Study On Generalized Intuitionistic Fuzzy Subsets In Ordered Γ-Semigroups

In this dissertation, we investigate generalized intuitionistic fuzzy subsets of ordered Γ-semigroups as follows:generalized intuitionistic fuzzy subsemigroups, generalized intuitionistic fuzzy left (right) ideal, generalized intuitionistic fuzzy bi-ideal, generalized intuitionistic fuzzy interior ideal, generalized intuinonistic fuzzy semiprime ideal, generalized intuitionistic fuzzy prime bi-ideal, and get some prop-erties of theirs. There are five sections, the main results are given as following:In the first section, we mainly give some basic concepts and symbols in this dissertation.In the second section, we mainly give the concepts of generalized intuition-istic fuzzy subsemigroups and generalized intuitionistic fuzzy left (right) ideal of ordered Γ-semigroups and some properties of theirs. The main results are given as following:Theorem 2.1 Let{Ai,i ∈I} be a family EIFSS of ordered Γ-semigroup S, then both ∩i∈I Ai and Ui∈IA are EIFSS of S.Theorem 2.2 Let U be a subsemigroup of ordered Γ-semigroup S, then U=(Xu,Xu)is an EIFSS of S.Theorem 2.3 Let U be a non-empty subset of ordered Γ-semigroup S. If U= (Xu, Xu) is an EIFSS1 or EIFSS2 of S, then U is a subsemigroup of S.Theorem 2.4 Let A=(μA,γA) be an EIFSS of an ordered Γ-semigroup S such that μA(x)≥μA(y)and γA(x)≤γA(y) for x,y∈S and x≤y. Then μA(a) ∨α>^AoA(α)∧β,γA(α)∧(1-α)≤γAoA(α) ∧ (1-β) for α∈ S S, α,β∈[0,1] and α<β.Theorem 2.5 Let A=(μA,γA) be an intuitionistic fuzzy subset of an ordered Γ-semigroup S. If μA(a)∨α≥μAoA(α)∧β,γA(α)∧(1-α)≤γAoA(α)∧(1-β) for every β∈S and α,β∈[0,1] and α<β, then A=(μA,γA) be an EIFSS of S.Theorem 2.6 Let the pair of mappings f1:M→M1,f2:Γ→Γ1 be a homomorphism from ordered Γ-semigroup M to ordered Γ1-semigroup M1. If A=(μA,γA) is a generalized intuitionistic fuzzy subsemigroup of M1, then the inverse image f1-1(A) of A=(μA-γA) under f1 is a generalized intuitionistic fuzzy subsemigroup of M.Theorem 2.7 Let U be a non-empty subset of ordered Γ-semigroup S, then U is left (right) ideal of S if and only if U=(XU,XU) is a generalized intuitionistic fuzzy left (right) ideal of S. Theorem 2.8 If A=(μA,γA) is an EIFRI1 of ordered Γ-semigroup S and U is a left-zero subsemigroup of S, then for any x,y ∈U, one of the following must holds:(Ⅰ) μA(x)=μA(y);(Ⅱ) μA(x)≠μA(y)(?)(μA(x) ∨μA≤α or μA(x)∧μA(y)≥ β).Theorem 2.9 If A=(μA,γA) is an EIFRI1 of ordered Γ-semigroup S and U is an right-zero subsemigroup of S, then for any x, y∈U, one of the following must holds:(Ⅰ) μA(x)＝μA(y)；(Ⅱ) μA(x)≠μA(y)(?)(μA(x)∨μA(y)≤α orμA(x)∧μA≥β).Theorem 2.10 If A=(μA,γA) is an EIFLI2 of ordered Γ-semigroup S and U is a left-zero subsemigroup of S, then for any x,y∈U, one of the following must holds:(Ⅰ)γA(x)= γA(y);(Ⅱ)γA(x)≠γA(y)(?)(γA(x)∨γA(y)≤α or γA(x)∧γA(y)≥β).Theorem 2.11 If A=(μA,γA) is an EIFRI2 of ordered Γ-emigroup S and U is an right-zero subsemigroup of S, then for any x,y∈U, one of the following must holds:(Ⅰ)γA(x)=γA(y);(Ⅱ)γA(x)≠γA(y)(?)(γA(x)∨γA(y)≤αor γA(x)∧γA(y)≥β).Theorem 2.12 Let the pair of mappings f1:M→M1,f2:Γ→Γ1 be a homomorphism from ordered Γ-semigroup M to ordered Γi-semigroup M1. If A=(μA,γA) is a generalized intuitionistic fuzzy left (right) ideal of M1, then the inverse imag f1-1(A) of A=(μA,γA) under f1 is a generalized intuitionistic fuzzy left(right)ideal of M.Theorem 2.13 Let S be a regular ordered Γ-semigroup,α∈S,α,β∈ [0,1],α<β.Then for every generalized intuitionistic fuzzy right ideal A=(μA,γA) and every intuitionistic fuzzy subset B=(μB,γB)of S,we have μAoB(α)∨α≥(μA∩μB)(α)∧β,γA为B(α)∧(1-α)≤(γA∩γB)(α)∨(1-β).Theorem 2.14 Let S be a rcgular ordered Γ-semigroup,α∈S,α,β∈ [0,1],α<β. Then for every intuitionistic fuzzy subset A=(μA,γA) and every generalized intuitionistic fuzzy left ideal B=(μB,γB) of S,we haveμAoB(α)∨α≥(μA∩μB)(a)∧β,γAoB(α)∧(1-α)≤(γA∩γB)(α)∨(1-β).Theorem 2.15 Let S be a regular ordered Γ-semigroup,α∈S,α,β∈ [0,1],α<β.Then for every generalized intuitionistic fuzzy right ideal A=(μA,γA) and every generalized intuitionistic fuzzy left ideal B=(μB,γB)of S,we have μAoB(α)∨α≥(μA∩μB)(α)∧β,γAoB(α)∧(1-α)≤(γA∩γB)(α)∨(1-β).Theorem 2.16 Let S be an ordered Γ-semigroup. IfA=(μA,γA) be an intuitionistic fuzzy subset of S,Then 1_o A(Ao1_)is a generalized intuitionistic fuzzy left (right) ideal of S.In the third section,,we mainly give the concept of generalized intuitionistic fuzzy bi-ideal of ordered Γ-semigroups and study some properties of theirs.The main results are given as following:Theorem 3.1 Let {A,i∈I} be a collection of generalized intuitionistic fuzzy bi-ideal of ordered Γ-semigroup S,then their intersection ∩i∈I Ai is a generalized intuitionistic fuzzy bi-ideal of S.Theorem 3.2 Let U be a non-empty subset of ordered Γ-semigroup S.Then U is a bi-ideal of S if and only if U=(XU,XU) is a generalized intuitionistic fuzzy bi-ideal of S.Theorem 3.3 Let S be a regular ordered Γ-semigroup.Then generalized intuitionistic fuzzy bi-ideals of S are generalized intuitionistic fuzzy subsemigroupTheorem 3.4 If S is a left simple ordered Γ-semigroup, then generalized intuitionistic fuzzy bi-ideal of S is generalized intuitionistic fuzzy right ideal of S.Theorem 3.5 Let the pair of mappings f1:M→M1,f2:Γ→Γ1 be a homomorphism from ordered Γ-semigroup M to ordered Γ-semigroup M1. If A=(μA,γA) is a generalized intuitionistic fuzzy bi-ideals of M1, then the inverse image f1-1(A) of A=(μA,γA) under f1 is a generalized intuitionistic fuzzy bi-ideals of M.In the fourth section, we define the generalized intuitionistic fuzzy interior ideal and generalized intuitionistic fuzzy semiprime ideal of ordered Γ-semigroups and give some properties of theirs. The main results are given as following:Theorem 4.1 Let U be an interior ideal of ordered Γ-semigroup S, then U=(XU,XU)is an EⅠFⅡ of S.Theorem 4.2 Let ordered Γ-semigroup S be regular and U be a non-empty subset of S. If U= (XU,XU) is an EⅠFⅡ1 or EⅠFⅡ2, then U is an interior ideal of S.Theorem 4.3 Let A=(μA,γA) be an intuitionistic fuzzy subset of an intra-regular ordered Γ-emigroup S, then A=(μA,γA) is a generalized intuitionistic fuzzy interior ideal of S if and only if A=(μA,γA) is a generalized intuitionistic fuzzy ideal of S.Theorem 4.4 Let the pair of mappings f1:M M→ M1, f2:Γ→Γ1 be a homomorphism from ordered Γ-semigroup M to ordered Γ1-semigroup M1. If A=(μA,γA) is a generalized intuitionistic fuzzy interior ideal of M1, then the inverse image f1-2(A) of A=(μA,γA) under f1 is a generalized intuitionistic fuzzy interior ideal of M.Theorem 4.5 Let U be a non-empty subset of an ordered Γ-semigroup. If U is semiprime, then U=(XU,XU) is EIFSP.Theorem 4.6 Let U be a non-empty subset of an ordered Γ-semigroup S. If U=(XU,XU) is EIFSP1 or EIFSP2, then U is semiprime.Theorem 4.7 Let S be an ordered Γ-semigroup. If S is left regular, then every generalized intuitionistic fuzzy left ideal of S is generalized intuitionistic fuzzy semiprime.Theorem 4.8 Let A=(μA,γA) be a generalized intuitionistic fuzzy ideal of an ordered Γ-emigroup S. If S is intra-regular, then A=(μA,γA) is generalized intuitionistic fuzzy semiprime.Theorem 4.9 Let A= (μA,γA) be a generalized intuitionistic fuzzy interior ideal of an ordered Γ-emigroup S. If S is intra-regular, then A= (μA,γA) is generalized intuitionistic fuzzy semiprime.In the fifth section, we mainly give the basic concepts of generalized intu-itionistic fuzzy prime (strongly prime) bi-ideal of ordered Γ-semigroups and some properties of theirs, the main results are given as following:Theorem 5.1 Let U be a non-empty subset of ordered Γ-semigroup S. If U is a prime bi-ideal of S, then U= (Xu, Xu) is a generalized intuitionistic fuzzy prime bi-ideal of S.Theorem 5.2 Let U be a non-empty subset of ordered Γ-semigroup S. If U is a strongly prime bi-ideal of S, then U= (Xu, Xu) is a generalized intuitionistic fuzzy strongly prime bi-ideal of S.