The Inner-shells Of Fuzzy Ordered Semigroups And Fuzzy S-partially Ordered Sets

Quantale,as a special ordered semigroup,was introduced by C.J.Mulvey in 1986,in order to provide a new lattice-characterization for studying non-commutative C*-algebras,as well as a new mathematical model for quantum mechanics.For non-commutative structures,the theory of quantales provides a new studying method.Therefore,many scholars have done an in-depth study on the theory of quantales in recent years.In 1965,the concept of fuzzy sets was proposed by L.A.Zadeh,as well as marked the birth of fuzzy mathematics.Due to its birth,the applied range of mathematics was expanded from accurate phenomenon to fuzzy phenomenon.This thesis is based on fuzzy ordered semigroups and the theory of fuzzy sets is applied to the S-posets and S-quantales.The injective hulls of fuzzy ordered semigroups and S-posets are studied.The structure of this thesis is organized as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and relevant conclusions used in this paper are given.Chapter Two:Injective hulls of fuzzy ordered semigroups.Firstly,we introduce the notions of fuzzy ordered semigroups and fuzzy quantales.Secondly,we disscuss the injective objects in the category of fuzzy ordered semigroups and prove that a fuzzy ordered semigroup S is ξ≤-injective if and only if it is a fuzzy quantale.At last,we show that,for a certain class of fuzzy ordered semigroups,ξ≤injective hulls exist,as well as a characterization of the ξ≤-injective hulls is given.Chapter Three:Injective hulls of fuzzy S-posets.Firstly,on the bases of fuzzy posets and fuzzy quantales,the concepts of fuzzy S-posets and fuzzy S-quantales are introduced.Secondly,we construct an ε≤-injective fuzzy S-poset starting from an arbitrary fuzzy S-poset,it is proved that ε≤-injective objects in the category of fuzzy S-posets axe precisely fuzzy S-quantales.Finally,a character-ization of the ε≤-injective hulls in the category of fuzzy S-posets is given.