Researches On The Theory Related To EQ-algebras

Type theory is a kind of higher-order logic, and fuzzy type theory is a fuzzy version of type theory. Fuzzy type theory becomes a kind of higher-order fuzzy logic. At first, fuzzy type theory is used in residuated lattice-or rather, IMTL-algebras (residuated lattice with prelinearity and double negation)-as an algebraic structure of truth values. EQ-algebra is a class of special algebraic structure which has been developed in the last few years. EQ-algebras can be seen as a generalization of residuated lattices in a certain sense. The original purpose of proposing EQ-algebras was to introduce a more general algebraic structure of truth values for fuzzy type theory. Hence so far fuzzy type theory has two kind of algebraic structure of truth values:IMTL-algebras and EQ-algebras. Since the concept of EQ-algebras was proposed, good EQ-algebras (a special and important EQ-algebra) have been widely studied. EQ-logic is a kind of higher-order fuzzy logic based on EQ-algebras. EQ△-logic is a generalization of EQ-logic. The research on EQ-algebras is going deeper. Based on the respective characteristics of fuzzy set theory, rough set theory, filer theory and semigroup rough ideals theory, the main work of this thesis is to find the juncture between EQ-algebras theory with the above theories and their applications in the theory related to EQ-algebras. The main research works are as follows:1. Filter theory plays an important role in studying various logical algebras. The preliminary study of filter theory of EQ-algebras had been done, and requires still more investigation. For the limitation of the existing definition of fuzzy filter of an EQ-algebra, the concept of fuzzy filter of an EQ-algebra is redefined. Further, the definition of fuzzy prime filter of an EQ-algebra is proposed and the connections between fuzzy filters and fuzzy prime filters of EQ-algebras are discussed. The properties of fuzzy filters and fuzzy congruences of EQ-algebras are studied in detail. Our research shows that there is a one-to-one correspondence between normal fuzzy filters and fuzzy congruences in a good lattice EQ-algebra. Some lattice properties about fuzzy filters and fuzzy congruences of EQ-algebras are discussed.2. An equivalence relation and a fuzzy equivalence relation can be respectively induced using a filter of an EQ-algebra and a fuzzy filter of an EQ-algebra. The core of rough set theory is an equivalence relation. We introduce for first time rough set models and rough fuzzy set models based on an equivalence relation which is induced by a filter of an EQ-algebra. Fuzzy rough set models are constructed by a fuzzy equivalence relation which is induced by a normal fuzzy filter of an EQ-algebra. The properties of these models are studied.3. An EQ-algebra includes two special semigroup (△-commutative idempotent monoid and (?)-commutative monoid), hence the problem of rough ideals in EQ-algebras can be studied by virtue of rough ideal theory in semigroup. Based on the congruence of an EQ-algebra, rough set models and rough fuzzy set models are constructed. We build fuzzy rough set models using the fuzzy congruence of an EQ-algebra. On this foundation, the definitions of ideal and fuzzy ideal in EQ-algebras are presented. Fur-thermore, we propose rough ideal, rough fuzzy ideal, fuzzy rough ideal and rough prime ideal in EQ-algebras and offer some basic properties of these ideals. The problems of homomorphism between two EQ-algebras are discussed.4. Making deep research and analysis to the existing notions of EQ-logics and EQ△-logics and their properties, some important new properties of EQ-logics, EQ△-algebras and EQ△-logics are introduced. IEQ△-logics (a kind of new and special EQ△-logics) are proposed. The basic properties of IEQ△-logics are provided.