A Few Class Order Homomorphism And Their Properties

An order homomorphism was defined between fuzzy lattices. On one hand, it preserves hights of fuzzy points. On the other hand it retains properties that maps molecular to molecular. Later, professor Wang Guojun removed the condition of involution and proposed the concept of generalized order homomorphisms in the realm of completely distributive lattices. In this paper, we consider to get rid of the prerequisite conditions of fuzzy lattices and completely distributive lattices to study wider range of generalized order homomorphisms. We generalize order homomorphisms to complete lattices, continuous lattices, domains and quasicontinous domains to explore all kinds of generalized order homomorphisms. To some extent, we expand the scope of applications of order homomorphisms. This article obtained certain conditions for a generalized order homomorphism being equal to the inverse of its inverse. We characterized a pseudo generalized order homomorphisms between complete lattices. We defined a0-minimal set and proved that a continuous lattice is a completely distributive lattice if (?)a, b∈L, one has B(aVb)=B(a)UB(b). We define Scott generalized order homomorphisms between domains and obtained some necessary and sufficient condition, as well as related properties for them. Proposed some equivalent conditions for maps injective or surjective into compact element between algebraic domains to be Scott generalized order homomorphism. At the same time by defining imitate directional minimal sets to obtain para Scott generalized order homomorphism concept between quasicontinuous domains. Through these studies, we can undersand and grasp the common essential characteristics of generalized order homomorphisms. This paper promotes studies of maps between completely distributive lattices, in particular, maps between lattices, even maps between posets.The first chapter as a preparation, with emphasis on partial order, lattices, molecular lattices, complete lattices, domains, generalized order homomorphisms and other related concepts and their properties.In chapter2, the properties of generalized order homomorphisms between completely distributive lattices are explored. The link between f and f-1are explored, conditions for (f-1)-1=fare given.In chapter3, the concept of pseudo-generalized order homomorphisms between complete lattices are defined. Tthe definition of prime upper set is proposed, a necessary and sufficient condition of a map between complete lattices being pseudo-generalized order homomorphisms is obatined. At the end, the concept of a θ-minimal set are defined, sufficient conditions of a continuous lattice to be a completely distributive lattice are given.In chapter4, the notion of Scott generalized order homomorphisms between domains are proposed. Characterizations and properties of Scott generalized order homomorphisms are obtained. By the set of compact elements in algebraic domain, we show that a map between sets of compact elements of algebraic domains is a Scott generalized order homomorphism under some assumption. By the notion of quasi-directed minimal set of quasicontinuous, we also show that para Scott generalized order homomorphisms have some equivalent conditions.Finally, we considered some possible future work on this subject.