On Topological Duality Of Several Special Posets

Order structures,topological structures and algebraic structures are called the three parent structures in mathematics.The intersection and merger of the three structures greatly promote the development of mathematics.Stone duality theory was established in the 1930s when M.H.Stone developed topological representation of Boolean algebras and distributive lattices.The theory reveals the profound relationship between order structures and topological structures,which makes the method of using order theory to study topology more mature and perfect.At the same time,it also provides a powerful tool for making the use of topological space to research order structures.Based on the work of M.H.Stone,the topological duality for lattice order structures has attracted many scholars'attention.The Stone-type duality of lattice order structures such as spatial frames,distributive continuous lattices,distributive join semi-lattices has been built successively.In recent years,researching on the related structures of posets has received extensive attention with the development of the study of order theory.The topological duality of distributive posets,posets and strongly Boolean posets has been provided.As a special kind of posets,directed complete posets play an important role in the study of denotational semantics for functional programming languages.In this paper,we focus on the topological duality of directed complete posets and related structures.The main contents are listed as follows:The first part studies the Stone-type topological duality for directed complete posets with top element and complete lattices.By defining the Scott open prime sub-sets and endowing the family of Scott open prime subsets with the Hull-Kernel topology,we obtain the dual space of a directed complete poset with the top element.The defini-tion of SF-sober spaces is introduced and the relationship between SF-sober and sober is discussed.It is proved that the dual space satisfies the property.Based on these results,the new concept of-spaces is introduced by analyzing the topological prop-erties of the dual space.With this concept,a topological duality for directed complete posets with top element is developed,and a dual equivalence between the category of directed complete posets with top element with corresponding mappings and the cate-gory of-spaces with corresponding maps is shown.Finally,following by this idea,a topological duality for complete lattices is established and the categorical equivalence between the category of complete lattices with complete lattices homomorphisms and the category of-spaces with corresponding maps is provided.In the second part,we focus on the topological duality for general directed com-plete posets.The dual space for directed complete posets is defined by endowing the ideal posets with Scott topology.The new concept of-spaces is introduced by s-tudying the topological properties of the dual space.With this concept,a topological duality for directed complete posets is proposed,and the dual equivalence between the category of directed complete posets with corresponding mappings and the category of-spaces with-continuous maps is demonstrated.In the third part,we study the topological duality of countably directed complete posets(c-dcpos,for short).The dual space for c-dcpos is defined by endowing the family of countable filters with-Scott topology.We obtained that the dual space of c-dcpos is a special-space by using the supremum of countably directed subsets of countable filters is a countable filter.The new concept of₍(8)-spaces is introduced by studying the topological properties of the dual space.With this concept,a topological duality for c-dcpos is built,and the dual equivalence between the category of c-dcpos with corresponding mappings and the category of₍(8)-spaces with corresponding maps is showed.