Research On Related Issues In Ordinal Algebra

After Banaschewski's study on the join completions of posets,more and more researchers have focused on the completions of various ordered structures.Just as complete lattices can be viewed as the completions of partially ordered sets,a series of complete ordered algebraic structures can also be viewed as the completions of corresponding ordered algebraic structures.Compared with the the completions of ordered structures,the completions of ordered algebraic structures are more extensive,and so it is meaningful for further study.The injective hull,though not under the name,was first obtained by Baer in 1940.The best-known early discussion of injective hulls was given by Bckmann and Schopf in 1953.Later,the injectivity in the categories of various ordered structures was intensively introduced and investigated by Banaschewski,Bruns,Lasker et al and so on.In recent years,the injectivity in the categories of various ordered algebraic structures has caused widespread concern by many researchers.On the one hand,we shall start to study the completions of several kinds of ordered algebraic structures.We propose the concepts of the completions of posemi-groups,left Q-posets,ordered algebras and the corresponding topological nuclei,and investigate the relationship between the above completions and topological nuclei.Si-multaneously,constructing the completions of such ordered algebraic structures is also the important content of this paper.For instance,we give the concrete structure of the least quantale(sup-algebra)completion of a posemigroup(an ordered algebra).Moreover,we shall investigate a special class of quantale completions of posemigroup-s,that is,the precoherent quantale completions of posemigroups,and give the concrete form of the least precoherent quantale completion of a posemigroup.In addition,we shall study the semiclosure semigroups and semitopological groups from the perspec-tive of quantales,obtain a categorical isomorphism between the category of strong semiclosure semigroups and the category of quantales,and give a characterization of quantales by the semitopological groups.On the other hand,we shall study the injectivity of the above ordered algebraic structures.We construct the injective hulls of posemigroups and ordered algebras,and study the injective objects in the category of algebraic posemigroups.This paper is divided into 5 chapters.In Chapter 1,we recall some basic concepts and results of lattices and quantales,and introduce some basic notions and results on category theory and general topology.In Chapter 2,we firstly consider the quantale completions of posemigroups,give the concrete form of the least quantale completion of a posemigroup,and obtain it-s corresponding application,that is,the least quantale completion is precisely the injective hull in the category of posemigroups.Moreover,the concept of the prequan-tale completions of posemigroups is introduced,and we prove that every posemigroup has such a completion.Finally,we investigate the left Q-module completions of left Q-posets.In chapter 3,we consider a special class of quantale completions of posemigroups,that is,the precoherent quantale completions,and construct three types of precoherent quantale completions of a posemigroup,including the least,the largest quantale com-pletions and Fd-quantale completion.Moreover,the notion of algebraic posemigroups is introduced,and we prove that the injective objects in the category of algebraic posemigroups are precisely the precoherent quantales.Chapter 4 is organized as follows.Firstly,we propose the notion of sm-ideals on a posemigroup S,discuss its several basic properties,and prove that the family K(S)of the sm-ideals of a posemigroup S is the sm-universal quantale completion and K(S)is also a new quantale completion of S.Furthermore,we study the the sm-universal conditional complete quantale completions of posemigroups,prove that the family of the upper bounded sm-ideals of S is just the sm-universal conditional complete quantale completion of S.Thus,we obtain that the category Quant of quantales and the category CQuant of conditional complete quantale are two full reflective subcategories of the category PoSgrv of posemigroups.Secondly,we give the concepts of(strong)complete semiclosure semigroups,obtain that the category SCG of strong semiclosure semigroups is a full reflective subcategory of the category STopo of T0-semiclosure semigroups,and prove that the categories PoSgrv(Quant)and CSS(SCG)are isomorphic.Similarly,we also consider the relationship between the category of conditional strong semiclosure semigroups and other categories.Fi-nally,we discuss several properties of semitopological groups from the perspective of quantales.In chapter 5,The notions of the sup-algebra completion of ordered algebra and topological nucleus are introduced.We prove that the sup-algebra completions of an ordered algebra and topological nuclei,up to isomorphism,are one-to-one correspond-ing,and obtain the concrete structures of three kinds of sup-algebra completions,that is,the least,the largest and the completion of all D-ideals the ordered algebras.Also,we give the concrete form of the injective hull in the category of ordered algebras,which is in fact the least sup-algebra completion.In addition,we propose the notions of ide-als and ideal conuclei of sup-algebras,prove that they are one-to-one corresponding,and consider the relationship between the ideals and congruences of sup-algebras.