Filters And Isomorphism Theorems In Heyting Algebra And Its Category Heyt

Abstract: Heyting algebra is proposed as an algebra model of intuitionsm prepositional logic, in which the law of excluded middle does not generally hold. Therefore Heyting algebras can be regarded as a generalization of Lindenbaum algebras. A Heyting algebra, from the logical standpoint, is essentially a generalization of the usual system of truth values. The usual two-valued logic system is the simplest example of a Heyting algebra, one in which the elements of the algebra are " true " and " false ". In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. Complete Heyting algebras are a central object of study in pointless topology.The followings are the construction and main contents of this paper:In Chapter 1, different kinds of filters in Heyting algebra are studied.Firstly, we review the definition and some properties of Heyting algebra, and the relationship with Boole algebra.Secondly, properties of filters in Heyting algebra are studied. The concrete construction of filter lattice of Heyting algebra and filters generelized by some subsets are given. Finally, we define some the special filter in Heyting algebra, such as maximal filters, submaximal filters, strong filters, and prime filters, etc. And we also prove implications among them if they exist, and give counterexamples if they do not exist. Besides, the properties of submaximal filters are studied specially. With bridge of submaximal filters, we prove that the filter lattice of Heyting algebra is a complete lattice generalized by its prime elements, that is spacial Frame.In chapter 2, we study the congruence relations derived from filters, and Heyting algebras' homomorphism as well as the isomorphic axioms. Firstly, an equivalent relation about filter in Heyting algebras is defined, and it is proved to be a congruence relation. The quotient about the congruence relation is still a Heyting algebra. There is a bijection between the set of all filters and the set of all congruence relations in Heyting algebras. Secondly, the definitions of Heyting homomorphisms and subHeyting algebras are given, and we point out that all...