Several Monographic Studies Of Basic Logic Algebras

Non-classical logics containing multi-valued logics and fuzzy logics are often used to deal with uncertain problems. In fuzzy logic, it is necessary to establish a strict system of fuzzy logic calculus before a basis of fuzzy logic inferences is established, and the completion of the job needs supports of algebra logic methods. Algebraic logic research focuses on two points:the first one is to study algebraic systems associated with logic systems; the second one is to establish relationships between the logical system and its matching algebraic system. We often use the research results of algebraic systems to solve logic problems, or use logic methods to solve algebraic problem.This paper studies basic logic from the algebraic point of view, it mainly fo-cuses on substructures and their uncertainty theories in BL-algebras, SBL-algebras and SBL-algebras. This thesis consists of the following parts:(1) Filter theories of BL-algebras, SBL (SBL)-algebras are investigated.The notion of II-special filters is introduced in BL-algebras, its extension as well as its relationships with quotient special BL-algebras are discussed. Through discussing properties of SBL-algebraic, we proved that if the quotient L/F induced by the perfect filter F of a BL-algebra L is a SBL-algebra, then F is an integral fil-ter, and then solve open problem raised by Borzooei and Paad. After strict filters are given, we display the relationships between strict filters and other types of fil-ters. We study properties of-filters in SBL-algebras, and obtain some character-izations of-filters. Meanwhile we introduce conjugate annihilators and radicals of-filters, and give a new expression form of-filters by conjugate annihilators and generated filters.(2) Implicative ideals of BL-algebras and-ideals of SBL-algebras are stud-ies, respectively.We introduce pseudo implication operation in BL-algebras and prove that a nonempty I of a BL-algebra L is an ideal if and only if for any x, y∈I, L(x, y)(?)I. With the help of pseudo implication operation, we introduce the concept of im-plicative ideal, and then we give some characterizations of implicative ideals. Through the analysis of characterizations of implicative ideals, we prove that im-plicative ideals coincide with Boolean ideals. We also establish relationships be-tween BL-algebras and Godel algebras via implicative ideals. In SBL-algebras, we introduces the concept of-ideals and pointed out that an-ideal is an ideal.(3) Fuzzy structures and soft structures of ideals in BL-algebras are investi-gated, respectively.In BL-algebras, we study generalized forms of ideals-(∈,∈∨q)-fuzzy ideals and falling fuzzy ideals, and give some characterizations of them. By using a fil-tered BL-algebra, we construct a fuzzy semi-rings. With the help of falling shadow theories, we discuss falling fuzzy ideals in BL-algebra. We not only establish the relationship between falling fuzzy ideals and fuzzy ideals, but also give some char-acterizations of falling fuzzy ideals. In order to investigate the fuzzy theory of soft BL-algebras, we first introduce the notion of (λ,μ)-soft union rings, and discuss some isomorphism theorems of (A,μ)-soft union rings by using invariant soft sets. Based on the,studies on rings, we introduce fuzzy ideals of soft BL-algebra, and then give fuzzy isomorphism theorems of soft BL-algebras via soft homomorphic mappings.