), 9-18.

[5] Dutta, T.K. and Sardar, S.K.: On Matrix Γ−semirings; Far East J.Math.

Sci.(FJMS) 7(1) (2002), 17-31.

[6] Dutta, T.K., Sardar, S.K. and Goswami, S.: An introduction to fuzzy

ideals of Γ−semirings; (To appear) Proceedings of National Seminar on

Algebra, Analysis and Discrete Mathematics., University of Kerala, India.

[7] Dutta, T.K., Sardar, S.K. and Goswami, S.: Operations on fuzzy ideals of

Γ−semirings; Communicated.

[8] Golan, J.S.: Semirings and their applications, Kluwer Academic Publish-

ers,1999.

[9] Rao, M.M.K.: Γ−semiring-1; Southeast Asian Bull. of Math. 19 (1995),

49-54.

[10] Sardar, S.K.: On Γ−semifields; Communicated.

[11] Zadeh, L.A.: Fuzzy sets; Information and Control 8( 1965 ), 338-353.

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