Regularities Of Ordered Ternary Semigroups And Soft Bi-hyperideals Of Ternary Semihypergroups

Von Neumann regularity of semigroups plays an important role in theory of semigroups.Recall that a semigroup S is called(von Neumann)regular if a ? aSa for all a ? S.Iseki in 1956 characterized the regularity of a semigroup in terms of left ideals and right ideals by proving that a semigroup S is regular if and only if AB = A ?B for every right ideal A and left ideal B of S.Since then the regularity was also characterized in Iseki's style by several authors in terms of other kinds of ideals such as two-sided ideal,bi-ideals and quasi-ideals.Other kinds of regularities of semigroups were also studied,for example,left regular and intra-regular.Such regularities were characterized in terms of one-sided ideals,two-sided ide-als,quasi-ideals and bi-ideals.Combining regularities such as regularity and left regularity,regularity and intra regularity,left regularity and intra-regularity,and so on were studied.Since it was introduced by Zadeh in 1965,fuzzy set theory has entered many fields of math-ematics.Among others,fuzzy algebraic systems,for example,fuzzy groups,fuzzy semigroups,fuzzy commutative algebras,fuzzy near-rings and semirings,fuzzy Lie algebras,fuzzy algebraic hyperstructures,and so on,have been developed rapidly.Rosenfeld in 1971 initiated the fuzzification of algebraic composition and pioneered the concept of fuzzy subgroups.The concept of a fuzzy set was applied to semigroups by Kuroki in 1978,and then theory of fuzzy ideals were developed.Several kinds of regularities were characterized in Iseki's style by several authors in terms of fuzzy ideals.Ternary algebraic operations and cubic associations were considered in the 19th century by mathematicians for example Cayley and Sylvester.In 1932,Lehmer worked upon triplexes certain ternary algebraic systems.The concept of ternary semigroups was set off by S.Banach and early researches were done by Los in 1955 and Lyapin in 1981.Studies on regularities of semigroups were extended to ternary semigroups and ordered semi-groups.Sioson in 1963 extended the classical result of Iseki to n-ary setting.Recently Dutta and others studied regularities of ternary semigroups.Kehayopulu,Shabir,Tang and Xie,and others studied regularities of ordered semigroups by means of fuzzy left ideals,fuzzy right ideals and fuzzy quasi-ideals.Ordered ternary semigroups provides a unified setting for ordered semigroups and ternary semigroups.It is natural to extend studies for regularity of ordered semigroups and ternary semigroups to ordered ternary semigroups.Several types of regularity of ordered ternary semi-groups have been illustrated in terms of fuzzy ideals,for example,von Neumann regularity and its generalizations and analogues such as weak regularity,intra-regularity and so on.In Chapter 3 we characterize left weakly regular ordered ternary semigroups.Combining regularities such as intra-regularity and weak regularity,regularity and weak regularity,and regularity and intra-regularity of ordered ternary semigroups are also characterized in terms of fuzzy ideals,respectively.Several results in literature are generalized.Theorem 3.1.7.An ordered ternary sermigroup S is left(resp.right)weakly regular if and only if every fuzzy left(resp.right)ideal f is idempotent.Theorem 3.1.14.Let S be an ordered ternary semigroup and U be a collection of fuzzy subsets of S containing all fuzzy left ideals of S.Then the following are equivalent.(1)S is left weakly regular;(2)f ? f' ? f"(?)f(?)f'(?)f" for any fuzzy left ideals f,f',f";(3)f?g?u(?)f(?)g(?)u for any fuzzy left ideal f and fuzzy two-sided ideal g,and for any u?U;(4)g?h?u(?)g(?)h(?)u for any fuzzy two-sided ideal g and fuzzy right ideal h,and for any u ? U;(5)g?u?f(?)g(?)u(?)f for any fuzzy two-sided ideal g and fuzzy left ideal f,and for any u?U;(6)g?u(?)1(?)g(?)u for any fuzzy two-sided ideal g and for any u?U;(7)h?u(?)1(?)h(?)u for any fuzzy right ideal h and for any u ?U;(8)u?f(?)1(?)u(?)f for any fuzzy left ideals f and for any u ?U.Theorem 3.2.2.Let S be a weakly regular ordered ternary sermigroup.Then a fuzzy subset f of S is a fuzzy quasi-ideal of S if and only if f =(1(?)1(?)f)?(f(?)1(?)1).Theorem 3.2.5.An ordered ternary senlmigroup S is weakly regular if and only if f =(1(?)1(?)f)3 ?(f(?)1(?)1)3 for any fuzzy quasi-ideal f of S.Theorem 3.2.6.An ordered ternary sermigroup S is weakly regular if and only if f =(f(?)1(?)1)3 ?(1(?)f(?)1)3 ?(1(?)1(?)f)3?(f(?)1(?)1)3 ?(1(?)1(?)f(?)1(?)1)3 ?(1(?)1(?)f)3 for every fuzzy quasi-ideal f of S.Theorem 3.3.5.Let S be an ordered ternary sermigroup and U be a collection of fuzzy subsets of S containing all fuzzy left ideals of S.Then S is intra-regular and left weakly regular if and only if one of the following assertions holds.(1)f?g?g'(?)f(?)g(?)g'for any fuzzy left ideal f and any bi-idecds g,g';(2)f?u?f(?)f(?)u(?)f' for fuzzy left ideals f,f' and for any fuzzy subset(resp.bi-ideal,quasi-ideal)u;(3)f?h?u(?)f(?)h(?)u for any fuzzy left ideal f and any fuzzy right ideal S,and for any u ? U;(4)h?u?g(?)h(?)u(?)g for any fuzzy two-sided ideal S,any fuzzy bi-ideal g and for any u ? U;(5)u?g(?)1(?)u(?)g for aty fuzzy bi-iccal g andfor any u?U.Theorem 3.4.4.Let S be an ordered ternary sermigroup.Let LU be a collection of fuzzy lubsets of S containing all fuzzy left ideals of S,U' a collection of fuzzy subsets of S containing all fuzzy right ideals of S and U" a collection of fuzzy subsets of S containing all fuzzy two-sided ideals of S.Then the following are equivalent.(1)S is regular and weakly regular(resp.left weakly regular,right weakly regular);(2)g?g'?u(?)g(?)g'(?)u for any fuzzy right ideals g,g' and for any u ? U;(3)u'?f'?f(?)u'(?)f'(?)f for any fuzzy left ideal f',f and for any u' ? U';(4)g ? u" ? f(?)g(?)u"(?)f for any fuzzy right ideal g,left ideal f and for any u"?U";(5)u'? h ? u(?)u'(?)h(?)u for any fuzzy two-sided ideal S and for any u'?U' and u ? U;(6)g?h?f(?)g(?)h(?)f for any fuzzy right ideal g,fuzzy two-sided ideal a and fuzzy left ideal f;(7)g?f(?)g(?)f(?)1(?)g(?)f for any fuzzy right ideal g and fuzzy left ideal f.Theorem 3.5.3.Let S be an ordered ternary semigroup.Then the following conditions are equivalent.(1)S is regular and intra-regular;(2)Every fuzzy bi-ideal f of S is idempotent;(3)Every fuzzy quasi-ideal f of S is idempotent.The concept of(?,?)-fuzzy subgroups was introduced by Bhakat and Das in 1992,which based on the relation of belong to(?)and quasi-coincident with relation(q)between a fuzzy point and a fuzzy subgroup.Jun et al in 2009 gave the concept of a generalized fuzzy bi-ideal in ordered semigroups and characterized regular ordered semigroups in terms of(?,??q)-ideals.Muhammad et al in 2010 studied the characterizations of regular semigroups using(?,? ?q)?fuzzy ideals.In 2012,Rehman and Shabir characterized ternary semigroups by using(?,?)?fuzzy ideals,and Zeb et al characterized ternary semigroups in terms of(?,??qk)ideals.The concepts of(?,??qk)-fuzzy ideals,(?,??qk)-fuzzy bi-ideals in an ordered semi-groups is introduced by Tang and Xie,and regular ordered semigroups were characterized in terms of(?,??qk)-fuzzy left ideals,(?,??qk)-fuzzy bi-ideals and(?,??qk)-fuzzy general-ized bi-ideals of S in 2014.In Chapter 4,we introduce concepts of(?,??qh)-fuzzy ideals,(?,??qh)-fuzzy bi-ideals and(?,??q)-fuzzy generalized bi-ideals in ordered ternary semigroups,which are generaliza-tions of corresponding(?,??qk)-fuzzy ideals.The characterizations of regular ordered ternary semigroups are given by(?,??qh)-fuzzy left ideals,(?,??qh)-fuzzy bi-ideal and(?,??qh)?fuzzy generalized bi-ideal of S.As an application of results of this chapter the corresponding results of semigroup(without order)are also obtained.Theorem 4.2.7.Let S be an ordered ternary semigroup and f be a fuzzy subset of S.Then f is an(?,? ?qh)-fuzzy bi-ideal of S if and only if f satisfies the following conditions.Theorem 4.2.12.Let S be an ordered ternary semigroup and f a strongly convex fuzzy subset of S.Then f is an(?,? ?qh)-fuzzy bi-ideal of S if and only if the(? ?qh)-level subset[f]t of f is a bi-ideal of S for all t ?(0,1].Theorem 4.2.14.Let S be an ordered ternary semigroup and f be a fuzzy subset of S.Then f is an(?,??qh)-fuzzy generalized bi-ideal of S if and only if f satisfies the following conditions.Theorem 4.2.15.Let S be an ordered ternary semigroup and f a strongly convex fuzzy subset of S.Then f is an(?,? ?qh)-fuzzy generalized bi-ideal of S if and only if the(? ?qh)-level subset[f]t of f is a generalized bi-ideal of S for all t ?(0,1].Theorem 4.3.5.Let S be an ordered ternary semigroup and A be a non-empty subset of S.Then A is a left(rep.right,lateral)ideal of S if and only if fA is an(?,? ?qh)-fuzzy left(resp.right,lateral)ideal of S.Soft set theory was proposed by Molodtsov in 1999 to deal with uncertainty.Aktas and Cagman in 2007 pioneered the fundamental idea of soft set theory.They also discussed the philosophy of soft groups and their essential properties.Marty pioneered the theory of hyperstructures in 1934.He considered a variety of properties of hypergroups and applied them to groups.Briefly,we can speak that hypergroups are exten-sions of classical groups.Many mathematicians have worked on algebraic hyperstructures and generalized diverse classical algebraic structures.Many papers and books have been written on hyperstructure theory.Hyperstructure theory has many applications in different domains of mathematics and computer science.In 2011,Yamak,Kazanci and Davvaz studied soft hyper-structure.In 2013,Hila,Naka,Leoreanu-Fotea,Sadiku,studied ternary semihypergroup,soft set,soft ternary semihypergroup,soft left(lateral,right)hyperideal,soft hyperideal,soft quasi(bi)-hyperideal.In 2014,Naz and Shabir studied prime soft bi-hyperideals of semihypergroups.In Chapter 5,we define and study primeness and semiprimeness of ternary semihypergroups.We generalize the results of Naz and Shabir.Theorem 5.2.6.A soft set is a ternary subsenlhihypergroup of S over U if and only if fA*fA*fA(?)fA.Theorem 5.3.9.For a ternary sermihypergroup S over U,the following assertions are equiv-alent.(1)Every soft bi-hyperideal of S over U is idempotent.(2)(gB*SC*iD)?(SC*iD*gB)(?)gB?SC?iD for every soft bi-hyperideals 9B,SC Sc and iD.(3)Each soft bi-hyperideal of S is semiprime.(4)Each proper soft bi-hyperideal of S over U is the intersection of all irreducible sermiprime soft bi-hyperideals of S over U which contain it.Theorem 5.3.10.Each soft bi-hyperideal of a ternary semihypergroup S over U is strongly prime if and only if each soft bi-hyperideal S over U is idempotent and the set of soft bi-hyperideals of S over U is totally ordered by inclusion.Theorem 5.3.11.If the set of soft bi-hyperideals of a ternary semihypergroup S over U is totally ordered,then each soft bi-hyperideal of S is idempotent if and only if each soft bi-hyperideal of S is prime.Theorem 5.3.12.If the set of soft bi-hyperideals of a ternary semihypergroup S over U is totally ordered,then the concepts of primeness and strongly primeness coincide.Theorem 5.3.13.For a ternary semihypergroup S,the following assertions are equivalent.(1)The set of soft bi-hyperideals of S over U is totally ordered by set inclusion.(2)Each soft bi-hyperideal of S over U i,s strongly irreducible.(3)Each soft bi-hyperideal of S over U is irreducible.