S - Topology And Pre - Quantale Model Of Topological Groups On Semigroups

Domain theory was established by D.Scott in the early 1970s, and it aims at providing mathematic model for semantics of programming languages. It is char-acterized by the close connections between order and topology, which make it a study field of experts on both theoretical computer science and mathematics and widely applied since it is established. The work of maximal point space of domain theory can date back to Scott, Kainura, Abramsky and others’work 20 years a-go. The maximal space is an important space which combines topology and order closely, one of the most central problems in domain theoretic studies of topological spaces. It provides continuous domain environment for some topologies, and it is the bridge between continuous domain environment and classical mathematics, in this paper, we discuss the topological structure on ordered semigroup,and the notion-s of S’-topology and strongly S-topology introduced. Moreover, research methods mentioned above can be generalized to the field of topological semigroup, which combine topological structure and the operation · of a (semi)group. The notions of maximal point topological (semi)group and prequantale model are proposed, and we investigate the prequantale model problems of topological (semi)groups.The arrangement of this paper is as follows:Chapter one:Preliminaries. We recall some basic notions and results of domain theory, quantale theory, topological groups and related structures.Chapter two:S-topology on ordered semigroups. Firstly, the notions of S-topology and strongly S-topology on ordered semigroups are proposed and its prop-erties are discussed. Secondly, equivalent characterizations of S’-closed subset and strongly S-closed subset are given. Lastly, it is proved that the family of all S’-closed sets forms an algebraic and completely distributive lattice under the set-inclusion order.Chapter three:The characterizations of semitopological semigroups and topo-logical groups. Firstly, we introduce the notions of prequantale and local prequantale and give the Quantale characterization of semitopological semigroup. Secondly, we propose notions of maximal point topological (semi)group and prequantale model, investigate the prequantale models of topological groups. It is proved that every topological group has a algebraic local prequantale model and every topological group has a prequantale model. Finally, we introduce the notions of bitopological semigroup and bisemitopological semigroup, we prove that:a multiplicative and continuous T1 topological semilattice (S, τ, ∧) has a continuous preframe model if and only if there is a T1 topology τ* on S such that (S, τ,τ*,∧) is a pairwise com-pletely regular topological semilattice.