Several Studies On Anti Fuzzy Subset With Thresholds (λ,μ) Of OrderedΓ-Semigroups

This dissertation gives the concepts of （E, E V^’（^,,））-anti fuzzy inverse sub-semigroups and anti fuzzy anti Covered-left （bi-, interior） ideals with thresholds （t,#） in ordered F-semigroups. It also studies the properties and characterizations of anti fuzzy inverse semigroups with thresholds （t,#） and anti fuzzy anti Covered-left （bi-, interior） ideals with thresholds （A,#） in ordered P-semigroups. It describes the regularity of the ordered F-semigroups taking advantage of anti fuzzy left （right） ideals with thresholds （t,#） of ordered F-semigroups. This dissertation is divided into four chapters, the main contents of each chapter are as follows:The first chapter mainly gives the basic concepts and the symbols used in this dissertation.The first section of the second chapter mainly gives the concept of anti fuzzy inverse subsemigroups with thresholds （t,#） of ordered F-semigroups, discussing its characterizations and related properties. The main results are as follows:Theorem2.6Let f be a fuzzy subset of ordered F-inverse semigroup S. f is an anti fuzzy inverse subsemigroup with thresholds （t,#） of S if and only if for all c^C [A,#) such that f^#, fⁱs an inverse subsemigroup of S.Theorem2.8Let F be a nonempty subset of ordered F-inverse semigroup S. F is inverse subsemigroup of S if and only if the complement （fF）c of characteristic function of F is an anti fuzzy inverse subsemigroup with thresholds （t,#） of S.Theorem2.9Let {fi I i E I} be a family of anti fuzzy inverse subsemigroups with thresholds （t,#） of an ordered F-inverse semigroup S, then U<I fⁱs an anti fuzzy inverse subsemigroup with thresholds （t,#） of S. Theorem2.10Let {fi|i∈I} be a family of anti fuzzy inverse subsemigroups with thresholds （λ,μ） of an ordered Γ-inverse semigroup S, then∪i∈I fi is an anti fuzzy inverse subsemigroup with thresholds （λ,μ） of S.The second section of the second chapter mainly gives the definition of （∈,∈∧q（λ,μ）-anti fuzzy inverse subsemigroups of an ordered Γ-inverse semigroup S, and proves that the anti fuzzy inverse subsemigroup with thresholds （λ,μ） of the S consists with the （∈,∈∨q（λ,μ））-anti fuzzy inverse subsemigroup of S. The main results are as follows:Theorem2.12Let f be a fuzzy subset of an ordered Γ-inverse semigroup S. f is an （∈,∈∨q（λ,μ））-anti fuzzy inverse subsemigroup of S if and only if f is an anti fuzzy inverse subsemigroup with thresholds （λ,λ） of S.The first section of the third chapter gives the definition,characterizations and some properties of anti fuzzy anti covered-left ideal with thresholds （λ,μ） of the ordered Γ-semigroups. The main results are as follows:Theorem3.2Let f be an order-preserving fuzzy subset of an ordered Γ-semigroup S, then the following conditions are equivalent:（1） f is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S;（2）（（?）x,y∈S,γ∈Γ） f（xγy）∧≤fc（y）∨λ;（3）（（?）x （SΓS]） f（x）∧μ≤[1-∨（y,z）∈Φ（x）f（z）]∨λ,here,（SΓS]={x∈S|（?） y,z∈S,γ∈Γ,x≤yγz}.Theorem3.3Let L be a nonempty subset of an ordered Γ-semigroup S. L is an anti covered-left ideal of S if and only if （fL）c is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S.Theorem3.4Let f1, f2are two fuzzy subsets of an ordered Γ-semigroup S, and f1（?）f2. If f1is order-preserving and f2is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S, then f1also is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S.Corollary3.5Let f1, f2are two fuzzy subsets of an ordered Γ-semigroup S. If one is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S and another is order-preserving, then f1A f2also is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S.Corollary3.6Let {fi,|i∈I} be a family of fuzzy subsets of an ordered Γ-semigroup S. If there exists io∈I such that fio is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S, and for arbitrary i∈I\{io}, fi is order-preserving, then∩i∈I fi also is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S.Theorem3.7Let f be an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of an ordered Γ-semigroup S,[0.5,1]∩[λ,λ）≠0, t∈[0.5,1]∩[λ,μ）, ft={x∈S|（?）（x）≤l}≠0. Then ft is anti covered-left ideal of S.Theorem3.8Let f be a fuzzy subset of an ordered Γ-semigroup S. f is an anti fuzzy anti covered-left ideal with thresholds （λ,μ） of S if and only if the following two conditions hold:（1）（（?）x,y∈S,γ∈Γ,t∈[0,μ）） yt∈fc（?）（xγy）t∈∨q（λ,μ）（?）（2）（（?）x,y∈S,x≤y） yt∈f（?）xt∈f.The second section of the third chapter gives the definition,characterization and some properties of anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of the ordered Γ-semigroups. The main results are as follows:Theorem3.13Let f be an order-preserving fuzzy subset of an ordered Γ-semigroup S, then the following conditions are equivalent:（1） f is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S;（2）（（?）x,x,z∈S,γ,γ2∈Γ） f（xγ1yγ2z）∧μfc（x）∨fc（z）∨λ.Theorem3.14Let f1, f2be order-preserving fuzzy subsets of an ordered Γ-semigroup S, and f1∈f2. If f2is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S, then fs also is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S.Corollary3.15Let f1, f2be order-preserving fuzzy subsets of an ordered Γ-semigroup S. If one is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S, then f1∩f2also is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S.Corollary3.16Let {fi|i∈I} be a family of order-preserving fuzzy subsets of an ordered Γ-semigroup S, If there exists io∈I such that fio is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S, then∩i∈I fi also is an anti fuzzy anti covered-bi-ideal with thresholds （λ,μ） of S.Theorem3.17Let f be an anti fuzzy anti covered-hi-ideal with thresholds （A,,） of the S,[0.^,1] v [A,,)#0, t c [0.^,1] v [A,,), f⁼{, c S I f（*）_&lt;t}#0, then fe is an anti covered-bi-ideal of S.Theorem3.19Let f be a fuzzy subset of an ordered F-semigroup S. f is an (E, E V^,)-anti fuzzy anti covered-bi-ideal of S if and only if f is an anti fuzzy anti covered-bi-ideal with thresholds （k,#） of S.The third section of the third chapter gives the definition,characterizations and some properties of anti fuzzy anti covered-interior ideal with thresholds （k,#） of the ordered F-semigroups. The main results are as follows:Theorem3.22Let f be an order-preserving fuzzy subset of an ordered F-semigroup S, then the following conditions are equivalent:（1） f is an anti fuzzy anti covered-interior ideal with thresholds （k,#） of S;（2）（Vz’,/,z c S,71,72c I2） f（x%/V2z） A#_&lt;f/ V A;（3）（Vs E （SFSFS]） f（s） A#_&lt;[1-V（.,>）65（.）f（/）] V A, here,（SFSFS]={5c s I3.,>=c s, ffs, ffs c r,5_&lt;ffsyffs=},Theorem3.23Let L be a nonempty subset of an ordered F-semigroup S. Then L is an anti covered-interior ideal of S if and only if （ff）c is anti fuzzy anti covered-interior ideal with thresholds （k,#） of S.Theorem3.24Let fl, f2be two fuzzy subsets of an ordered F-semigroup S, and fl C_f2. If fl is order-preserving and f2is an anti fuzzy anti covered-interior ideal with thresholds （k,#） of S, then fl also is an anti fuzzy anti covered-interior ideal with thresholds （A,#） of S.Corollary3.25Let fl, f2be two fuzzy subsets of an ordered F-semigroup S. If one is an anti fuzzy anti covered-interior ideal with thresholds （k,#） of S and another is order-preserving, then fl r] f2also is an anti fuzzy anti covered-interior ideal with thresholds （A,#） of S.Corollary3.26Let {fi [i E I} be a family of fuzzy subsets of an ordered F-semigroup S. If there exists io C I such that rio is an anti fuzzy anti covered-interior ideal with thresholds （k,#） of S, and for arbitrary i C I\{io}, L is order-preserving, then [-]<I fi also is an anti fuzzy anti covered-interior ideal with thresholds （k,#） of S.Theorem3.27Let f be an anti fuzzy anti covered-interior ideal with thresh-olds （A,#）of an ordered I?-semigroup S,[0.5,1]n[l,,）#0, t[0.5,1]n[A,,）, ff= {z c S I f（z）_&lt;t}(3, then A is an anti covered-interior ideal of S.Theorem a.ao Let S be an ordered F-semigroup. Consider the following conditions:（1）（W’,V,z C S,1,2C r） f（zlW2z） A,_&lt;f（v） V L（V） V A;（2）(Vs E （SFSFS]） f（s） A#_&lt;v A. If f is complete anti fuzzy interior ideal with thresholds （/,#） of S, then condition （1） holds, condition （1） and condition （2） are equivalent.The fourth chapter gives the concept of the anti fuzzy left （right） ideal with thresholds （A,#） of the ordered F-semigroups, and discusses its properties, describes regular ordered F-semigroups taking advantage of them. The main results are as followsTheorem4.1Let fl, f2,91,92be fuzzy subsets of an ordered F-semigroup S, and fl_&lt;91, f2_&lt;92, then （Vz C S）（fl’6f2）（:r） A#<_91"92（:r） V,.Theorem4.4Let f be a fuzzy subset of an ordered F-semigroup S. f is an anti fuzzy left ideal with thresholds （k,#） of S if and only if the following two conditions hold:（1）（VzC S） f（z） A#_&lt;（S’f）（z） V X;（2）（Vz,/CS） z_&lt;/=>f（z）_&lt;f（/）.Theorem4.5Let f be a fuzzy subset of an ordered F-semigroup S. f is an anti fuzzy right ideal with thresholds （A,#） of S if and only if the following two conditions hold:（1）（VzcS） f（z） A#_&lt;（faS）（z）Vk;（2）（Vz,/CS）,_&lt;/=>f（z）_&lt;f（/）.Theorem4.6Let f be an anti fuzzy right ideal with thresholds （k,#） of an ordered F-semigroup S,9is an anti fuzzy left ideal with thresholds （k,#） of S, then （x C S） If（x） V g（x）] A#<_f"@（x） V,.Theorem4.8An ordered r-semigroup S is regular if and only if every fuzzy subset f of S satisfies:（z c S）（f"6S’f）（z） A#<_f（z） V,. Theorem4.9Let an ordered Γ-semigroup S be regular, f is an anti fuzzy right ideal with thresholds （λ,μ） of S, g is a fuzzy subset of S, then （x∈S）（fog）（x）∧μ≤f（x）∨g（x）∨λ.Theorem4.10Let an ordered Γ-semigroup S be regular, g is an anti fuzzy left ideal with thresholds （λ,μ） of S, f is a fuzzy subset of S, then （x∈S）（fog）（x）∧μ≤f（x）∨g（x）∨λ.Theorem4.15Let S be an ordered Γ-semigroup, then the following conditions are equivalent:（1） S is regular;（2） every anti fuzzy right ideal f with thresholds （λ,μ） and every fuzzy subset g of S satisfy （x∈S）（fog）（x）∧μ≤f（x）∨g（x）∨λ;（3） every anti fuzzy left ideal g with thresholds （λ,μ） and every fuzzy subset f of S satisfy （x∈S）（foy）（x）∧μ≤f（x）∨g（x）∨λ.</sup>