Abstract
Fuzzy Inf. Eng. (2012) 3: 261-272 DOI 10.1007/s12543-012-0115-z ORIGINAL ARTICLE Role of Operator Semirings in Characterizing Γ−semirings in Terms of Fuzzy Subsets Sujit Kumar Sardar · Sarbani Goswami · Y. B. Jun Received: 4 July 2011/ Revised: 10 July 2012/ Accepted: 1 August 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The operator semirings of a Γ-semiring have been brought into use to studyΓ-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 semiring such as lattice isomorphism between the sets of fuzzy ideals of a Γ-semiring and its operator semirings. Keywords 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. Be- sides, it also provides an algebraic home to the nonpositive cones of the totally or- dered rings (It may be recalled here that the nonnegative cones of the totally ordered rings form semirings but the nonpositive cones cannot form semirings as the multi- plication is no longer deﬁned). Γ-semiring theory has been enriched with the help of operator semirings of aΓ-semiring by Dutta and Sardar [3]. To make operator semir- ings effective in the study of Γ-semirings, Dutta et al [3] established correspondence between the ideals of aΓ-semiring 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 Sujit Kumar Sardar () Department of Mathematics, Jadavpur University, Kolkata-700032, India email: sksardarjumath@gmail.com Sarbani Goswami () Lady Brabourne College, Kolkata, W.B., India email: sarbani7 goswami@yahoo.co.in Y. B. Jun () Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Korea email: skywine@gmail.com 262 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) 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 aΓ-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 matrixΓ -semiring S . n n 2. Preliminaries We recall the following deﬁnitions and results for their use in the sequel. Deﬁnition 1 [11] Let S be a nonempty set. A mappingμ : S → [0, 1] is called a fuzzy subset of S. Deﬁnition 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) 0 αx = 0 = xα0 ; S S S (ii) x0 y = 0 = y0 x for all x, y ∈ S and for all α ∈ Γ, Γ S 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 0 . Example 1 [3] Let S be the additive commutative semigroup of all m × n matrices over the set of all non-negative integers andΓ be the additive commutative semigroup of all n× m matrices over the same set. Then S forms aΓ-semiring if xαy denotes the usual matrix multiplication of x,α, y where x, y ∈ S andα ∈ Γ. Deﬁnition 3 [3] Let SbeaΓ−semiring and F be the free additive commutative semi- group generated by S ×Γ. Then the relationρ on F, deﬁned by m n m n (x,α )ρ (y ,β ) if and only if xα a = y β a i i j j i i j j i=1 j=1 i=1 j=1 for all a ∈ S (m, n ∈ Z ), is a congruence on F. The congruence class containing m m (x,α ) is deﬁned by [x,α ]. Then F/ρ, the set of congruence classes, is an ad- i i i i i=1 i=1 ditive commutative semigroup and this also forms a semiring with the multiplication deﬁned by m n ( [x,α ])( [y ,β ]) = [xα y ,β ]. i i j j i i j j i=1 j=1 i, j This semiring is denoted by L and called the left operator semiring of theΓ−semiring S. Fuzzy Inf. Eng. (2012) 3: 261-272 263 Dually the right operator semiring R of theΓ−semiring S is deﬁned. Deﬁnition 4 [3] LetSbea Γ−semiring and L be the left operator semiring and R m n be the right one. If there exists an element [e,δ ] ∈ L [γ , f ] ∈ R such i i j j i=1 j=1 m n that eδ a = a aγ f = a for all a ∈ S , then S is said to have the left unity i i j j i=1 j=1 m n [e,δ ] (resp. the right unity [γ , f ]). i i j j i=1 j=1 Deﬁnition 5 [3] LetSbea 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 m ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ Q := [x,α ] ∈ L : [x,α ] S ⊆ Q , ⎪ ⎪ i i i i ⎪ ⎪ ⎩ ⎭ i=1 i=1 where [x,α ] S denotes the set of all ﬁnite sums xα s , s ∈ S and i i i i k k i=1 i,k ⎧ ⎫ m m ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ Q := [α, x ] ∈ R : S ([α, x ]) ⊆ Q , ⎪ i i i i ⎪ ⎪ ⎪ ⎩ ⎭ i=1 i=1 where S [α, x ] denotes the set of all ﬁnite sums s α x, s ∈ S. i i k i i k i=1 i,k Deﬁnition 6 [3] A Γ−semiring S is said to be zero-divisor free (ZDF) if aαb = 0 implies that either a = 0 orα = 0 or b = 0 for a, b ∈ S,α ∈ Γ. Deﬁnition 7 [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 exist b ∈ S,β ∈ Γ such that aαbβd = d for all d ∈ S. + 2 + + Example 2 [10] Let S = {rω : r ∈ Q ∪{0}} andΓ= {rω : r ∈ Q ∪{0}}, where Q is the set of all positive rational numbers. Then S forms a Γ-semiﬁeld. Deﬁnition 8 [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 9 [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. 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 the set of all fuzzy left ideals, the set of all fuzzy right ideals and the set of all fuzzy ideals 264 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) of theΓ-semiring S . Similar is the meaning of FLI(L), FRI(L), FI(L), where L is the left operator semiring of theΓ−semiring S . Also throughout we assume thatμ(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 10 [7] LetSbeaΓ-semiring andμ ,μ ∈ FLI(S)[FRI(S ), FI(S )]. Then 1 2 the sum μ ⊕μ ofμ andμ is deﬁned as follows: 1 2 1 2 sup [min[μ (u),μ (v)]], ⎪ 1 2 x=u+v (μ ⊕μ )(x) = 1 2 ⎪ 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. Proposition 1 [6] Let I be a non-empty subset of a Γ−semiring S and λ be the characteristic function of I. Then I is a left ideal (right ideal, ideal) of S if and only ifλ is a fuzzy left ideal (resp. fuzzy right ideal, fuzzy ideal) of S . Deﬁnition 11 [5] Let S be aΓ-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 S and Γ respectively. Let A, B ∈ S and Δ ∈ Γ . Then AΔB ∈ S and A+ B ∈ S . n n n n n n Clearly, S forms a Γ -semiring with these operations. This is called the matrix Γ- n n semiring over S or the matrix Γ -semiring S or simply theΓ -semiring S . n n n n The right operator semiring of the matrix Γ -semiring S is denoted by [Γ , S ] n n n n and the left one by [S ,Γ ]. If x ∈ S , the notation xE will be used to denote a matrix n n ij in S with x in the (i, j)-th entry and zeros elsewhere. αE ∈ Γ , where α ∈ Γ, will n ij n have a similar meaning. If P ⊆ S, P will denote the set of all n × n matrices with entries from P. Similar is the meaning of Δ , where Δ ⊆ Γ. Proposition 2 [5] LetSbea Γ-semiring and R (resp. L) be its right (resp. left) operator semiring. Let R (resp. L ) denote the semiring of all n× n matrices over R n n (resp. L). Then (i) The right operator semiring [Γ , S ] of the Γ -semiring S is isomorphic with R n n n n n p p n i i i i via the mapping [[γ ], [x ]] → ( [γ , x ]) . 1≤ j, v≤n uv tv jk jt i=1 i=1 t=1 (ii) The left operator semiring [S ,Γ ] of the Γ -semiring S is isomorphic with L n n n n n p p n i i i i via the mapping [[x ], [γ ]] → ( [x ,γ ]) . 1≤u, k≤n uv ut jk tk i=1 i=1 t=1 In view of the above proposition, we henceforth identify [S ,Γ ] and R ;[S ,Γ ] n n n n n and L . 3. Corresponding Fuzzy Ideals and Their Applications in Γ-semirings Throughout this section, S denotes aΓ-semiring and L denotes the left operator semir- ing of the Γ-semiring S . Fuzzy Inf. Eng. (2012) 3: 261-272 265 + + Deﬁnition 12 Letμ be a fuzzy subset of L, we deﬁne a fuzzy subsetμ of S byμ (x) = inf μ([x,γ]), where x ∈ S. If σ is a fuzzy subset of S, we deﬁne a fuzzy subset σ of γ∈Γ Lbyσ [x,α ] = inf σ xα s where [x,α ] ∈ L. i i i i i i s∈S i i i By routine veriﬁcation, we obtain the following lemma: + + Lemma 1 If{μ : i ∈ I} is a collection of fuzzy subsets of L, then μ = ( μ ) . i i i∈I i∈I Proposition 3 Suppose σ,σ ,σ ∈ FI(S ) andμ ∈ FI(L). Then 1 2 (i)σ ∈ FI(L), + + (ii) (σ ) = σ, + + (iii) σ σ implies that σ σ , 1 2 1 2 + + + (iv) (σ ⊕σ ) = σ ⊕σ , 1 2 1 2 + + + (v) (σ ∩σ ) = σ ∩σ , 1 2 1 2 + + (vi)σ ⊆ σ implies thatσ ⊆ σ , 1 2 1 2 (vii)μ ∈ FI(S ). + + (viii) (μ ) = μ, + + (ix)μ ⊆ μ implies that μ ⊆ μ . 1 2 1 2 + + (x) If σ is non-constant, then σ is non-constant and if μ is non-constant, then μ is non-constant. Proof (i) Let σ ∈ FI(S ). Then σ(0) = 1. Letγ ∈ Γ. Then σ ([0,γ]) = inf[σ(0γs)] = σ(0) = 1,γ ∈ Γ. s∈S + + 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 [x,α ], [y ,β ] ∈ L. Then i i j j i j σ ( [x,α ]+ [y ,β ]) = inf[σ( xα s+ y β s)] i i j j i i j j s∈S i j i j ≥ inf[min[σ( xα s),σ( y β s)]] i i j j s∈S i j = min[inf[σ( xα s)], inf[σ( y β s)]] i i j j s∈S s∈S i j + + = min[σ ( [x,α ]),σ ( [y ,β ])]. i i j j i j Again + + σ ( [x,α ] [y ,β ]) = σ ( [xα y ,β ]) i i j j i i j j i j i, j = infσ( xα y β s) i i j j s∈S i, j ≥ [min[σ( xα y ),σ( xα y ),σ( xα y ),··· ]] i i 1 i i 2 i i 3 i i i ≥ inf[σ( (xα s))] = σ ( [x,α ]). i i i i s∈S i i 266 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) Similarly, we can show that + + σ ( [x,α ] [y ,β ]) ≥ σ ( [y ,β ]). i i j j j j i j j Consequently, σ ∈ FI(L). (ii) Let x ∈ S . Then + + + ((σ ) )(x) = inf[σ ([x,γ])] = inf[inf[σ(xγs)]] ≥ σ(x). γ∈Γ γ∈Γ s∈S + + Henceσ ⊆ (σ ) . Let [γ, f ] be the right unity of S . Then xγ f = x for all x ∈ S . i i i i i i Now σ(x) = σ( xγ f ) i i ≥ min[σ(xγ f ),σ(xγ f ),··· ] 1 1 2 2 + + ≥ inf[inf[σ(xγs)]] = (σ ) (x). γ∈Γ s∈S + + + + Therefore (σ ) ⊆ σ and hence (σ ) = σ. + + + + + + (iii) Let σ σ . If possible, let σ = σ . Then (σ ) = (σ ) , i.e., σ = σ , (by 1 2 1 2 1 2 1 2 + + (ii)), which contradicts our assumption. Hence σ σ . 1 2 (iv) Let [a,α ] ∈ L. Then i i ((σ ⊕σ ) )( [a,α ]) = inf(σ ⊕σ )( aα s) 1 2 i i 1 2 i i s∈S i i = inf[sup[min[σ ( x δ s),σ ( y β s)] : 1 k k 2 j j s∈S aα s = x δ s+ y β s]] i i k k j j i j = sup[min[infσ ( x δ s), infσ ( y β s)]] 1 k k 2 j j s∈S s∈S k j + + = sup[min[σ ( [x ,δ ]),σ ( [y ,β ])]] k k j j 1 2 k j + + = (σ ⊕σ )( [a,α ]). i i 1 2 + + + Thus (σ ⊕σ ) = σ ⊕σ . 1 2 1 2 (v) Let [a,α ] ∈ L. Then i i (σ ∩σ ) ( [a,α ]) = inf[(σ ∩σ )( aα s)] 1 2 i i 1 2 i i s∈S i i Fuzzy Inf. Eng. (2012) 3: 261-272 267 = inf[min[σ ( aα s),σ ( aα s)]] 1 i i 2 i i s∈S i i = min[infσ ( aα s), infσ ( aα s)] 1 i i 2 i i s∈S s∈S i i + + = min[σ ( [a,α ]),σ ( [a,α ])] i i i i 1 2 i i + + = (σ ∩σ )( [a,α ]). i i 1 2 + + + Hence (σ ∩σ ) = σ ∩σ . 1 2 1 2 (vi) Let σ ,σ ∈ FI(S ) be such that σ ⊆ σ . Then 1 2 1 2 + + σ ( [x,α ]) = inf[σ ( xα s)] ≤ inf[σ ( xα s)] = σ ( [x,α ]) i i 1 i i 2 i i i i 1 2 s∈S s∈S i i i i for all [x,α ] ∈ L. i i + + Thusσ ⊆ σ . 1 2 (vii) Let μ ∈ FI(L). Then μ(0) = 1. Now μ (0) = inf[μ([0,γ])] = 1. Since for all γ ∈ Γ, [0,γ] is the zero element of γ∈Γ L. This also shows that μ is non-empty. Let x, y ∈ S and α ∈ Γ. Then μ (x+ y) = inf[μ([x+ y,γ])] γ∈Γ = inf[μ([x,γ]+ [y,γ])] ≥ inf[min[μ([x,γ]),μ([y,γ])]] = min[inf[μ([x,γ])], inf[μ([y,γ])] γ∈Γ γ∈Γ + + = min[μ (x),μ (y)]. + + + Therefore μ (x+ y) ≥ min[μ (x),μ (y)]. Again μ (xαy) = inf[μ([xαy,γ])] = inf[μ([x,α][y,γ])] γ∈Γ γ∈Γ ≥ infμ[y,γ] = μ (y) γ∈Γ and μ (xαy) = inf[μ([xαy,γ])] = inf[μ([x,α][y,γ])] γ∈Γ γ∈Γ ≥ μ([x,α]) ≥ inf[μ([x,δ])] = μ (x). δ∈Γ Consequently,μ ∈ FI(S ). (viii) Let μ ∈ FI(L). Then for [x,α ] ∈ L, i i + + + ((μ ) )( [x,α ]) = inf[μ ( xα s)] i i i i s∈S i i 268 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) = inf[inf[μ( [xα s,γ])]] i i s∈S γ∈Γ = inf[inf[μ( [x,α ][s,γ])]] i i s∈S γ∈Γ ≥ μ( [x,α ]). i i + + Henceμ ⊆ (μ ) . Let [e,δ ] be the left unity of S. Then i i μ( [x ,α ]) = μ( [x ,α ] [e,δ ]) j j j j i i j j i ≥ min[μ( [x ,α ][e ,δ ]),μ( [x ,α ][e ,δ ]),··· ] j j 1 1 j j 2 2 j j ≥ inf[inf[μ( [x ,α ][s,γ])]] j j s∈S γ∈Γ + + = (μ ) ( [x ,α ]). j j + + + + Therefore (μ ) ⊆ μ and so (μ ) = μ. (ix) Proof is similar to that of (vi). (x) Let σ be a non-constant fuzzy ideal of S.Ifσ is constant, then σ ( [x,α ]) = 1 for all [x,α ] ∈ L. i i i i i i Then + + + σ(x) = (σ ) (x) = infσ ([x,γ]) = 1 for all x ∈ S , γ∈Γ whereσ is constant. This contradicts thatσ is non-constant. Consequently,σ is non-constant. + + Next, let μ be a nonconstant fuzzy ideal of L.If μ is constant, then μ (x) = 1 for all x ∈ S asμ (0 ) = 1. Suppose [x,α ] ∈ L. Then S i i + + + μ( [x,α ]) = ((μ ) )( [x,α ]) = infμ ( xα s) = 1 i i i i i i s∈S i i i for all [x,α ] ∈ L. This implies that μ is constant, which contradicts our assump- i i tion. This completes the proof. 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. Theorem 1 The lattices of all fuzzy ideals [fuzzy right ideals] of S and L are isomor- phic via the inclusion preserving bijectionσ → σ , whereσ ∈ FI(S ) [resp. FRI(S)] and σ ∈ FI(L) [resp. FRI(L)]. Fuzzy Inf. Eng. (2012) 3: 261-272 269 Corollary 1 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]. Lemma 2 Let I be an ideal (left ideal, right ideal) of a Γ−semiring S and λ be the + + characteristic function of I. Then (λ ) = λ + . Moreover, I is an ideal of L. + + Proof Let [x,α ] ∈ L. Then either [x,α ] ∈ I or [x,α ] I .If i i i i i i i i i [x,α ] ∈ I , thenλ + ( [x,α ]) = 1 and xα s ∈ I for all s ∈ S . Hence i i I i i i i i i i (λ ) ( [x,α ]) = infλ ( xα s) = 1. I i i I i i s∈S i i Again, if [x,α ] I , then λ ( [x,α ]) = 0 and xα s I for some i i i i i i i i i + + s ∈ S . Hence (λ ) ( [x,α ]) = infλ ( xα s) = 0. Thus we obtain (λ ) = λ + . I i i I i i I I s∈S i i Now by Proposition 3(i), (λ ) is a fuzzy ideal of L. Henceλ is a fuzzy ideal of L. Consequently, I is an ideal of L (cf. Lemma 2.3 [1]). Lemma 3 Let I be an ideal (left ideal, right ideal) of L and λ be the characteristic + + ﬁnction of I. Then (λ ) = λ . Moreover, I is an ideal of S . I I + + + Proof Let x ∈ S . Then either x ∈ I or x I .If x ∈ I , then λ (x) = 1. Again + + x ∈ I implies that [x,γ] ∈ I for allγ ∈ Γ. Hence (λ ) (x) = infλ ([x,γ]) = 1. I I γ∈Γ + + Again, if x I , then λ (x) = 0 and [x,γ] I for some γ ∈ Γ. Hence (λ ) (x) = I I infλ ([x,γ]) = 0. γ∈Γ + + Consequently, (λ ) = λ +. Now by Proposition 3(vii), (λ ) is a fuzzy ideal of S. I I I Hence I is an ideal of S (cf. Proposition 1). Remark The last parts of Lemma 2 and Lemma 3 are originally due to Dutta and Sardar [3]. These 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 1 and by using Lemma 2 and Lemma 3. Theorem 2 The lattices of all ideals (right ideals ) of S and L are isomorphic via the mapping I → I , where I denotes an ideal (right ideal) of S. Proof That I → I is a mapping which follows from Lemma 2. Let I and I be two 1 2 ideals of S such that I I . Thenλ andλ are fuzzy ideals of S , whereλ andλ 1 2 I I I I 1 2 1 2 are characteristic functions of I and I respectively (cf. Proposition 1). Evidently, 1 2 + + λ λ . Then by Theorem 1, λ λ . Hence by Lemma 2, λ + λ + whence I I 1 2 I I I I 1 2 1 2 + + + I I . Consequently, the mapping I → I is one-one. 1 2 270 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) Next let J be an ideal of L. Then λ is a fuzzy ideal of L (cf. Lemma 2.3 [1]). By + + + Proposition 3 (vii), (λ ) is a fuzzy ideal of S and by Theorem 1 we obtain ((λ ) ) = J J λ . Now by successive use of Lemma 3 and Lemma 2, we obtain λ + + = λ and J J (J ) + + consequently, (J ) = J. Hence the mapping is surjective. Let I , I be two ideals of S such that I ⊆ I . Then λ ⊆ λ . Hence by Theorem 1 2 1 2 I I 1 2 + + + + 1, λ ⊆ λ whence by Lemma 2, λ + ⊆ λ + . Consequently I ⊆ I . Thus the I I I I 1 2 1 2 1 2 mapping is inclusion preserving. This completes the proof. Theorem 3 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} andμ a non-constant fuzzy ideal of −1 −1 −1 M. Then μ(y) = μ(yxx ) ≥ μ(xx ) ≥ μ(x), where x is 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. Letλ be the characteristic function of I. Thenλ (x) = 0 1 = λ (0). This implies I I I that λ is a non-constant fuzzy ideal of M. Suppose y( 0) ∈ I. Then by hypothesis, λ (x) = λ (y)butλ (x) = 0 andλ (y) = 1. Thus we get a contradiction. Hence I = M. I I I I Thus M has no non-zero proper ideals. Consequently, M is a semiﬁeld [8]. Theorem 4 A ZDF commutativeΓ-semiring S is aΓ-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 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 and by using Theorem 9.7 of [3]. Using the above theorem and Proposition 3 (hence Theorem 1), we give a new proof of the following result of Dutta and Sardar [3]. Theorem 5 Let S be 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 be a Γ-semiﬁeld. Let μ be a non-constant fuzzy ideal of L. Then by Proposition 3 (vii), μ is a non-constant fuzzy ideal of S. Hence by Theorem 4, Fuzzy Inf. Eng. (2012) 3: 261-272 271 + + + μ (x) = μ (y)<μ (0 ) for all x, y ∈ S \{0 }. Let S S [x,α ], [x ,β ] ∈ L\{0 }. i i j j L i j Then + + + μ( [x,α ]) = ((μ ) )( [x,α ]) = infμ ( xα s) i i i i i i s∈S i i i + + + = infμ ( x β s) = ((μ ) )( [x ,β ])<μ(0 ). j j j j L s∈S j j Hence by Theorem 3, 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 (i), μ is a non-constant fuzzy ideal of L. Hence by Theorem 3, + + + μ ( [x,α ]) = μ ( [y ,α ])<μ (0 ) i i j j L i j for all [x,α ], [y ,α ] ∈ L\{0 }. Now let x, y ∈ S \{0 }. Then i i j j L S i j + + μ(x) = (μ ) (x) + + = infμ ([x,γ]) = infμ ([y,γ]) γ∈Γ γ∈Γ = μ(y)<μ(0 ). Hence S is a Γ-semiﬁeld. The following result is also a consequence of Theorem 1. Theorem 6 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 matrixΓ -semiring S . n n 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(L ). In view of Theorem 1, there is an inclusion preserving bijection between FI(S ) and FI([S ,Γ ]). Again by n n n Proposition 2, FI([S ,Γ ]) and FI(L ) are isomorphic. Further, there is an inclusion n n n preserving bijection between FI(S) and FI(L) (cf. Theorem 1). Combining all these, we obtain the theorem. Remark The above theorem can also be obtained directly via the mapping μ → μ , whereμ ∈ FI(S ) andμ ∈ FI(S ) andμ is deﬁned by n n n μ ([a ]) = min[μ(a ): a ∈ [a ]; 1 ≤ i, j ≤ n]. n ij ij ij ij Remark 1 Throughout the paper, we deﬁne the concepts and obtain 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. 272 Sujit Kumar Sardar· Sarbani Goswami· Y. B. Jun (2012) Remark 2 From Theorem 1 and its dual (cf. Remark ) for right operator semiring R, we easily obtain lattice isomorphism of all fuzzy ideals of R and that of L. 4. Conclusion It is well known that operator semirings of aΓ-semiring are very effective in the study of Γ-semirings. Besides, Lemma 2, Lemma 3, Theorems 2 and 6 illustrate that it can also be made effective in the study of Γ-semiring in terms of fuzzy subsets. Acknowledgments The authors are thankful to the learned referees for their valuable comments and suggestions for improving the paper. References 1. Dutta T K, Biswas B K (1994) Fuzzy prime ideals of a semiring. Bull. Malaysian Math. Soc. (Second Series) 17: 9-16 2. Dutta T K, Biswas B K (1997) Structure of fuzzy ideals of semirings. Bull. Calcutta Math. Soc. 89(4): 271-284 3. Dutta T K, Sardar S K (2002) On the operator semirings of a Γ-semiring. Southeast Asian Bull. of Math. 26: 203-213 4. Dutta T K, Chanda T (2005) Structures of fuzzy ideals of Γ-Ring. Bull. Malays. Math. Sci. Soc. (2) 28(1): 9-18 5. Dutta T K, Sardar S K (2002) On matrixΓ-semirings. Far East J.Math. Sci.(FJMS) 7(1): 17-31 6. Dutta T K, Sardar S K, Goswami S (2008) An introduction to fuzzy ideals of Γ-semirings. Pro- ceedings of National Seminar on Algebra, Analysis and Discrete Mathematics. University of Kerala, India: 47-58 7. Dutta T K, Sardar S K, Goswami S (2011) Operations on fuzzy ideals ofΓ-semirings. Communicated 8. Golan J S (1999) Semirings and their applications. Kluwer Academic Publishers 9. RaoMMK (1995)Γ-semiring-I. Southeast Asian Bull. of Math. 19: 49-54 10. Sardar S K (2011) A note on Γ-semiﬁelds. International Mathematical Fourm 6(18): 875-879 11. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353
Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Sep 1, 2012
Keywords: Γ-semiring; Operator semiring; Fuzzy left(right) ideal; Fuzzy k -ideal; Γ-semifield