Research On Roughness And Topological Properties Of Logical Algebra System

Rough set theory is a theoretical tool for dealing with incomplete and uncertain information in information systems to discover implicit knowledge and to reveal poten-tial laws. It is well-known that the most important concepts in the classical rough set theory are the upper and lower approximations derived from an equivalence relation on a universal set without any algebraic structure or partial-ordering. However, as pointed out by many scholars, the requirement of an equivalence relation in rough set models of Pawlak seems to be a very restrictive condition that might limit the appli-cations of rough set theory. To address this issue, many authors have generalized the classical rough set theory by using different methods such as substituting an algebraic system or a partial-ordered set for the universal set. This thesis aims at generalizing rough sets into serval algebras such as MTL-algebras and quantales and constructing the upper and lower approximate operators derived from equivalence relations, which, are induced by ideals of MTL-algebras and quantales, respectively. Based on fuzzy set theory of Zadeh, the notions of fuzzy ideals and rough fuzzy ideals in quantales are introduced. As a consequence, ideals and rough ideals of quantales can be brought into a unified framework.The upper and lower approximations in the classical rough sets are just as a pair of topological closure and interior, from which topologies could be naturally obtained. Thus, using topological techniques to describe rough approximate spaces has generated increasing interest in the research community of rough set theory. The other aim of this thesis is to construct topologies on effect algebras, the set of all formulae in the classical propositional logic system and R0-algebras using weakly algebraic ideals, theories and filters, respectively. Some properties of such uniform topologies are studied. The continuity of operators in these systems w.r.t. aforementioned uniform topologies are also discussed.Furthermore, the structure and topological properties of maximal filters in BL-algebras are studied in this dissertation. The concrete structures of all maximal filters in Boolean algebras which are generated by infinitely countable basic elements are presented.The present paper is divided into5chapters.Chapter1reviews some preliminaries about several commonly used classes of logical algebras and topology. The first section recalls syntax, semantics as well as completeness of propositional logic systems. Then we recall in the second section the definitions and basic properties of Boolean algebras,Ro-algebras and MTL-algebras, which are necessary for the rest chapters.Chapter2first introduces the notion of ideals in MTL-algebras and gives several characterizations of them. It is pointed out that every ideal of a MTL-algebra is a lattice ideal. The converse, however, is not true in general, and counterexamples are constructed. Then it proves that if the MTL-algebra is a BL-algebra, the the equiva-lence relations induced by ideals of the MTL-algebra are congruences. The properties of upper and lower approximate operators induced by the aforementioned equivalence relations are studied. Lastly, the relationship between the upper (resp. lower) approx-imations of ideals of MTL-algebras and the upper(resp. lower) approximations of their homomorphic images are also discussed.Chapter3recalls first the notions of ideals, rough ideals and congruences in quantales. As a generalizations of a congruence, weak congruences in quantales are introduced. The concrete methods for inducing weak congruences by ideals and gen-erating ideals from non-empty subsets of quantales are also presented. Given ideal I, every ideal is a rough ideal in the approximate spaces induced by I if and only if I={0}. Based on the classical theory of fuzzy sets, fuzzy ideals are defined, and several characterizations of such ideals are presented. It is proved that the set of all fuzzy ideals of a quantale, under the order of set inclusion is a complete lattice. In par-ticular, if the quantale is a frame, the set consisting of all fuzzy ideals of the quantale is also a frame. Moreover, the concepts and characterizations of fuzzy prime ideals, fuzzy semi-prime ideals and fuzzy primary ideals of quantales are proposed. By apply-ing rough set theory to fuzzy ideals of quantales, the notions of rough fuzzy (prime, semi-prime, primary) ideals in quantales are initiated. The sufficient conditions under which fuzzy prime ideals become rough fuzzy prime ideals are given. Finally, the re-lationship between the upper (resp. lower) rough ideals and the upper (resp. lower) approximations of their homomorphic images are also discussed.Chapter4reviews some preliminaries of (weak) congruences as well as weakly algebraic ideals in effect algebras. Then the uniformity and uniform topology (weakly algebraic ideal topology, for short) are established in effect algebras based on a weakly algebraic ideal. It is obtained that every weakly algebraic ideal of an effect algebra E induces a weakly algebraic ideal topology, with which E is a first-countable, zero-dimensional, disconnected, locally compact and completely regular topological space, and the operation (?) of effect algebras is continuous with respect to these topologies. In addition, it proves that the operations’and (?) of effect algebras and the operations A and V of lattice effect algebras are continuous with respect to the weakly algebraic ideal topology generated by a Riesz ideal.Based on the congruence induced by theory Γ in the set F(S) of all formulae in the classical propositional logic system, Chapter5defines first the uniformities on F(S) and proves that the uniform topologies induced by the uniformities are second countable, zero-dimensional and complete regular spaces without isolated points, and the logic connections(?) and→are continuous with respect to the uniform topology induced by a theory. The relationship between the aforementioned uniformities and the uniformity induced by the pseudo-metric on F(S) is discussed. As an application, it is proved that F(S) is divided into2n pairwise disjoint areas by n maximal consistent theories and the diameter of each area is equal to{1} in the logic metric space. Then the structure of each maximal filter in Boolean algebras which are finite or generated by infinitely countable basic elements is clearly described. Then two types of topologies are established on the set of all maximal filters in a BL-algebra. It is proved that if the related universe is either finite or infinite linearly ordered, then these two types of topologies are equal. In particular, if the BL-algebra is a Boolean algebra generated by infinitely countable basic elements, the obtained topological space is homeomorphic to the Cantor ternary set. It is proved that the uniform topological space induced by a filter in a.Ro-algebra is a To space if and only if the filter is equal to1, and the negative operation, the join operation and the implication operation are continuous in the uniform topological space. Finally, some properties of the uniform topology on the quotient R0-algebra under the equivalence relation induced by a filter are discussed.