aa r X i v : . [ m a t h . G M ] S e p On Fuzzy Γ -Hypersemigroups R. Ameri † , R. Sadeghi ‡ † School of Mathematics, Statistic and Computer Sciences, College of Science,University of Tehran, P.O. Box 14155-6455, Tehran, Iran, E-mail: [email protected], ‡ Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University,Aliabad Katoul, Iran, e-mail: razi − sadeghi @ yahoo.com Abstract
We introduced and study fuzzy Γ-hypersemigroups, according to fuzzy semihyper-groups as previously defined [33] and prove that results in this respect. In this regardfirst we introduce fuzzy hyperoperation and then study fuzzy Γ-hypersemigroup. Wewill proceed by study fuzzy Γ-hyperideals and fuzzy Γ-bihyperideals. Also we studythe relation between the classes of fuzzy Γ-hypersemigroups and Γ-semigroups. Pre-cisely, we associate a Γ-hypersemigroup to every fuzzy Γ-hypersemigroup and viceversa. Finally, we introduce and study fuzzy Γ-hypersemigroups regular and fuzzystrongly regular relations of fuzzy Γ-hypersemigroups.
Keywords: fuzzy Γ-hyperoperation, Γ-hypersemigroup, fuzzy Γ-hypersemigroup, fuzzyΓ-hyperideals, fuzzy regular relation 1
Introduction
Hyperstructure theory was born in 1934 when Marty [31] defined hypergroups, beganto analysis their properties and applied them to groups. Algebraic hyperstructures area suitable generalization of classical algebraic structures. In 1986, M.K. Sen and Saha[34] defined the notion of a Γ-semigroup as a generalization of a semigroup. Ameri [12]introduced and study fuzzy ideals of gamma-hyperrings, after that Davvaz et. al. [26]studied Γ-semihypergroup as a generalization of a semihypergroup and then many classicalnotions of semigroups and semihypergroups have been extended to Γ-semihypergroups.Zadeh [35] introduced the notion of a fuzzy subset of a non-empty set X, as a functionfrom X to [0 , Definition 2.1.
Let M = { a, b, c, ... } and Γ = { α, β, γ, ... } be two non-empty sets. Then M is called a Γ-semigroup if there exists a mapping M × Γ × M −→ M written as( a, γ, b ) aγb satisfying the following identity2 aαb ) βc = aα ( bβc ) for all a, b, c ∈ M and for all α, β ∈ Γ . Let N be a non-empty subset of M . Then N is called a sub Γ-semigroup of M if aγb ∈ N for all a, b ∈ N and γ ∈ Γ.Let H be a non-empty set and let P ∗ ( H ) be the set of all non-empty subsets of H . Ahyperoperation on H is a map ◦ : H × H −→ P ∗ ( H ) and the couple ( H, ◦ ) is calleda hypergroupoid. If A and B are non-empty subsets of H , then we denote A ◦ B = S a ∈ A,b ∈ B a ◦ b , x ◦ A = { x } ◦ A and A ◦ x = A ◦ { x } . Definition 2.2.
A hypergroupoid ( H, ◦ ) is called a semihypergroup if for all x, y, z of H we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) which means that S u ∈ x ◦ y u ◦ z = S v ∈ y ◦ z x ◦ v A semihypergroup ( h, ◦ ) is called a hypergroup if for all x ∈ H , we have x ◦ H = H ◦ x = H . Definition 2.3.
Let M and Γ be two non-empty sets. M is called a Γ-semihypergroupif every γ ∈ Γ be a hyperoperation on M , i.e, xγy ⊆ M for every x, y ∈ M , and for every α, β ∈ Γ and x, y, z ∈ M we have xα ( yβz ) = ( xαy ) βz. If every γ ∈ Γ is an operation, then M is a Γ-semigroup.Let A and B be two non-empty subsets of M and γ ∈ Γ we define:
AγB = ∪ { aγb | a ∈ A, b ∈ B } . Also A Γ B = ∪ { aγb | a ∈ A, b ∈ B and γ ∈ Γ } = S γ ∈ Γ AγB. Fuzzy Γ -hyperoperations Definition 3.1.
Let M and Γ be two non-empty sets. F ( M ) denote the set of all fuzzysubsets of M . A fuzzy Γ-hyperoperation on M is a mapping ◦ : M × Γ × M −→ F ( M )written as ( a, γ, b ) a ◦ γ ◦ b . M together with a fuzzy Γ-hyperoperation is called afuzzy Γ-hypergroupoid. Definition 3.2.
A fuzzy Γ-hypergroupoid ( M, ◦ ) is called a fuzzy Γ-hypersemigroup iffor all a, b, c ∈ M, α, β ∈ Γ , ( a ◦ α ◦ b ) ◦ β ◦ c = a ◦ α ◦ ( b ◦ β ◦ c ) where for any fuzzysubset µ of M ( a ◦ α ◦ µ )( r ) = ∨ t ∈ M (( a ◦ α ◦ t )( r ) ∧ µ ( t )) , µ = 00 , otherwise and ( µ ◦ α ◦ a )( r ) = ∨ t ∈ M (( µ ( t ) ∧ ( t ◦ α ◦ a )( r )) , µ = 00 , otherwise Definition 3.3.
Let µ, ν be two fuzzy subsets of a fuzzy Γ-hypergroupoid ( M, ◦ ), then wedefine µ ◦ γ ◦ ν by ( µ ◦ γ ◦ ν )( t ) = ∨ p,q ∈ M ( µ ( p ) ∧ ( p ◦ γ ◦ q )( t ) ∧ ν ( q )), for all t ∈ M, γ ∈ Γ. Definition 3.4.
A fuzzy Γ-hypersemigroup ( M, ◦ ) is called a fuzzy Γ-hypergroup if x ◦ γ ◦ M = M ◦ γ ◦ x = χ M , for all x ∈ M, γ ∈ Γ. Example 3.5.
Define a fuzzy Γ-hyperoperation on a non-empty set M by a ◦ γ ◦ b = χ { a,γ,b } , where χ { a,γ,b } denotes the characteristic function of the set { a, γ, b } , for all a, b ∈ M, γ ∈ Γ, or a ◦ γ ◦ b = χ { a,b } . Then ( M, ◦ ) is a fuzzy Γ-hypersemigroup. Example 3.6.
Let M be a Γ-semigroup (Γ-semihypergroup). Define a fuzzy Γ-hyperoperationon M by a ◦ γ ◦ b = χ aγb , for all a, b ∈ M, γ ∈ Γ, where χ aγb is the characteristic functionof the element M , then ( M, ◦ ) is a fuzzy Γ-hypersemigroup. If M be a Γ-group then( M, ◦ ) is a fuzzy Γ-hypergroup. 4 xample 3.7. Let M be a Γ-semigroup and µ = 0 be a fuzzy Γ-semigroup on M . Let a, b ∈ M and γ ∈ Γ. Define a fuzzy Γ-hyperoperation ◦ on M by( a ◦ γ ◦ b )( t ) = µ ( a ) ∧ µ ( b ) , if t = aγb , otherwise. Then ( M, ◦ ) is a fuzzy Γ-hypersemigroup. Example 3.8.
For each positive integer n consider the set X n = {−∞ , , , ..., n } . Definefuzzy Γ-hyperoperation on X n and non-empty set Γ by a ◦ γ ◦ b = χ max { a,b } for all a, b ∈ X n , γ ∈ Γ and −∞ is assumed to satisfying the conditions that −∞ ≤ a for all a ∈ X n and −∞ ◦ γ ◦ a = −∞ . This gives X n the structure of a fuzzy Γ-hypersemigroup. Proof.
Let a, b, c ∈ X n , α, β ∈ Γ. (( a ◦ α ◦ b ) ◦ β ◦ c )( t ) = ( χ max { a,b } ◦ β ◦ c )( t ) = W r ∈ X n ( χ max { a,b } ( r ) ∧ ( r ◦ β ◦ c )( t ))= ( r ◦ β ◦ c )( t ) , r = max { a, b } , otherwise. = χ max { r,c } ( t ) , r = max { a, b } , otherwise. = , t = max { a, b, c } , otherwise. Similarly, we can show that( a ◦ α ◦ ( b ◦ β ◦ c ))( t ) = , t = max { a, b, c } , otherwise. In this example, if we consider fuzzy Γ-hyperoperation on X n by5 a ◦ γ ◦ b )( t ) = / , t = min { a + b, n } , otherwise, where + is integers additive operation, then ( X n , ◦ ) is a fuzzy Γ-hypersemigroup. (cid:3) Example 3.9.
Let S ( M ) be set of all subsets nonempty set M and Γ be a non-emptyset. Define fuzzy Γ-hyperoperation ◦ on S ( M ) by( A ◦ γ ◦ B )( C ) = / , C ⊆ A ∪ B , otherwise. Let M, Γ be nonempty sets, and M endowed with a fuzzy Γ-hyperoperation ◦ and for all a, b ∈ M, γ ∈ Γ, consider the p -cuts ( a ◦ γ ◦ b ) p = { t ∈ M : ( a ◦ γ ◦ b )( t ) ≥ p } of a ◦ γ ◦ b ,where p ∈ [0 , p ∈ [0 , M : a ◦ p γ ◦ p b =( a ◦ γ ◦ b ) p . Theorem 3.10.
For all a, b, c, u ∈ M and α, β ∈ Γ and for all p ∈ [0 ,
1] the followingequivalence holds:( a ◦ α ◦ ( b ◦ β ◦ c )) ≥ p ⇐⇒ u ∈ a ◦ p α ◦ p ( b ◦ p β ◦ p c ) . Theorem 3.11. ( M, ◦ ) is a fuzzy Γ-hypersemigroup if and only if ∀ p ∈ [0 , , ( M, ◦ p ) isa Γ-hypersemigroup. Proof.
It is obvious. (cid:3) . Theorem 3.12.
For all a ∈ M , the following equivalence holds: a ◦ γ ◦ M = χ M ⇐⇒ ∀ p ∈ [0 , , a ◦ p γ ◦ p M = M. Proof. If a ◦ γ ◦ M = χ M , then for all t ∈ M and p ∈ [0 , W u ∈ M ( a ◦ γ ◦ u )( t ) =1 ≥ p , whence there exists m ∈ M such that ( a ◦ γ ◦ m )( t ) ≥ p , which means that t ∈ a ◦ p γ ◦ p m . Hence, ∀ p ∈ [0 , , a ◦ p γ ◦ p M = M . Conversely, for p = 1 we have6 ◦ γ ◦ M = M , whence for all t ∈ M , there exists u ∈ M , such that t ∈ a ◦ γ ◦ u ,which means that ( a ◦ γ ◦ u )( t ) = 1. In other words, a ◦ γ ◦ M = χ M . (cid:3) Theorem 3.13.
Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup. Then χ a ◦ γ ◦ χ b = a ◦ γ ◦ b ,for all a, b ∈ M, γ ∈ Γ. Proof. ( χ a ◦ γ ◦ χ b )( t ) = ∨ p ∈ M ( χ a ( p ) ∧ ( p ◦ γ ◦ χ b )( t )) = ( a ◦ γ ◦ χ b )( t ) = ∨ q ∈ M (( a ◦ γ ◦ q )( t ) ∧ χ b ( q )) = ( a ◦ γ ◦ b )( t ), for all t ∈ M, γ ∈ Γ. Therefore χ a ◦ γ ◦ χ b = a ◦ γ ◦ b , for all a, b ∈ M, γ ∈ Γ. Hence the result. (cid:3)
Theorem 3.14.
Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup. Then( i ) a ◦ α ◦ ( b ◦ β ◦ µ ) = ( a ◦ α ◦ b ) ◦ β ◦ µ, for all a, b ∈ M, α, β ∈ Γ and for all µ ∈ F ( M ).( ii ) a ◦ α ◦ ( µ ◦ β ◦ b ) = ( a ◦ α ◦ µ ) ◦ β ◦ b, for all a, b ∈ M, α, β ∈ Γ and for all µ ∈ F ( M ).( iii ) µ ◦ α ◦ ( a ◦ β ◦ b ) = ( µ ◦ α ◦ a ) ◦ β ◦ b ,for all a, b ∈ M, α, β ∈ Γ and for all µ ∈ F ( M ).( iv ) µ ◦ α ◦ ( a ◦ β ◦ ν ) = ( µ ◦ α ◦ a ) ◦ β ◦ ν for all a ∈ M, α, β ∈ Γ and for all µ, ν ∈ F ( M ).( v ) a ◦ α ◦ ( µ ◦ β ◦ ν ) = ( a ◦ α ◦ µ ) ◦ β ◦ ν for all a ∈ M, α, β ∈ Γ and for all µ, ν ∈ F ( M ).( vi ) µ ◦ α ◦ ( ν ◦ β ◦ a ) = ( µ ◦ α ◦ ν ) ◦ β ◦ a for all a ∈ M, α, β ∈ Γ and for all µ, ν ∈ F ( M ).( vii ) µ ◦ α ◦ ( ν ◦ β ◦ δ ) = ( µ ◦ α ◦ ν ) ◦ β ◦ δ for all µ, ν, δ ∈ F ( M ). Proof.
It is straight forward. Γ -hyperideals Definition 4.1.
A fuzzy subset µ of a fuzzy Γ-hypersemigroup ( M, ◦ ) is called a fuzzy sub Γ -hypersemigroup of ( M, ◦ ) if µ ◦ γ ◦ µ ⊆ µ . Theorem 4.2.
If ( M, ◦ ) is a fuzzy Γ-hypersemigroup and µ, ν are two fuzzy sub Γ-hypersemigroups of ( M, ◦ ), then µ ∩ ν is also a fuzzy sub Γ-hypersemigroup of ( M, ◦ ). Definition 4.3.
A fuzzy subset µ of a fuzzy Γ-hypersemigroup is called a left fuzzy Γ -hyperideal if a ◦ γ ◦ µ ⊆ µ , for all a ∈ M, γ ∈ Γ.Similarly we can define a right fuzzy Γ -hyperideal of a fuzzy Γ-hypersemigroup ( M, ◦ ).7 heorem 4.4. A fuzzy subset µ of a fuzzy Γ-hypersemigroup ( M, ◦ ) is a left fuzzyΓ-hyperideal if and only if M ◦ γ ◦ µ ⊆ µ . Theorem 4.5.
Let µ and ν be two left fuzzy Γ-hyperideal of a fuzzy Γ-hypersemigroup( M, ◦ ), then µ ∪ ν and µ ∩ ν are also left fuzzy Γ-hyperideals of ( M, ◦ ). Theorem 4.6.
Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup. Then( i ) χ M is a left fuzzy Γ-hyperideal of ( M, ◦ ),( ii ) χ M ◦ γ ◦ m = M ◦ γ ◦ m , for all m ∈ M, γ ∈ Γ,( iii ) M ◦ γ ◦ m is a left fuzzy Γ-hyperideal of ( M, ◦ ), for all m ∈ M, γ ∈ Γ,( iv ) For any fuzzy subset µ = 0 on M , M ◦ γ ◦ µ is a left fuzzy Γ-hyperideal of ( M, ◦ ) forall γ ∈ Γ. Proof. ( i ) m ◦ γ ◦ χ M ⊆ χ M , for all m ∈ M, γ ∈ Γ.( ii ) ( χ M ◦ γ ◦ m )( t ) = ∨ p ∈ M ( χ M ( p ) ∧ ( p ◦ γ ◦ m )( t )) = ∨ p ∈ M ( p ◦ γ ◦ m )( t ) = ( M ◦ γ ◦ m )( t ),for all m ∈ M, γ ∈ Γ.( iii ) x ◦ α ◦ ( M ◦ γ ◦ m ) = x ◦ α ◦ ( χ M ◦ γ ◦ m ) = ( x ◦ α ◦ χ M ) ◦ γ ◦ m ⊆ χ M ◦ γ ◦ m = M ◦ γ ◦ m, for all x, a ∈ M, α, γ ∈ Γ.( iv ) x ◦ α ◦ ( M ◦ γ ◦ µ ) = x ◦ α ◦ ( χ M ◦ γ ◦ µ ) = ( x ◦ α ◦ χ M ) ◦ γ ◦ µ ⊆ M ◦ γ ◦ µ = M ◦ γ ◦ µ ,for all x ∈ M, α, γ ∈ Γ. (cid:3) Theorem 4.7. If µ is a left fuzzy Γ-hyperideal of a fuzzy Γ-suhhypersemigroups ( M, ◦ ),then( i ) µ ◦ γ ◦ m is a left fuzzy Γ-hyperideal of ( M, ◦ ), for all m ∈ M, γ ∈ Γ.( ii ) µ ◦ γ ◦ M is a left fuzzy Γ-hyperideal of ( M, ◦ ). Proof. ( i ) x ◦ α ◦ ( µ ◦ γ ◦ m ) = ( x ◦ α ◦ µ ) ◦ γ ◦ m ⊆ µ ◦ γ ◦ m , for all x ∈ M, α ∈ Γ.( ii ) x ◦ α ◦ ( µ ◦ γ ◦ M ) = x ◦ α ◦ ( µ ◦ γ ◦ χ M ) = ( x ◦ α ◦ µ ) ◦ γ ◦ χ M ⊆ µ ◦ γ ◦ χ M = µ ◦ γ ◦ M ,for all x ∈ M, α ∈ Γ. (cid:3) Definition 4.8. If µ = 0 is a fuzzy subset of a fuzzy Γ-semihypergroup ( M, ◦ ), then the8ntersection of all left fuzzy Γ-hyperideals of ( M, ◦ ) containing µ ( χ M itself being one suchis a left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ and contained in every other such leftfuzzy Γ-hyperideal of ( M, ◦ ). We call it the left fuzzy Γ-hyperideal of ( M, ◦ ) generatedby µ . Theorem 4.9. If µ = 0 is a fuzzy subset of a fuzzy Γ-hypersemigroup ( M, ◦ ), then µ ∪ ( M ◦ γ ◦ µ ) is the smallest left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ . Proof.
It is obvious that µ ∪ ( M ◦ γ ◦ µ ) is a left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ . Let ν be a left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ . Then µ ⊆ ν = ⇒ M ◦ γ ◦ µ ⊆ M ◦ γ ◦ ν ⊆ ν = ⇒ µ ∪ M ◦ γ ◦ µ ⊆ µ ∪ ν = ν .This shows that µ ∪ ( M ◦ γ ◦ µ ) is the smallest left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ . (cid:3) . Definition 4.10.
For any fuzzy subset µ = 0 of a fuzzy Γ-hypersemigroup of ( M, ◦ ), µ ∪ M ◦ γ ◦ µ is the smallest left fuzzy Γ-hyperideal of ( M, ◦ ) containing µ . It is calledthe left fuzzy Γ-hyperideal of ( M, ◦ ) generated by µ . Similarly we can define the rightfuzzy Γ-hyperideal of ( M, ◦ ) generated by µ . Definition 4.11.
A fuzzy sub Γ-hypersemigroup µ of a fuzzy Γ-hypersemigroup ( M, ◦ )is called a fuzzy Γ-hyper bi-ideal of ( M, ◦ ) if µ ◦ α ◦ y ◦ β ◦ µ ⊆ µ , for all y ∈ M, α, β ∈ Γ. Theorem 4.12. If µ is a fuzzy sub Γ-hypersemigroup of a fuzzy Γ-hypersemigroup ( M, ◦ ),then µ is also a fuzzy Γ-hyper bi-ideal of ( M, ◦ ) if and only if µ ◦ α ◦ M ◦ β ◦ µ ⊆ µ , forall α, β ∈ Γ. Proof.
Let µ be a fuzzy Γ-hyper bi-ideal of ( M, ◦ ). Then µ ◦ α ◦ y ◦ β ◦ µ ⊆ µ , forall y ∈ M . Now ( µ ◦ α ◦ M ◦ β ◦ µ )( t ) = ∨ x,y,z ∈ M ( x ◦ α ◦ y ◦ β ◦ z )( t ) ∧ µ ( x ) ∧ µ ( z ) = ∨ y ∈ M ( µ ◦ α ◦ y ◦ β ◦ µ )( t ) ≤ ∨ y ∈ M µ ( t ) = µ ( t ), for all t ∈ M, α, β ∈ Γ. Therefore µ ◦ α ◦ M ◦ β ◦ µ ⊆ µ . Conversely, we suppose µ ◦ α ◦ M ◦ β ◦ µ ⊆ µ . Then clearly µ ◦ α ◦ y ◦ β ◦ µ ⊆ µ , for all y ∈ M, α, β ∈ Γ. Hence the result. (cid:3) . heorem 4.13. If µ and ν are two fuzzy Γ-hyper bi-ideals of ( M, ◦ ), then µ ∩ ν is alsoa fuzzy Γ-hyper bi-ideal of ( M, ◦ ). Proof.
It is obvious. (cid:3) . Theorem 4.14. If µ is a fuzzy subset of a fuzzy Γ-hypersemigroup ( M, ◦ ) and ν be anyfuzzy Γ-hyper bi-ideal of ( M, ◦ ), then µ ◦ γ ◦ ν and ν ◦ γ ◦ µ are both fuzzy Γ-hyper bi-idealof ( M, ◦ ), for all γ ∈ Γ. Proof. ( µ ◦ γ ◦ ν ) ◦ α ◦ ( µ ◦ γ ◦ ν ) = µ ◦ γ ◦ ν ◦ α ◦ µ ◦ γ ◦ ν ⊆ µ ◦ γ ◦ ν ◦ α ◦ M ◦ γ ◦ ν ⊆ µ ◦ γ ◦ ν .This implies that ( µ ◦ γ ◦ ν ) is a fuzzy Γ-subhypersemigroup of ( M, ◦ ), for all α ∈ Γ.Also, ( µ ◦ γ ◦ ν ) ◦ α ◦ M ◦ β ◦ ( µ ◦ γ ◦ ν ) = µ ◦ γ ◦ ν ◦ α ◦ ( M ◦ β ◦ µ ) ◦ γ ◦ ν ⊆ µ ◦ γ ◦ ν ◦ α ◦ M ◦ γ ◦ ν ⊆ µ ◦ γ ◦ ν . Therefore, µ ◦ γ ◦ ν is a fuzzy Γ-hyper bi-ideal of ( M, ◦ ), for all α, β ∈ Γ. Similarly,we can show that ν ◦ γ ◦ µ is a fuzzy Γ-hyper bi-ideal of ( M, ◦ ). (cid:3) . Definition 4.15.
A fuzzy subset µ of a fuzzy Γ-hypersemigroup ( M, ◦ ) is called Γ-hyperinterior ideal of ( M, ◦ ) if x ◦ α ◦ µ ◦ β ◦ y ⊆ µ , for all x, y ∈ M, α, β ∈ Γ. Theorem 4.16.
A fuzzy subset µ of a fuzzy Γ-hypersemigroup ( M, ◦ ) is Γ-hyper interiorideal of ( M, ◦ ) if and only if M ◦ α ◦ µ ◦ β ◦ M ⊆ µ , for all α, β ∈ Γ.We can associate a Γ-hyperoperation on a fuzzy Γ-hypersemigroup ( M, ◦ ), as follows: ∀ a, b ∈ M, γ ∈ Γ , a ∗ γ ∗ b = { x ∈ M | ( a ◦ γ ◦ b )( x ) > } . Theorem 4.17.
If ( M, ◦ ) is a fuzzy Γ-hypersemigroup, then ( M, ∗ ) is a Γ-hypersemigroup.On the other hand, we can define a fuzzy Γ-hyperoperation on a Γ-hypersemigroup ( M, ∗ ),as follows: ∀ a, b ∈ M, γ ∈ Γ , a ◦ γ ◦ b = χ a ∗ γ ∗ b . Theorem 4.18.
If ( M, ∗ ) is a Γ-hypersemigroup, then ( M, ◦ ) is a fuzzy Γ-hypersemigroup.Denote by F HSG the class of all fuzzy Γ-hypersemigroups and by
HSG the class of allΓ-hypersemigroup. We define the following two maps: ϕ : HSG −→ F HSG , ϕ (( M, ∗ )) = ( M, ◦ ) , where for all a, b of M and γ ∈ Γ we have a ◦ γ ◦ b = χ a ∗ γ ∗ b and ψ : F HSG −→ HSG , ψ (( M, ◦ )) = ( M, ∗ ) , where for all a, b of M γ ∈ Γ, we have a ∗ γ ∗ b = { x | ( a ◦ γ ◦ b )( x ) > } . Definition 4.19. If µ , µ are fuzzy sets on M , then we say that µ is smaller than µ and we denote µ ≤ µ iff for all m ∈ M , we have µ ( m ) ≤ µ ( m ).Let f : M −→ M be a map. If µ is a fuzzy set on M , then we define f ( µ ) : M −→ [0 , f ( µ ))( t ) = W r ∈ f − ( t ) µ ( r ) if f − ( t ) = φ ,otherwise we consider ( f ( µ ))( t ) = 0 . Remark 4.20. If f : M −→ M is a map and m ∈ M , then f ( χ m ) = χ f ( m ) . Indeed, forall t ∈ M , we have( f ( χ m ))( t ) = W r ∈ f − ( t ) χ m ( r ) = f ( m ) = t otherwise. = χ f ( m ) ( t ) . We can introduce now the fuzzy Γ-hypersemigroup homomorphism notion, as follows:
Definition 4.21.
Let ( M , ◦ ) and ( M , ◦ ) be two fuzzy Γ-hypersemigroups and f : M −→ M be a map. We say that f is a homomorphism of fuzzy Γ-hypersemigroups iffor all a, b ∈ M, γ ∈ Γ, we have f ( a ◦ γ ◦ b ) ≤ f ( a ) ◦ γ ◦ f ( b ) . The following two theorems present two connections between fuzzy Γ-hypersemigrouphomomorphisms and Γ-hypersemigroup homomorphism.
Theorem 4.22.
Let ( M , ◦ ) and ( M , ◦ ) be two fuzzy Γ-hypersemigroups and ( M , ∗ ) = ψ ( M , ◦ ) , ( M , ∗ ) = ψ ( M , ◦ ) be the associated Γ-hypersemigroups. If f : M −→ M is a homomorphism of fuzzy Γ-hypersemigroups, then f is a homomorphism of the asso-ciated Γ-hypersemigroups, too. Proof . For all a, b ∈ M , γ ∈ Γ, we have f ( a ◦ γ ◦ b ) ≤ f ( a ) ◦ γ ◦ f ( b ). Let x ∈ a ∗ γ ∗ b ,which means that ( a ◦ γ ◦ b )( x ) > t = f ( x ). We have( f ( a ◦ γ ◦ b ))( t ) = W r ∈ f − ( t ) ( a ◦ γ ◦ b )( r ) ≥ ( a ◦ γ ◦ b )( x ) > , f ( a ) ◦ γ ◦ f ( b ))( t ) >
0. Hence t ∈ f ( a ) ∗ γ ∗ f ( b ). We obtain f ( a ∗ γ ∗ b ) ⊆ f ( a ) ∗ γ ∗ f ( b ). (cid:3) Theorem 4.23.
Let ( M , ∗ ) and ( M , ∗ ) be two Γ-hypersemigroups and ( M , ◦ ) = ϕ ( M , ∗ ) , ( M , ◦ ) = ϕ ( M , ∗ ) be the associated fuzzy Γ-hypersemigroups. The map f : M −→ M is a homomorphism of Γ-hypersemigroups iff it is a homomorphism offuzzy Γ-hypersemigroups. Proof . (= ⇒ ) Suppose that f is a homomorphism of Γ-hypersemigroups. Let a, b ∈ M, γ ∈ Γ. For all t ∈ Imf , we have( f ( a ◦ γ ◦ b ))( t ) = W r ∈ f − ( t ) ( a ◦ γ ◦ b )( r ) = W r ∈ f − ( t ) χ a ∗ γ ∗ b ( r )= a ∗ γ ∗ b ) ∩ f − ( t ) = φ otherwise = t ∈ f ( a ∗ γ ∗ b )0 otherwise = χ f ( a ∗ γ ∗ b ) ( t ) ≤ χ f ( a ) ∗ γ ∗ f ( b ) ( t ) = ( f ( a ) ◦ γ ◦ f ( b ))( t ). If t Imf , then ( f ( a ◦ γ ◦ b ))( t ) = 0 ≤ ( f ( a ) ◦ γ ◦ f ( b ))( t ). Hence, f ( a ◦ γ ◦ b ) ≤ f ( a ) ◦ γ ◦ f ( b ).Hence, f is a homomorphism of fuzzy Γ-hypersemigroups.( ⇐ =) Conversely, suppose that f is a homomorphism of fuzzy Γ-hypersemigroups and a, b ∈ M . Then, for all t ∈ M ,we have( f ( a ◦ γ ◦ b ))( t ) ≤ ( f ( a ) ◦ γ ◦ f ( b ))( t ),whence we obtain χ f ( a ∗ γ ∗ b ) ( t ) ≤ χ f ( a ) ∗ γ ∗ f ( b ) ( t ) , which means that f ( a ∗ γ ∗ b ) ⊆ f ( a ) ∗ γ ∗ f ( b ) . Hence f is a homomorphism of Γ-hypersemigroups.12 Fuzzy fuzzy (strongly) regular relations
In [12] fuzzy regular relations are introduced in the context of fuzzy hypersemigroups. Wedefine these relations on a fuzzy Γ-hypersemigroup:Let ρ be an equivalence relation on a fuzzy Γ-hypersemigroup ( M, ◦ ) and let µ, ν be twofuzzy subsets on M . We say that µρν if the following two conditions hold:(1) if µ ( a ) >
0, then there exists b ∈ M , such that ν ( b ) > aρb ;(2) if ν ( x ) >
0, then there exists y ∈ M , such that µ ( y ) > xρy .An equivalence relation ρ on a fuzzy Γ-hypersemigroup ( M, ◦ ) is called a fuzzy Γ -regular re-lation (or a fuzzy Γ -hypercongruence ) on ( M, ◦ ) if, for all a, b, c ∈ M, γ ∈ Γ, the followingimplication holds: aρb = ⇒ ( a ◦ γ ◦ c ) ρ ( b ◦ γ ◦ c ) and ( c ◦ γ ◦ a ) ρ ( c ◦ γ ◦ b ).This condition is equivalent to aρa ′ , bρb ′ implies ( a ◦ γ ◦ b ) ρ ( a ′ ◦ γ ◦ b ′ ) for all a, b, a ′ , b ′ of M and γ ∈ Γ.Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup and let ψ ( M, ◦ ) = ( M, ∗ ) be the associated Γ-hypersemigroup, where, for all a, b ∈ M, γ ∈ Γ, we have a ∗ γ ∗ b = { x ∈ M | ( a ◦ γ ◦ b )( x ) > } . Theorem 5.1.
An equivalence relation ρ is a fuzzy Γ-regular relation on ( M, ◦ ) if andonly if ρ is a Γ-regular relation on ( M, ∗ ). Definition 5.2.
An equivalence relation ρ on a fuzzy Γ-hypersemigroup ( M, ◦ ) is calleda fuzzy Γ -strongly regular relation on ( M, ◦ ) if, for all a, a ′ , b, b ′ of M and for all γ ∈ Γ,such that aρb and a ′ ρb ′ , the following condition holds: ∀ x ∈ M such that ( a ◦ γ ◦ c )( x ) > ∀ y ∈ M such that ( b ◦ γ ◦ d )( y ) >
0, we have xρy .Notice that if ρ is a fuzzy Γ-strongly relation on a fuzzy Γ-hypersemigroup ( M, ◦ ), thenit is fuzzy Γ-regular on ( M, ◦ ). 13 heorem 5.3. An equivalence relation ρ is a fuzzy Γ-strongly regular relation on ( M, ◦ )if and only if ρ is a Γ-strongly regular relation on ( M, ∗ ).Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup. Let ρ be a fuzzy Γ-hypercongruence on M . Wedefine a Γ-hyperoperation ∗ on M by a ∗ γ ∗ b = { x ∈ M : ( a ◦ γ ◦ b )( x ) > } .Therefore, ( M, ∗ ) is a Γ-hypersemigroup and ρ is a Γ-hypercongruence on ( M, ∗ ).Let M/ρ = { aρ : a ∈ M } . We define a Γ-hyperoperation ⊗ on M/ρ by aρ ⊗ γ ⊗ bρ = { cρ : c ∈ a ∗ γ ∗ b } = { cρ : ( a ◦ γ ◦ b )( c ) > } ,then ( M/ρ, ⊗ ) is a Γ-hypersemigroup.The Γ-hypersemigroup ( M/ρ, ⊗ ) is called the quotient Γ-hypersemigroup induced by theΓ-hypercongruence ρ on ( M, ∗ ). Theorem 5.4. If ρ is a fuzzy Γ-strong hypercongruence on a fuzzy Γ-hypersemigroup( M, ◦ ), then M/ρ = { aρ : a ∈ M } is a fuzzy Γ-hypersemigroup. Proof.
We define a fuzzy Γ-hyperoperation ∗ on M/ρ by ( aρ ∗ γ ∗ bρ )( cρ ) = ∨ a ′ ∈ aρ,b ′ ∈ bρ,c ′ ∈ cρ ( a ′ ◦ γ ◦ b ′ )( c ′ ), for all aρ, bρ, cρ ∈ M/ρ, γ ∈ Γ. Clearly ∗ is well-defined.Now (( aρ ∗ γ ∗ bρ ) ∗ γ ∗ cρ )( dρ ) = ∨ pρ ∈ M/ρ (( aρ ∗ γ ∗ bρ )( pρ ) ∧ ( pρ ∗ γ ∗ cρ )( dρ )) = ∨ pρ ∈ M/ρ ( ∨ a ′ ∈ aρ,b ′ ∈ bρ,p ′ ∈ pρ ( a ′ ◦ γ ◦ b ′ )( p ′ ) ∧ ( ∨ p ∈ pρ,c ′ ∈ cρ,d ′ ∈ dρ ( p ◦ γ ◦ c ′ )( d ′ )) = ∨ p ∈ M ( ∨ a ′ ∈ aρ, b ′ ∈ bρ, c ′ ∈ cρ, d ∈ dρ,p ′ ∈ pρ (( a ′ ◦ γ ◦ b ′ )( p ′ ) ∧ ( p ′ ◦ γ ◦ c ′ )( d ))) = ∨ a ′ ∈ aρ, b ′ ∈ bρ, c ′ ∈ cρ, d ∈ dρ (( a ′ ◦ γ ◦ b ′ ) ◦ γ ◦ c ′ )( d ′ )) = ∨ a ′ ∈ aρ,b ′ ∈ bρ,c ′ ∈ cρ,d ∈ dρ ( a ′ ◦ γ ◦ b ′ ◦ γ ◦ c ′ )( d ) , for all dρ ∈ M/ρ .Similarly, we show that ( aρ ∗ γ ∗ ( bρ ∗ γ ∗ cρ ))( dρ ) = ∨ a ′ ∈ aρ,b ′ ∈ bρ,c ′ ∈ cρ,d ∈ dρ ( a ′ ◦ γ ◦ b ′ ◦ γ ◦ c ′ )( d ),for all dρ ∈ M/ρ .Therefore ( aρ ∗ γ ∗ bρ ) ∗ γ ∗ ( cρ ) = aρ ∗ γ ∗ ( bρ ∗ γ ∗ cρ ). Therefore, ( M/ρ, ∗ ) is a fuzzyΓ-hypersemigroup. (cid:3) heorem 5.5. Let ( M, ◦ ) be a fuzzy Γ-hypersemigroup and ψ ( M, ◦ ) = ( M, ∗ ) be theassociated Γ-hypersemigroup. Then we have:( i ) The relation ρ is a fuzzy Γ-regular relation on ( M, ◦ ) if and only if ( M/ρ, ⊗ ) is aΓ-hypersemigroup.( ii ) The relation ρ is a fuzzy Γ-strongly regular relation on ( M, ◦ ) if and only if ( M/ρ, ⊗ )is a Γ-semigroup. Proof.
Straightforward. (cid:3)
Acknowledgements.
The first author partially has been supported by the ”Research Center in Algebraic Hyper-structures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran” and ”Alge-braic Hyperstructure Excellence, Tarbiat Modares University, Tehran, Iran”.
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