| Quantales were introduced by Mulvey in 1986 with the purpose of studying the spectrum of C*-algebra, in order to provide constructive foundations for quantum mechanics. In 1990, the publication of Rosenthal's research monograph on quantale greatly promotes the development of quantale theory. Because of combining the ordered structure and the algebraic structure, quantale plays a crucial role in pro-moting the inter-development of other branches of Mathematics. Founded by Scott in the late 1960s, domain theory consists of a wide range of content and structure, which makes the theoretical computer science and mathematics researchers showing common interest in it. In this paper, the concept of continuous directed-complete ordered semigroup as the common promotion of quantale and meet continuous semi-lattice's structure is introduced. What's more, continuous directed-complete ordered semigroup's categorical properties and the directed-completion of ordered semigroup are further investigated. The structure of this thesis is organized as follows:In the first part, we have studied continuous directed-complete ordered semi-group and its categorical properties. Firstly, the concept of continuous directed-complete ordered semigroup is introduced. Secondly, some equivalent characteriza-tions of continuous directed-complete ordered semigroup are obtained. Finally, it is proved that the category CDOSG with objects all continuous directed-complete ordered semigroups and morphisms all ordered semigroup homomorphisms preserv-ing directed sups between them is finite complete. Moreover, its full subcategory CCDOSG with objects all commutative continuous directed-complete ordered semi-groups is finite complete and closed for function space. All of these make prepare for the research of the cartesian of CDOSG.In the second part, the directed-completion of ordered semigroup is discussed. Firstly, the concept of directed-completion of ordered semigroup is given. Secondly, it is proved that for every ordered semigroup, the set of all its ordered ideals and the set of all its ideals are both continuous directed-complete ordered semigroups. Moreover, the set of all the ordered ideals is the directed-completion for every ordered semigroup and the set of all the ideals is the directed-completion for every negative ordered semigroup. Finally, the reflective relationship among some full subcategories of the category of ordered semigroups is obtained. |
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