| In this dissertation, we first study the anti fuzzy left ideals （right ideals,two-sidesideals,interior ideals, bi-ideals,quasi-ideals） with threshods （λ, ） of ordered semigroupsin the view of anti-subseting,and give the characterizations of several special classes ofordered semigroups. Then we discuss the anti t fuzzy fuzzy bi-deals and anti t fuzzyfuzzy quasi-deals of ordered semigroups by introducing anti t fuzzy operators. In theend, we give the （?） fuzzy （fuzzy weakly） regular subsemigroups and（∈,∈∨q（λ,μ）） fuzzy （fuzzy weakly） complete regular subsemigroups of ordered semi-groups,and their characterizations .There are three chapters in this paper, the mainresults are given as following.In Chapter 1 , we mainly give the intruction and preliminaries in this paper.In Chapter 2 , we mainly define the anti fuzzy left ideals （right ideals,two-sides ide-als,interior ideals,bi-ideals,quasi-ideals） with threshods （λ, ） of ordered semigroups,discuss the properties of fACand fα≤, and give the characterizations of regular （intra-regular,quasi-regular,semisimple） ordered semigroups. The main results are given asfollowing:Theorem 2.1.1 Let S be ordered semigroups. f is the anti fuzzy left（right） idealswith threshods （λ, ） of S if and only if for allα∈[λ, ), fα≤≠φis left （right） idealsof S.Theorem 2.1.2 Let S be ordered semigroups. f is the anti fuzzy bi-ideals withthreshods （λ, ） of S if and only if for allα∈[λ, ), fα≤≠φis bi-ideals of S.Theorem 2.1.3 Let S be ordered semigroups. f is the anti fuzzy interior idealswith threshods （λ, ） of S if and only if for allα∈[λ, ), fα≤≠φis interior ideals of S.Theorem 2.1.4 Let S be ordered semigroup. f is the anti fuzzy quasi-ideals withthreshods （λ, ） of S if and only if for allα∈[λ, ), fα≤≠φis quasi-ideals of S.Theorem 2.1.5 Let S be ordered semigroup. A be left （right） ideals of S if andonly if fACis the anti fuzzy left（right） ideals with threshods （λ, ） of S.Theorem 2.1.6 Let S be ordered semigroup. A be bi-ideals of S if and only if fACis the anti fuzzy bi-ideals with threshods （λ, ） of S.Theorem 2.1.7 Let S be ordered semigroup. A be interior ideals of S if and onlyif fACis the anti fuzzy interior ideals with threshods （λ, ） of S.Theorem 2.1.8 Let S be ordered semigroup. A be quasi-ideals of S if and only iffACis the anti fuzzy quasi-ideals with threshods （λ, ） of S.Theorem 2.1.9 Let S be ordered semigroup. If f is the anti fuzzy quasi-idealswith threshods （λ, ） of S,Then f is the anti fuzzy bi-ideals with threshods （λ, ） ofS.Theorem 2.1.10 Let S be ordered semigroup. If f is the anti fuzzy bi-ideals withthreshods （λ, ） of S,Then f is the anti fuzzy interior ideals with threshods （λ, ） ofS.Theorem 2.2.1 Let S be ordered semigroup. The following statements are equiv-alent:（1）S is regular ordered semigroups;（2）（f*Θf* ）∧≤f∨λfor every anti fuzzy quasi-ideal f of S with threshods（λ, ） ;（3）（f*Θf* ）∧≤f∨λfor every anti fuzzy bi-ideal f with threshods （λ, ）of S .Theorem 2.2.2 Let S be ordered semigroup.The following statements are equiv-alent:（1）S is regular ordered semigroups;（2）（f g）∧≤（f∨g）∨λfor every anti fuzzy right ideal f with threshods（λ, ） and every anti fuzzy left ideal g with threshods （λ, ） of S. Theorem 2.2.3 Let S be ordered semigroup.The following statements are equiv-alent:（1）S is regular ordered semigroups;（2）（f*g*h）∧≤（f∨g∨h）∨λfor every anti fuzzy right ideal f with threshods（λ, ） , every anti fuzzy quasi-ideal g with threshods （λ, ） and every anti fuzzy leftideal h with threshods （λ, ） ;（3）（f*g*h）∧≤（f∨g∨h）∨λfor every anti fuzzy right ideal f with threshods（λ, ） , every anti fuzzy bi-ideal g with threshods （λ, ） and every anti fuzzy left idealh with threshods （λ, ） .Theorem 2.2.4 Let S be ordered semigroup with an identity element .S is semisim-ple if and only if （f g）∧≤（f∨g）∨λfor every anti fuzzy interior ideals f and gwith threshods （λ, ） .Theorem 2.2.5 Let S be ordered semigroup.S is intra-regular if and only if（f*g）∧≤（f∨g）∨λfor every anti fuzzy left ideal f with threshods （λ, ）and every anti fuzzy right ideal g with threshods （λ, ） .Theorem 2.2.6 Let S be ordered semigroup.S is left （right） quasi-regular if andonly if （f*f ）∧≤f∨λfor every anti fuzzy left ideal f with threshods （λ, ） .In Chapter 3 ,we mainly define the anti t fuzzy fuzzy bi-ideals and anti t fuzzyfuzzy quasi-ideals , and discuss their e normality , The main results are given asfollowing:Theorem 3.1.1 Let S be ordered semigroup, T is an anti t fuzzy operator, B isan unempty subsets of S, Then B is bi-ideals of S if and only if fBCis anti t fuzzyfuzzy bi-deals.Theorem 3.1.2 Let S be ordered semigroup, T be an anti t fuzzy operator, Q isan unempty subsets of S,Then Q is quasi-ideals of S if and only if fQCis anti t fuzzyfuzzy quasi-deals.Theorem 3.1.3 Let S be ordered semigroup, f∈F （S）, T be anti t fuzzy oper-ator. f is anti t fuzzy fuzzy bi-ideals if and only if for every x, y∈S:（1）x≤y = f （x）≤f （y）; （2）f（?）fΘf.Theorem 3.1.4 Let S be ordered semigroup, f∈F （S）, T be an anti t fuzzyoperator and T be idempotent , if f is anti t fuzzy fuzzy quasi-ideals , then f is antit fuzzy fuzzy bi-ideals.Theorem 3.1.5 Let S be ordered semigroup, T be an anti t fuzzy operator andT be idempotent , The following statements are equivalent:（1） every bi-ideal is a right （ left ） ideal of S;（2） every anti t fuzzy fuzzy bi-ideal is an anti fuzzy right （ left ） ideal .Theorem 3.1.6 Let S be ordered semigroup, T be an anti t fuzzy operator andT be idempotent .Then S is regular if and only if f g = f∪Tg, for every anti fuzzyright ideal f and anti fuzzy left ideal g of S.Theorem 3.1.7 Let S be ordered semigroup, T be an anti t fuzzy operator andT be idempotent. S is regular if and only if f = fΘf , for every anti t fuzzyfuzzy bi-ideals f .Theorem 3.2.1 Let B be bi-ideals （ quasi-ideals ）, and fBC（e） = 0. Then fBCisanti t fuzzy fuzzy bi-deals （ anti t fuzzy fuzzy quasi-ideals ） with e normality ofS, and SfBC= B.Theorem 3.2.2 Let f, g be anti t fuzzy fuzzy bi-deals （ anti t fuzzy fuzzyquasi-ideals ） , Then Sf∩Sg Sf∪Tg.Theorem 3.2.3 Let f, g be anti t fuzzy fuzzy bi-deals （ anti t fuzzy fuzzyquasi-ideals ） with e normality of S, Then Sf∩Sg= Sf∪Tg.Theorem 3.2.4 Let f∈F （S）, T be an anti t fuzzy operator , and T （f （x）, f （y））α≤T （f （x）α, f （y）α）, for all x, y∈S,α∈[0, 1].if f is anti t fuzzy fuzzy bi-dealsof S,then f+is anti t fuzzy fuzzy bi-deals with e normality , and f+contains in f .Theorem 3.2.5 Let f∈F （S）, T be an anti t fuzzy operator , and T （f （x） x∈Aα,f （x）α）≥T （f （x）,f （y））α. for every x, y∈S, A, B S, andy∈Bx∈Ay∈BA, B≠φ,α∈[0, 1]. If f is anti t fuzzy fuzzy quasi-deals of S, then f+is antit fuzzy fuzzy quasi-deals with e normality , and f+contains in f .Theorem 3.2.6 Let f be anti t fuzzy fuzzy bi-deals （ anti t fuzzy fuzzy quasi- ideals ） of S. if there exist anti t fuzzy fuzzy bi-deals （ anti t fuzzy fuzzy quasi-ideals） g of S such that f g, then f has e normality .Theorem 3.2.7 Let S be ordered semigroups, f be anti t fuzzy fuzzy bi-deals,θ: [0, 1]â†'[0, 1] is monotone increasing function, fθ∈F （S）, and define fθ（x） =θ（f （x））, for all x∈S, T is anti t fuzzy operator, and satisfiesθ（T （f （x）, f （y））） =T （θ（f （x））,θ（f （y）））, then fθis anti t fuzzy fuzzy bi-deals of S.Theorem 3.2.8 Let S be ordered semigroups, f be anti t fuzzy fuzzy quasi-deals,θ: [0, 1]â†'[0, 1] is monotone increasing function, fθ∈F （S）,define fθ（x） =θ（f （x））,for all x∈S, T is anti t fuzzy operator,and satisfies T (θ（f （x））,θ（f （y）） =x∈Ay∈Bθ(T （（f （x））,（f （y））, in which A, B is not empty, and A, B S, Then fθis antix∈Ay∈Bt fuzzy fuzzy quasi-deals of S.Theorem 3.2.9 Let S be ordered semigroups, f be an unconstant anti t fuzzyfuzzy bi-deal with e normality （ on inclusion relations of anti t fuzzy fuzzy bi-dealwith e normality ）, T is anti t fuzzy operator andα（T （f （x）, f （y）））≤T （αf （x）,αf （y））,for everyα∈[0, 1]. Then Imf = {0, 1}.Theorem 3.2.10 Let S be ordered semigroups, f be an unconstant anti t fuzzyfuzzy quasi-deal with e normality （ on inclusion relations of anti t fuzzy fuzzy quasi-deal with e normality ）, T is anti t fuzzy operator andα（T （f （x）,f （y））≤x∈Ay∈AT （αf （x）,αf （y））, for everyα∈[0, 1], Then I mf = {0, 1}.x∈Ay∈BIn Chapter 4,we mainly gives （∈,∈∨q（λ,μ）） fuzzy （weakly） regular subsemi-groups and （∈,∈∨q（λ,μ）） fuzzy （weakly） complete regular subsemigroups of S,andtheir characterizations,The main results are given as following:Theorem 4.1.1 The fuzzy subset f of ordered semigroup S is generalized antifuzzy regular subsemigroups if and only if fα≤≠φis regular subsemigroups of S, forallα∈[λ, ）.Theorem 4.1.2 The unempty subset A of ordered semigroup S is regular subsemi-groups if and only if fACis generalized anti fuzzy regular subsemigroups of S.Theorem 4.1.3 The fuzzy subset f of ordered semigroup S is a generalized antifuzzy regular subsemigroup if and only if f is a （∈,∈∨q（λ,μ）） fuzzy regular subsemi- group.Theorem 4.1.4 The fuzzy subset f of ordered semigroup S is generalized antifuzzy weakly regular subsemigroups if and only if fα≤is regular subsemigroups of S,for allα∈[λ, ）.Theorem 4.1.5 The unempty subset A of ordered semigroup S is regular subsemi-groups if and only if fACis generalized anti fuzzy weakly regular subsemigroups of S.Theorem 4.1.6 The fuzzy subset f of ordered semigroup S is a generalized antifuzzy weakly regular subsemigroup if and only if f is a （（?）（λ,μ）） fuzzy weaklyregular subsemigroup.Theorem 4.2.1 Let S be ordered semigroups,fθ∈F （S）. Then the following state-ments are equivalent:（1）f is a （∈,∈∨q（λ,μ）） fuzzy complete regular subsemigroup of S;（2）f is a generalized fuzzy completely regular subsemigroup;（3）fα≠φ{x∈S|f （x）≤α}≠φis a completely regular subsemigroup of S, for allα∈(λ, ].Theorem 4.2.2 The unempty subset A of ordered semigroup S is completelyregular subsemigroups if and only if fACis generalized fuzzy completely regular sub-semigroups of S.Theorem 4.2.3 The fuzzy subset f of ordered semigroup S is a generalized fuzzyregular subsemigroup if and only if the unempty subset fα>is a subsemigroup of S,for allα∈(λ, ].Theorem 4.2.4 The fuzzy subset f of ordered semigroup S is a generalized fuzzyweakly completely regular subsemigroup if and only if the unempty subset fα>is acompletely subsemigroup of S, for allα∈(λ, ].Theorem 4.2.5 The unempty subset A of ordered semigroup S is completely reg-ular subsemigroups if and only if fACis generalized fuzzy weakly completely regularsubsemigroups of S.Theorem 4.2.6 Let f be fuzzy subset of S, then the following statements areequivalent: （1）f is a （∈,∈∨q（λ,μ）） fuzzy weakly complete regular subsemigroup of S;（2）f is a generalized fuzzy weakly completely regular subsemigroup;（3）fα>= is a completely regular subsemigroup of S,for allα∈(λ, ]. |
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