The Theory Of Semicontinuous Lattices

The content of this paper consist of three parts.The first part of the paper includes Chapters II, III, IV and V. In the 1970sScott was led by problems of semantics for computer languages to initiate thestudy of continuous lattices. Almost at the same time, in the fields of pureMathematics and Computer Sciences, Lawson, Ho?man, Stralka, etc. defined akind of complete lattices with special properties to characterize a class of compactsemilattices. They soon recognized that the complete lattice defined by themwas precisely the continuous lattice given by Scott. From the different areas,they discussed the same thing that led to the growth of continuous lattice theory.With continuous lattice theory development, it has been studied extensively bymany people from various fields due to its close connection to algebra, logics,general topology and computer science and a great many results about continuouslattices and domain theory were obtained.Later the continuous lattices were generalized to various classes of orderedstructures for di?erent motivations, for example, Gierz, Lawson, Jung, etc. intro-duced the concepts of generalized continuous lattices and hypercontinuous lat-tices, the L-domains, and the most general, continuous dcpos. With the deeperresearch, Bandelt and Erne introduced the concepts of Z-continuous posets. In1997, in order to get more general results on the relationship between primeand pseudo-prime elements in a complete lattice, two new classes of lattices re-lated to continuous lattices, namely the semicontinuous lattices and the stronglycontinuous lattices, were introduced by D. Zhao.In Chaper II, we study some basic properties of semicontinuous latticesand introduce semialgebraic lattices, quasi-semicontinuous lattices and meet-semicontinuous lattices. At the same time, we study some properties of them andthe relationship among them. Since there are two important intrinsic topologiesin continuous lattices: the Scott topology and the Lawson topology and thesetopologies have many good properties.In Chapter III, we introduce two new intrinsic topologies on complete lat-tices, which are then used to formulate new characterizations for semicontinuouslattices and strongly continuous lattices. Moveover, we give the new characteri-zations of strongly continuous lattices based on convergence of nets. In Chapter IV, we define several new types of mappings between completelattices and study the relationship among S-continuous mappings, Scott con-tinuous mappings, strongly continuous mappings and semicontinuous mappings.We also study the properties about fixed points of semicontinuous self-mappingsbetween semicontinuous lattices.In the application of computer science, Cartesian closedness of the domaincategory is usually the required property. Thus in Chapter V, we will consider thecategory of strongly continuous lattices and prove that the strongly continuouslattice category is Cartesian closed. It is also pointed out that the subcategoryof distributive continuous lattices is cartesian closed. We discuss the conditionsunder which the function space of semicontinuous lattices is semicontinuous.In the second part, i.e., Chapter VI, we mainly study the meet continuousposets. We will firstly extend these classical results of meet continuous dcpos tomeet continuous posets. And we establish the relations between meet continu-ous posets and W-continuous posets. Moveover, we will introduce the lim-inf*convergence in a poset which generalized the lim-inf convergence introduced byB. Zhao etc. We obtain the conditions under which the lim-inf* convergence istopological.In the third part, i.e., Chapter VII, we mainly study the characterizations ofcontinuous lattices and Z-continuous posets. We firstly introduce quasi-compactsin continuous lattices, which are different with meet-irreducible elements andjoin-irreducible elements. And then we give quasi-bases of continuous latticesand a procedure for generating continuous lattices using the quasi-bases. At last,we will introduce the concept of Z-embedded basis for posets and establish somecharacterization theorems for Z-continuous posets by the technique of embed-ded Z-basis. Moreover, we will discuss the conditions under which the Z-idealcompletion of a abstract Z-basis is Z-algebraic.