| Quantale theory, established in the early 1980’s, is one of mathematical foun-dations of theoretical computer science. Based on order theory from the viewpoint of the common mathematical foundation, quantale theory makes close relationship with topology, algebra, logic and so on. As a relevant structure of quantale theory, m-semilattices connect the structures of V-semilattices with the multiplications of semigroups, and hence there are abundant contents in the structure of m-semilattices, and m-semilattices can be regarded as generalizations of residuated lattices, frames, quantales and lattice-ordered semigroups. Moreover, m-semilattices play an impor-tant role in the research of quantale theory, since each coherent quantale is isomorphic to a quantale consisting of all V-semilattice ideals of an m-semilattice with a top element. Rough set theory, a new mathematical approach to deal with inexact and uncertain knowledge in information systems, was originally introduced by Pawlak. It has turned out to be fundamentally important in artificial intelligence, data analysis and cognitive sciences. With the development of rough set theory, many mathemati-cians have considered to apply rough set theory and the research methods to studies of various algebraic structures. The first part of this paper is to further investigate prop-erties of rough sets on non-empty sets and lattices. The second part is to apply rough set theory and fuzzy set theory to m-semilattices, and study properties of Pawlak rough sets, rough fuzzy sets, approximation operators based on coverings, fuzzy rough sets based on fuzzy coverings and fuzzy ideals. This paper is organized as follows:Chapter One:Preliminaries. In this chapter, some basic notions and results of lattice theory, fuzzy set theory and rough set theory are given, which will be used throughout the paper.Chapter Two:Some properties of approximation operators. Firstly, some prop-erties of rough sets on non-empty sets are studied. The relationships between upper and lower approximation operators of rough sets and equivalence relations are dis-cussed. It is proved that the set of all equivalence relations on an non-empty set is a complete lattice, and is isomorphic to the set of all upper (lower) approximation operators induced by equivalence relations. Secondly, using the research methods of topology theory, we introduce definitions of T0a L-FA spaces, T1a L-FA spaces and regular L-FA spaces in L-fuzzy approximation spaces, and discuss the relationships among them.Chapter Three:Fuzzy ideals of m-semilattices. Firstly, the definitions of fuzzy ideals and fuzzy prime ideals of m-semilattices are introduced, the relationship-s between fuzzy (prime) ideals and (prime) ideals are discussed and properties of the sets of all fuzzy ideals are studied. The equivalent characterizations of fuzzy (prime) ideals and (prime) ideals are given. It is proved that the set of all fuzzy ideals with 1 in their images of a positive m-semilattice with a bottom element is a distributive l-semigroup. Secondly, constructions of fuzzy ideals and congruences with each other are studied, and the properties of congruences induced by fuzzy ideals are also investi-gated. Finally, by constructing V-semigroup congruences, equivalent characterization of the distributivity of V-semigroups are obtained. Moreover, a lattice homomorphis-m between the set of all fuzzy m-semilattice ideals of a V-semigroup and the set of all V-semigroup congruences is presented.Chapter Four:Rough sets on lattices. Firstly, properties of the sets of fixed points under upper and lower approximation operators on the sets of all ideals (fil-ters) of lattices are studied. It is shown that the set of all fixed points under an upper approximation operator on the set of all ideals (filters) of a distributive lattice is a coherent frame. A characterization for fixed points under a lower approximation operator on the set of all ideals (filters) of a finite lattice is given. Secondly, proper-ties of L-fuzzy rough sets on lattices are investigated and the relationships between L-relations and L-fuzzy rough sets are discussed. Finally, the definitions of S-fuzzy rough sublattices and S-fuzzy rough ideals (filters) are given, and some properties of them are investigated.Chapter Five:Rough sets on m-semilattices. Firstly, some kinds of equiv-alence relations are introduced, relationships among them are discussed and some relative properties of Pawlak rough sets based on them are studied. Secondly, a kind of approximation operators based on coverings, called minimal neighborhood approx-imation operators, are introduced and studied on m-semilattices. It is proved that when the covering is composed of upper sets, the set of all fixed points under a min-imal neighborhood upper approximation operator-aprN on the set of all ideals is an algebraic lattice. Finally, the definitions of Φ-upper and Φ-lower fuzzy rough ap-proximation operators based on a covering Φ are given, and the properties of them are investigated.Chapter Six:Rough fuzzy ideals of m-semilattices. Firstly, based on the con- gruences induced by fuzzy (prime) ideals of m-semilattices proposed in chapter three, properties of the upper (lower) rough fuzzy approximation operators with respect to these congruences are studied. Secondly, the notions of rough fuzzy (prime) ideals of m-semilattices are introduced. Relationships between rough fuzzy (prime) ideals and fuzzy (prime) ideals and relationships between rough fuzzy (prime) ideals and (prime) ideals of m-semilattices are established. Finally, relationships between upper (lower) rough fuzzy ideals and upper (lower) approximations of their homomorphic images are discussed, and properties of the sets of upper and lower rough fuzzy ideals are studied. It is proved that under some conditions, every lower rough fuzzy ideal of m-semilattices can be represented as the join of a directed subset of the set of all lower rough fuzzy ideals.
|
PDF Full Text Request