Some Properties Of Several Special Posets

Boolean posets, distributive posets, and modular posets are three special posetsand have some special structures and properties. The research based on thesestructures and properties can give us a better understanding of posets.In Chapter one, we introduce the development of lattices and posets, theorigin of lattice theory is very early, at the end of nineteenth century, Dedekinddefined the concept of lattice, by1930people started to pay attention to it，it is analgebra system followed the introduction of the development of algebra anduniversal algebra of classical logic. Especially, G.Birkhoffs artile《Lattice Theory}laid the base of the modern lattice theory，it is sum up of the lattice theory, theapplication of mathematical logic, universal algebra, general topology, functionalanalysis and probability theory and other branches of mathematics. In recent years,posets and lattice theory is used not only in combinatorial mathematics, fuzzymathematics and theoretical computer science, even in social science have beenwidely used, but also greatly promoted the development of the discipline’s own view,which will be an important research object of mathematics and theoretical computerscience.and then we review some important work on this field. Also, some results onthe structures of posets are described. At last, we give some preliminary knowledgesuch as upper cone, lower cone, complement, pseudocomplement, distributivity,ideal, and so on.In Chapter two, we first focus on the properties of pseudocomplemented posetsand its dual version. Then, the characterization and representation of Boolean posetsare obtained. Also, we build some equivalent forms of distributive posets. Specially,we investigate the relationship of distributive posets and modular posets, as well assome properties of modular posets.