Semigroup On The Topology And Partial Order And The Associated Domain

Algebra structures, topology structures and ordered structures are the three root structures of mathematics. Interactions of various structures are motivations for deeply developing mathematics. Groups are basic algebra structures. Semigroups are more general than groups. In this paper, topological method and domain theory are used to study semigroups.In Chapter two, the semigroup topology O[G] on a semigroup G is defined, properties of semigroup endowed with semigroup topology are investigated. Main results are: (1) The set O of semigroup G is O[G]-open iff for all g∈O one has ? O; (2) The semigroup topology is closed under arbitrary intersections; (3) (G,O[G]) is a T1- space iff O[G] is discrete, iff every element of G is idempotent. In this chapter we also characterize the closure of every element in a semigroup which is endowed with semigroup topology and explore characters of continuous mappings and continuous open mappings between semigroups. It is proved that every homomorphism of semigroups is a continuous open mapping.In Chapter three the concept of semigroup partial order on a semigroup is introduced. Relationships between closure of an element and its generating subsemigroup in a semigroup are explored. It is proved that for all elements x and y in a semigroup G, conditions that x∈{y}—, y∈ and ? are mutually equivalent and that {x}—= {y}—iff = . If (G,O[G]) is a T0-space, then O[G] is the dual Alexandrov topology on (G,≤G). Open sets in (G,O[G]) are lower sets in poset (G,≤G) and closed sets in (G,O[G]) are upper sets in poset (G,≤G). In Chapter four domain properties of semigroups are investigated. Firstly the concept of finite cyclic semigroup is introduced and properties of such kind of semigroups are explored. A sufficient condition for generating elements of a finite cyclic semigroup is obtained. Secondly domain properties of the semigroup topology and the semigroup partial order are investigated. Main results are: (1) If every generating subsemigroup of elements in a semigroup G is a finite cyclic subsemigroup, then (G,≤G) is an algebraic poset; (2) The topology O[G] is an algebraic and completely distributive complete lattice in the set-inclusion order; (3) A semigroup G is pseudo finite iff poset (G,≤Gop) is an algebraic domain.In the last chapter the semigroup topology and the cyclic group topology, the semigroup partial order and the cyclic group partial order are compared. The cyclic group topology and cyclic group partial order are further investigated. It is proved that if (G,O [G]) is a T0-space then the cyclic group topology O [G] on group G is the dual Alexandrov topology on (G,≤G).