Theory Research On Filters Of Residuated Lattices And Derivations Of Partially Ordered Sets

The residuated lattice is a kind of algebraic structure closely connecting with mathematical logic. It not only provides algebraic semantics for almost all substructural logics, but also generalizes some widely used algebraic structure. In the logical algebra systems, such as MV-algebras, BL-algebras, R0-algebras, MTLalgebras, Heyting algebras, etc. are particular residuated lattices. Filters play a vital role in the study of logical algebra systems. Since the notion of fuzzy set is proposed, from the perspective of fuzzification and application, the notions of fuzzy filters,(∈, ∈ ∨q)-fuzzy filters and intuitionistic fuzzy filters are subsequently introduced. In this paper, we in a different light investigate fuzzy filters,(∈, ∈ ∨q)-fuzzy filters and intuitionistic fuzzy filters on residuated lattices. We try to unify the above-mentioned fuzzy filters and provide a more general method for the study of fuzzy filters,(∈, ∈ ∨q)-fuzzy filters and intuitionistic fuzzy filters on the logical algebra systems.In addition, the derivation originated from the analytical theory is a mapping on an algebraic system. We introduce the notion of derivation on partially ordered sets and R0-algebras, discuss the properties of the derivation and try to characterize the algebraic structure of posets and R0-algebras. The detailed contents are listed below:1. In the first chapter, the research history and current situation of the filter theory, fuzzy filter theory on residuated lattices and the derivation on lattices,logical algebra systems are introduced.2. In the second chapter, we introduce some knowledge about residuated lattices and universal algebras. We specially study the relations among filters and give the simple method for judging their relations on residuated lattices.3. In the third chapter, we unify the fuzzy filters on residuated lattices.Firstly, we define the fuzzy filters and the fuzzy V-filters from the perspective of the cut-sets. Then we obtain the necessary and sufficient conditions under which fuzzy filters become fuzzy V-filters. We also get the common properties of fuzzy V-filters(Triple of equivalent characteristics; Quotient characteristics; Extension property). Finally, we give the general principle of studying the relations among fuzzy V-filters.4. In chapter 4, we unify the(∈, ∈ ∨q)-fuzzy filters on residuated lattices.Firstly, we define the(∈, ∈ ∨q)-fuzzy filters from the viewpoint of cut-sets and give several kinds of equivalent characterizations of(∈, ∈ ∨q)-fuzzy filters. Then we introduce the notion of(∈, ∈ ∨q)-fuzzy V-filters. We give the necessary and sufficient conditions under which fuzzy(∈, ∈ ∨q)-filters become(∈, ∈ ∨q)-fuzzy V-filters. We also find the common properties of(∈, ∈ ∨q)-fuzzy V-filters. Finally,we investigate the relations among(∈, ∈ ∨q)-fuzzy V-filters.5. In chapter 5, we discuss the intuitionistic fuzzy filter theory on residuated lattices. Firstly, we give the equivalent characterizations of intuitionistic fuzzy filters. Then we put forward the concept of intuitionistic fuzzy V-filters. We obtain the necessary and sufficient conditions under which intuitionistic fuzzy filter become intuitionistic fuzzy V-filters. In particular, we study the quotient structure and the properties of intuitionistic fuzzy V-filters. Finally, we find the law of judging the relations among intuitionistic fuzzy V-filters.6. In chapter 6, we study the derivation of posets. We introduce the notion of derivations on posets and study the basic properties of these derivations. We show that the concept of derivation on posets generalizes the corresponding concept on lattices. Finally, we discuss the properties of the fixed point and operations related with derivations.7. In chapter 7, we propose the notions of((?), ⊕)-derivations and isotone((?), ⊕)-derivations on R0-algebras. We investigate the basic properties of these derivations. We especially give the equivalent characterizations of isotone((?), ⊕)-derivations.