On Rough Set In Lattices

Rough set theory was introduced by Z.Pawlak (1982), which is a new math-ematical approach to deal with uncertain information. In this paper, the authordiscusses the properties of rough sets and generalized rough sets which are con-structed by congruences and set-valued homomorphisms respectively in latticesand quantales, then the algebraic structure of lattices and quantales is explored indetail.A congruence relation is an equivalence relation consistent with the algebraicstructure. In the paper, the upper and lower approximations are constructed bycongruence relations. The concepts of rough ideal and rough prime ideal are de-fined which are the extended notion of ideal and prime ideal respectively, and theinclusion relations of the upper and lower approximations are discussed. The lat-tice structures of the lower and upper approximations are studied. We obtain theresults that the collection of all definable sets in a lattice is a topped intersectionstructure and also a complete field of sets. And the collection of all definable idealsin a lattice with a bottom element is a topped algebraic intersection structure. Wealso discuss the relations among products、quotients of the lower、upper approx-imations and the lower、upper approximations of products、quotients. And theproblems of homomorphism of the lower and upper approximations are studied.There is a special class of join congruence relations determined by ideals oflattice, it is a congruence relation if the lattice is distributive. We study thespecial properties of the upper and lower approximations constructed by means ofthese special congruence relations on a lattice. By using the properties of ideals,we analyze the inclusion relation of the lower and upper approximations. Thedefinable sets with respect to the special congruences are studied. We obtain thatthe collection of definable ideals with respect to the special congruence determinedby an ideal is that of all ideals which contained this ideal, and the set is an algebraiclattice. The lower and upper approximations of some particular ideals are discussedsuch as lower set and kernel ideal, etc.In an algebraic system, we consider the rough set in the case of two universes,the congruence relation needs to be generalized. Chapter3and chapter4intro-duce the notion of set-valued homomorphism of lattices and quantales respectively.We probe into the properties of generalized rough subsets and generalized roughideals. We also propose the special set-valued homomorphisms with respect to ideals of lattice, the properties of the generalized rough sets constructed by themare discussed. And the example in the application of formal concept analysis isgiven.The notion of derivation is a function on an algebraic system introduced fromthe analytic theory, is helpful to the research of structure and property in al-gebraic systems. Chapter5, the concept of derivation on quantales is defined,simple derivation、identity derivation and zero derivation are proposed, the prop-erties of derivations are discussed in detail. The image of derivation of (strict)left、right、two sided elements of quantales are studied. The structure of the col-lection of all fixed points of a derivation is explored. The collection of all derivationson a quantale which satisfies some conditions is a quantale. The quantic nuclei playan important role in quantale theory, by using the relation between pre-quanticnuclei and quantic nuclei, the relations between derivation and quantic nuclei arestudied.This dissertation is typeset by software LATEX2ε.