| In this dissertation, we mainly study some intuitionistic fuzzy subsets with thresh-olds (α, β) of semigroups, give some results about intuitionistic fuzzy subscmigroups with thresholds(α, β), intuitionistic fuzzy bi-ideals with thresholds (α, β), intuition-istic fuzzy interior ideals with thresholds (α, β), intuitionistic fuzzy quasi-idcals with thresholds (α, β) of semigroups. In the end, give characterizations and properties of regular semigroups, intra-regular semigroups, quasi-regular semigroups, completely regular semigroups, group semilattices, groups, semisimple semigroups and intuition-istic fuzzy duo semigroups by intuitionistic fuzzy subsets. There are three chapters, the main results are as follows.In Chapter One, we mainly give the instruction and preliminaries in this paper.In Chapter Two, there are four sections, the main results are as follows.In Section2.1, we mainly study the properties and charaterizations of intuition-istic fuzzy subsemigroups with thresholds (α, β). The main results are as follows:Theorem2.1.1Let{Ai=(μAi,γAi)} be a family intuitionistic fuzzy sub-semigroups with thresholds (α, β) of a semigroup S. Then both (?)\i€ΛAi and Ui€GΛ.Ai; are intuitionistic fuzzy subscmigroups with thresholds (α, β) of S.Theorem2.1.2Let A=(μA.γA) be an intuitionistic fuzzy subset of a semigroup S, then A is an intuitionistic fuzzy subsemigroup with thresholds(α, β) of S if and only if non-empty setsμ(?)r={x∈S|μA(x)≥r} and γ(?),t={x∈S|,γA(x)≤t} are subsemigroups of S, for all r∈(αβ] and t∈[1-β，,1-α).Theorem2.1.3Let A=(μA,γa) be an intuitionistic fuzzy subset of a semigroup S, then A is an intuitionistic fuzzy subsemigroup with thresholds (αβ) of s if and only if fuzzy subsets μA and (?)7are fuzzy subsEmigroups with thresholds (a, β) of S. Theorem2.1.4A non—empty subset U in semigroup S is a susemigroup of S if and only if U=(Xu,(?))is an intuitionistic fuzzy subsemigroup with thresholds (α,β) of S,Theorem2.1.5Let A=(μA,γA)be an intuitionistic fuzzy,subset of a.semigroup S,then A=(μA,γA)is an intuitionistic fuzzy subsemigroup with thresholds(α,β)of S if and only if(AοA)∩β～(?)A∪α～Theorem2.1.6Let(S,·),(T,*)bc semigroups, f a homomorphism from S onto T,A=(μA,γA)and B=(μB,γB)be intuitionistic fuzzy subsemigroups with thresholds(α,β)of S and T.Then(1)f(A)(?)(f(μA),f(γA))is an intuitionistic fuzzy subsemigroup with thresholds (α,β) of T,and for any x’∈T, f(μA):x’(?) Vx∈f-1(x’)μA(x),f(γA):x’(?)∧x∈f-1(x’)γA(x).(2)f-1(B)(?)(f-1(μB),,f-1(γB))is an intuitionistic fuzzy subsemigroup with thresholds(α,β)of S,and for any x∈S, f-1(μB)(x)=μBf(x),f-1(γB)(x)=γBf(x).(3)If.f is isomorphism from s onto T,then the mapping Aâ†',f(A)defines a one-to-one correspondence between the set of all intuitionistic fuzzy subsemigroups with thresholds(α,β) of S and the set of all intuitionistic fuzzy subsemigroups with thresholds(α,β) of T.In Section2.2,we mainly give the concepts of intuitionistic fuzzy(generalized) bi-ideals with thresholds(α,β) of semigroups,and study the properties and charateri-zations of intuitionistic fuzzy bi-ideals with thresholds(α,β) in semigroups.The main results are as folows:Theorem2.2.1Let{Ai=(μAi.γAi)}i∈A be a family intuitionistic fuzzy bi-ideals with thresholds(α,β) of a semigroup S.Then both∩i∈A Ai and∪i∈A Ai are intuitionistic fuzzy bi-idcals with thresholds(α,β)of S.Theorem2.2.2Let A=(μA,γA)be an intuitionistic fuzzy subset of a semigroup S,then A is an intuitionistic fuzzy bi-ideal with thresholds f(α,β)of S if and only if non-empty sets μ(?)r={x∈S|μA(x)≥r}and γ(?),t={x∈S|γA(x)≤t}are bi-ideals of S, for all r∈(α,β) and t€[1-β,1-α).Theorem2.2.3Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S, then A is an intuitionistic fuzzy bi-ideal with thresholds (α,β) of S if and only if fuzzy subsets μA and (?) are fuzzy bi-ideals with thresholds (α,β) of S.Theorem2.2.4A non-empty subset U in semigroup S is a bi-idcal of S if and only if U=(xu,(?)) is an intuitionistic fuzzy bi-idcal with thresholds (α,β)of S.Theorem2.2.5Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S, then A=(μA,γA) is an intuitionistic fuzzy bi-idcal with thresholds (α,β) of S if and only if (Ao A) n β～(?)A∪α～Ao1～oA)∩β～(?)A∪α～Corollary2.2.1Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S, then A=(μA,γA) is an intuitionistic fuzzy generalized bi-ideal with thresholds (α,β) of S if and only if (A o1～oA)∩β～(?)A∪α～Theorem2.2.6A semigroup S is regular if and only if every intuitionistic fuzzy (generalized) bi-ideal A=(μA,γA) with thresholds (α,β) of S satisfies A∪α～=(Ao1～oA)∩β～.Theorem2.2.7Let A=(μA,γA) and B=(μB,γB) be an intuitionistic fuzzy (generalized) bi-ideals with thresholds (α,β) of a semigroup S, then A o B=(μA o μBγA*γB)(B o A) is an intuitionistic fuzzy (generalized) bi-ideal with thresholds (α,β) of a semigroup S.Theorem2.2.8Let S be a regular semigroup, the following conditions are equivalent:(1)Every bi-ideal of S is a right (left) ideal;(2))Every intuitionistic fuzzy bi-ideal of S is an intuitionistic fuzzy right (left) ideal. Theorem2.2.9Let S be a regular semigroup and A==(μA,γA) be an intuition-istic fuzzy subset of S. The following conditions arc equivalent:(I)A==(μA,γA) is an intuitionistic fuzzy (generalized) bi-ideal of S;(2)A=BoC, and B is an intuitionistic fuzzy right ideal of S, C is an intuitionistic fuzzy left ideal of S. Theorem2.2.10Let(S,.),(T,})be semigroups,f a homomorphism from^S onto T,A=(μA,γA)and B=(μB,γB)be intuitionistic fuzzy bi-ideals with thresholds (α.β)of S and T.Then(1)f(A)(?)(f(μA),f(γA))is an intuitionistic fuzzy bi-ideal with thresholds(α.β) of T,and for any x’∈T, f.(μA):x’(?) Vz∈f-1(x’)μA(x),f(γ.A):x’(?)∧x∈f-1(x’).γA(x).(2)f-1(B)(?)(f-1(μB),f-1(γB))is an intuitioistic fuzzy bi-ideal with thresholds (α.β)of S,and for any x∈S, f-1(μB)(x)=μBf(x),f-1(γB)(x)=γBf.(x).(3)If f is isomorphism from S onto T,then the mapping A(?)f(A)defines a one-to-one correspondence between the set of all intuitiomistie fuzzy bi-ideals with thresholds(α.β)of.S and the set of all intuitionistic fuzzy bi-ideals with thresholds (α.β) of T.In Section2.3,we mainly give the concepts of intuitiOnistic fuzzy(generaliZed) interior ideals with thrcsholds(α.β)of semigroups,and study their properties and charaterizations in semigroups.The main results are as follows:Theorem2.3.1Let A=(μA,γA)be an intuitionistic fuzzy subset of a semigroup S,then A=(μA,γA)is an intuitionistic fuzzy interior ideal with thrcsholds(α.β) of S if and only if(A o A)∩β～(?).∈A∪α～,(1～oA o～)∩β (?)A∪α～Corollary2.3.1Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S,then A=(μA,γA)is an intuitionistic fuzzy generalized interior ideal with thresholds (α.β) of S if and only if(1～oA o.l～)nβ～(?)A∪a～Theorem2.3.2Let{Ai=(μAi,γAi)i∈A be a family intuitionistic fuzzy interior ideals with thresholds(α.β) of a semigroup S.Then both∩i∈A Aiand∪i∈A Ai are intuitionistic fuzzy interior ideals with threshholds(α.β) of S.Theorem2.3.3Let A=A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S,then A is an intuitionistic fuzzy interior idcal with thresholds(2111of S if and only if non-empty sets,μ(?),r={x∈s S|,A(x)≥r)and γ(?),t={x∈S|γA(x)≤t}are interior ideals of S,for all r’∈(α.β) and t∈[1-β,1-α). Theorem2.3.4Let A=(μA,γA)be an intuitionistic fuzzy subset of a semigroup S,then A is an intuitionistic fuzzy interior ideal with thresholds(α,β)of S if and only if fuzzy Subsets μA and γA are fuzzy interior ideals with thrcsholds(α,β)of S.Theorem2.3.5A non-empty subset U in semigroup S is a intcrior ideal of S if and only if U=(χU,χU)is an intuitionistic fuzzy interior ideal with thresholds(α,β) of S.Theorem2.3.6Let(S,-),(T,*)be Semigroups,f a homomorphism from S onto t,A=(μA,γA) and B=(μB,γB)be intuitionistic fuzzy interior idcals with thresholds (α,β) of S and T.Then(1)f(A)(?)(f(μA),f(γA))is an intuitionistic fuzzy interior ideal with thresholds (α,β) of T,and for any x’∈T, f(μA):x’(?)∨x∈f-1(X’)μA(x),f(γA):x’(?)∧x∈f-1(X’)γA(x),f(γA)(x).(2)f-1(B)(?)(f-1(μB),f-(γB))is an intuitionistic fuzzy interior ideal with thresholds(α,β)of S,and for any x∈S, f-1(μB)(x)=μBf(x),f-1(γB)(z)=γBf(x).(3)If f is isomorphism from S onto T,then the mapping Aâ†'f(A)defines a one-to-one corr-spondence between the set of all intuitionistic fuzzy interior ideals with thresholds(α,β) of S and the set of all intuitionistic fuzzy interior ideals with throsholds (α,β) of T.In Section2.4,we mainly give tree concepts of intuitionistic fuzzy quasi-ideals with thresholds(α,β) of semigroups,and study their properties and charaterizations in scmigroups.The main results are as follows:Theorem2.4.1Let A=(μA,γA)be an intuitionistic fuzzy subset of a scmigroup S,then A=(μA,γA) is an intuitionistic fuzzy quasi-idcal with thresholds(α,β) of S if and only if for every a∈S,there exist b,s,t,c∈S such that x=bs,x=tc,then (1)μA(x)∨α≥min{μA(b),μA(c),β),(2)γA(x)∧(1-α)≤max{γA(b)),γA(c),1-β).Theorem2.4.2Let{Ai=(μAi,γAi)i∈A b a family intuitionistic fuzzy quasi- ideals with thresholds (α,β) of a semigroup S. Then both∩i∈Λ Ai and∪i∈ΛAAi are intuitionistic fuzzy quasi-ideals with thresholds(α,β) of S.Theorem2.4.3Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S, thenA is an intuitionistic fuzzy quasi-ideal with thresholds (α,β) of S if and only if non-empty setsμ(?),r={x€S|μZa(x)> r} and γ(?),t={x€S|γA(x)≤are quasi-ideals of S, for all r€(α,β) and t€[1-β,1-α).Theorem2.4.4A non-empty subset U in semigroup S is a quasi-ideal of S if and only if U=(xu,(?)) is an intuitionistic fuzzy quasi-ideal with thresholds (α,β) of S.Theorem2.4.5Let A=(μA,γA) be an intuitionistic fuzzy subset of a semigroup S. If A is an intuitionistic fuzzy quasi-ideal with thresholds (α,β) of S, then A is an intuitionistic fuzzy bi-ideal with thresholds (α,β) of S.Theorem2.4.6Let S be a regular semigroup and A=(μA,γA) be an intu-itionistic fuzzy subset of a semigroup S. If A is an intuitionistic fuzzy bi-ideal with thresholds (α,β) of S, then A is an intuitionistic fuzzy quasi-ideal with thresholds (α,β) of S.Theorem2.4.7Let (S,),(T,*) be semigroups,f homomorphism from S onto T, A=(μA,γA) and B=(μB,γB) be intuitionistic fuzzy quasi-ideals with thresholds (a,0) of S and T. Then(1)f(A)(?){f(μA),f(γA)) is an intuitionistic fuzzy quasi-ideal with thresholds (α,β) of T, and for any x’€T, f(μA):x’(?)Vx∈f-1(x’)μA (X),.f (γA):x’(?)Vx∈f-1(x’)γA (X).(2) f-1(B)(?)f-1(μB), f-1(γb)) is an intuitionistic fuzzy quasi-ideal with thresh-olds (α,β)of S, and for any x∈S, f-(μB)（x)=μbf(x),f-1(γB)(x)=γBf(x).(3) If f is isomorphism from5onto T, then the mapping Aâ†'f(A) defines a one-to-one correspondence between the set of all intuitionistic fuzzy quasi-ideals with thresholds (α,β) of S and the set of all intuitionistic fuzzy quasi-ideals with thresholds (α,β) of T. In Chapter Three, we mainly discuss the structures and properties of regular semigroups, intra-regular semigroups, quasi-regular semigroups, completely regular semigroups, group semilattices, groups, semisimple semigroups and intuitionistic fuzzy duo semigroups by intuitionistic fuzzy left, right ideals, intuitionistic fuzzy bi-idcals, intuitionistic fuzzy interior ideals, intuitionistic fuzzy quasi-idcals. The main results are as follows:Theorem3.1.1Let S be a semigroup. Then the following statements arc equivalent:(1)S is regular;(2)A∩B C AoB, for every intuitionistic fuzzy right ideal A=(μA,γA) and every intuitionistic fuzzy quasi-ideal B=(μB,γB) of S;(3)A∩B C AoB, for every intuitionistic fuzzy right ideal A=(μA,γA) and every intuitionistic fuzzy bi-ideal B=(μB,γB) of S;(4)A∩B C A o B, for every intuitionistic fuzzy right ideal A=(μA,γA) and every intuitionistic fuzzy generalized bi-ideal B=(μB,γB) of S;(5)A∩B C AoB, for every intuitionistic fuzzy right ideal A=(μA,γA) and every intuitionistic fuzzy left ideal B=(μB,γB) of S.Corollary3.1.1Let S be a semigroup. Then the following statements are equivalent:(1)S is regular;(2)A∩B C A o B, for every intuitionistic fuzzy quasi-ideal ideal A=(μA,γA) and every intuitionistic fuzzy left ideal B=(μB,γB) of S;{3)A∩B (?)C A o B, for every intuitionistic fuzzy bi-idcal A=(μA,γA) and every intuitionistic fuzzy left ideal B=(μB,γB) of S;(4)A∩B (?) Ao B, for every intuitionistic fuzzy generalized bi-idcal A=(μA,γA) and every intuitionistic fuzzy left ideal B=(μB,γB) of S.Theorem3.1.2Let S be a semigroup. Then the following statements are equivalent:(1)S is regular; (2)A∩B=AοBοA,for every intuitionistic fuzzy quasi-ideal A=(μA,γA) and every intuitionistic fuzzy ideal B=(μB,γB)of Sï¼›(3)A∩B=AοBοA,for every intuitionistie fuzzy quasi-ideal A=(μA,γA) and every intuitionistic fuzzy interior ideal B=(μB,γB) of Sï¼›(4)A∩B=AοBοA,for every intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy ideal B=(μB,γB)of Sï¼›(5)A∩B=AοBοA,for every intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy interior ideal B=(μB,γB) of Sï¼›(6)A∩B=AοBοA,for every intuitionistic fuzzy generalized bi-ideal A=(μA,γA) and every intuitionistic fuzzy ideal B=(μB,γB) of.Sï¼›(7)A∩B=AοBοA,for every intuitionistic fuzzy generalized bi-idealA=(μA,γA) and every intuitionistic fuzzy interior ideaI B=(μB,γB) of Sï¼›(8)A∩B=AοBοA,for every intuitionistic fuzzy generalized bi-ideal A=(μA,γA) and every intuitionistic fuzzy generalized interior ideal B=(μB,γB)of Sï¼›(9)A∩B∩C(?) CοAοB,for every intuitionistic fuzzy bi-ideal A=(μA,γA) intuitionistic fuzzy left ideal B=(μB,γB)of S and every intuitionistic fuzzy right ideal C=(μC,γC)of Sï¼›(10)A∩B∩C(?) CοAοB,for every intuitionistic fuzzy generalized bi-ideal A=(μA,γA),every intuitionistic fuzzy left idea B=(μB,γB)of S and every intuitionistie fuzzy right ideal C=(μC,γC)of STheorem3.1.3Let S be a semigroup. Then the following statements are equivalent:(1)S is regular and intra-regularï¼›(2)every intuitionistic fuzzy quasi-ideal of S is idempotentï¼›(3)every intuitionistic fuzzy bi-ideal of S is idempotent.Theorem3.1.4Let S be a semigroup. Then the follwing statements are equivalent:(1)S is regular and intra-regularï¼›(2))A∩B(?)(AοB)∩(BοA),for all intuitionistic fuzzy bi-ideals A=(μA,γA) and B=(μB,γB) of Sï¼›(3)A∩B(?)(A o B)∩(B o A),for every intuitionistic fuzzy left ideal A=(μA,γA) and every intuitionistic fuzzy bi-ideal.B=(μB,γB) of.Sï¼›(4)A∩B(?)(A o B)∩(B o A),for overy intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy right ideal B=(μB,γB) of Sï¼›(5)A∩B(?)(A o B)∩(B o A),for every intuitionistic fuzzy left ideal A=(μA,γA) and every intuitionistic fuzzy right ideal B=(μB,γB) of S.Theorem3.1.5Let S be a semigroup. Then the following statements are equivalent:(1)S is regular and intra.regularï¼›(2)A∩B(?)Ao Bo A,for every intuitionistic fuzzy quasi-ideal’A=(μA,γA) and every intuitionistic fuzzy left idcal B=(μB,γB) of Sï¼›(3)A∩B(?)Ao Bo A,for every intuitionistic fuzzy quasi-ideal A=(μA,γA)and every intuitionistic fuzzy right ideal B=(μB,γB)of Sï¼›(4)A∩B(?)(A o B)∩(B o A),for and intuitionistic fuzzy quasi-ideals A=(μA,γA) and B=(μB,γB) of Sï¼›(5)A∩B(?)(A o B)∩(B o A),for every intuitionistic fuzzy quasi-ideaJ A=(μA,γA) and every intuitionistic fuzzy bi-ideal B=(μB,γB) of Sï¼›(6)A∩B(?)(A o B)∩(B o A),for every intuitionistie fuzzy quasi-ideal A=(μA,γA) and every intuitionistic fuzzy generalized bi-ideal B=(μB,γB) of Sï¼›(7)A∩B(?)(A o B)∩(B o A),for every intuitionistic mzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy left ideal B=(μB,γB)of Sï¼›(8)A∩B(?)(A o B)∩(B o A),for every intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy right ideal B=(μB,γB)of Sï¼›(9)A∩B(?)(A o B)∩(B o A) for every intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy quasi—ideal B=(μB,γB)of Sï¼›(10)A∩B(?)(A o B)∩(B o A),for all intuitionistic fuzzy bi-ideals A=(μA,γA) and B=(μB,γB)of Sï¼›(11)A∩B(?)(A o B)∩(B o A),for every intuitionistic fuzzy bi-idea A=(μA,γA) and every intuitionistic fuzzy geneltalized bi-ideal B=（μB,γB)of Sï¼›(12)A∩B(?)(A o B)∩(B o.A),for every intuitionistic fuzzy generalized bi-ideal A=(μA,γA)and every intuitionistic fuzzy left ideal B=(μB,γB)of Sï¼›(13)A∩BE(?)(A o B)∩(B o A),for every intuitionistic fuzzy generalized bi—ideal A=(μA,γA)and every intuitionistic fuzzy right ideal B=(μB,γB) of Sï¼›(14)A∩BE(?)(A o B)∩(B o A),for every intuitionistic fuzzy generalizcd bi-ideal A=(μA,γA)and every intuitionistic fuzzy quasi-idcal B=(μB,γB) of Sï¼›(15)A∩BE(?)(A o B)∩(B o A),for every intuitionistic fuzzy generalized bi—ideal A=(μA,γA)and every intuitionistic fuzzy bi-ideal B=(μB,γB)of sï¼›(16)A∩BE(?)(A o B)∩(B o A),for all intuitionistic fuzzy gcneralized bi-ideals A=(μA,γA)and B=(μB,γB)of S.Theorem3.1.6A semigroup S is quasi-regular if and only if every intuitionistic fuzzy quasi-ideal A=(μA,γA)of.S satisfies A=(A o1~)2n(1~o A)2.Theorem3.1.7Let S be a semigroup. Then the following statements are equivalent:(1)S is intra-regular and left quasi-regularï¼›(2)B∩C∩A(?)B o C o A,for every intuitionistic fuzzy quasi—ideal A=(μA,γA) every intuitionistic fuzzy left ideal B=(μB,γB) and every intuitionistic fuzzy right ideal C=(μC,γC) of Sï¼›(3)B∩C∩4(?)B.C o A,for every intuitionistic fuzzy bi—ideal A=(μA,γA) every intuitionistic fuzzy left ideal B=(μB,γB) and every intuitionistic fuzzy right ideal C=(μC,γC) of Sï¼›(4)B∩C∩A(?)B o C o A,for every intuitionistic fuzzy generalized bi-ideal A=(μA,γA),every intuitionistic fuzzy left ideal B=(μB,γB) and every intuitionistic fuzzy right ideal C=(μC,γC) of S.Theorem3.1.8Let S bc a semigroup. Then the following statements are equiValent:(1)S is right quasi-regularï¼›(2)A n B(?) A o B,for every intuitionistic fuzzy bi-ideal A=(μA,γA) and every intuitionistic fuzzy ideal B=(μB，γB)of Sï¼›(3)A∩B(?)AoB,for every intuitionisti’c fuzzy right ideal A=(μA，γA)and every intuitionistic fuzzy ideal B=(μB，γB)of Sï¼›(4)A∩B∩C(?)∈A o B o C,for every intuitionistic fuzzy generalized bi-ideal A=(μA，γA)every intuitioniste fuzzy ideal B=(μB，γB)and every intuitionistic fuzzy right ideal C=(μC，γC)of Sï¼›(5)A∩B∩C (?) Ao Bo C,for every intuitionistic fuzzy bi—ideal A=(μA，γA) every intuitionistic fuzzy ideal B=(μB，γB)and every intuitionistic fuzzy right ideal C=(μC，γC) of S.Theorem3.2.1A semigroup S is eompletely regular if and only if every in-tuitionistic fuzzy quasi—ideal A=(μA，γA) of S satisfies A(a)=A(an+1),for all a∈S.n∈N.Theorem3.2.2Let S be a semigroup.Then S is a group semilattice if and only if every intuitionistic fuzzy quasi-ideal A=(μA，γA) of S satisfies A(a)=A(a2),A(ab)=4(ba),for all a,b∈S.Theorem3.2.3Let S be a scmigroup. Then the following statements are equivalent:(1)S is a group semilatticeï¼›(2)B∩A=B o A,for every i’ntuitionistic fuzzy left ideal A=(μA，γA) and every intuitionistic fuzzy right ideal B=(μB，γB)of Sï¼›(3B∩A=B o A,for every intuitionistic fuzzy left idcal A=(μA，γA) and every intuitionistic fuzzy bi-idcal B=(μB，γB) of Sï¼›(4)B∩A=B o A, for every intuitionistic fuzzy bi-idcal A=(μA，γA)and every intuitionistic fuzzy right ideal B=(μB，γB)of Sï¼›(5)B∩A=B o A,for all intuitionistic fuzzy bi-idcals A=(μA，γA)and B=(μB，γB)of S.Theorem3.2.4Lct S be a semigroup.Then S is a group Semilattice if and only if the set of all intuitionistic fuzzy bi-idealsA=(μA，γA)of S is a semila,ttiee under the product of intuitionistic fuzzy subsets. Theorem3.2.5Let S be a regular semigroup. Then the following statements are equivalent:(l)E(S)={e∈S|e2=e} is a subsemigroup without zero of S;(2)cvery intuitionistic fuzzy left ideal A=(μA,γA) of5satisfies A(e1)=A(e2), for all e1,e2∈E(S).Corollary3.2.1Let S be a band, then the following statements are equivalent:(1)S is a semigroup without zero;(2)every intuitionistic fuzzy left ideal A=(μA,γA) of S satisfies A(x)=A(y), for all x,y∈S.Theorem3.2.6Let S be a semigroup, then the following statements are equiv-alent:(1)S is a group;(2)for every intuitionistic fuzzy bi-ideal A=(μA,γA) of S we have μA and γA are constant.(3)for every intuitionistic fuzzy quasi-ideal A=(μA,γA) of S we haveμAand γA are constant.Theorem3.2.7Let S be a regular semigroup, then the following statements are equivalent:(1)5is a group;(2) every intuitionistic fuzzy bi-ideal A=(μA,γA) of S satisfies A(e1)=A(e2), for all e1, e2€E(S).Theorem3.2.8A semigroup5is left and right simple if and only if for every intuitionistic fuzzy quasi-ideal A=(μA,γA) of S, we have μA and γA are constants.Theorem3.2.9Let S be a semigroup, then the following statements are equiv-alent:(1)5is scmisimplc;(2)every intuitionistic fuzzy ideal of S is idempotent;(3)A∩B=AοB, for all intuitionistic fuzzy ideals A=(μA,γA) and B=(μB,γB) of S; (4)The set of all all intuitionistic fuzzy ideals of S is a distributive lattice under the union and product of intuitionistic fuzzy subsets.Theorem3.2.10Let S be a regular semigroup. Then S is left duo if and only if S is intuitionistic fuzzy left duo.Corollary3.2.2Let5be a regular semigroup. Then S is duo if and only if S is intuitionistic fuzzy duo.Theorem3.2.11Let S be a regular and right duo semigroup. Then every in-tuitionistic fuzzy bi-ideal of S is an intuitionistic fuzzy left ideal. |
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