Roughness In Ordered Semigroups

Rough set theory, a new mathematical approach to deal with uncertain knowl-edge in information systems, was originally introduced by Polish mathematician Pawlak in1982. It has turned out to fundamentally important in artificial intel-ligence, machine learning, pattern recognition and so on. With the development of rough set theory, possible connections between rough sets and various algebraic systems were considered by many authors. Inspired by the construction of Pawlak rough set algebras and the investigation in algebraic properties of rough set, re-searchers have applied notions and methods of rough set theory to various algebraic structures, this not only enriched the theory the theory of rough sets but also pro-vided new ideals in the study of pure algebraic. In this paper, we introduce rough set theory into ordered semigroups, propose the notions of rough (prime, semiprime, primary) ideals and rough fuzzy (prime, semiprime, primary) ideals of ordered semi-groups, and investigate their properties.The arrangement of this paper is as follows:Chapter one:Preliminaries. We recall some basic notions and results of ordered semigroups, rough set theory, Lattice theory and fuzzy set theory.Chapter two:Rough ideals in ordered Semigroups. Firstly, we introduce notions of complete congruences on ordered semigroups, and upper and lower approxima-tions in ordered semigroups, and discuss their properties. Then, we propose concepts of (upper, lower) rough (prime, semiprime, primary) ideals (multiplicative sets, m-systems), study their properties and the links among them. It is proved that under some conditions,(prime, semiprime, primary) ideals must be (upper, lower) rough (prime, semiprime, primary) ideals. We show that the set of all lower rough ideals is an algebraic lattice under the set inclusion order if the ordered semigroup has a bottom element and for every element of it, the equivalence class is finite. Finally, we investigate properties of images and inverse images of rough (prime, semiprime, primary) ideals under ordered semigroup homomorphisms.Chapter three:Rough fuzzy ideals in ordered Semigroups. Firstly, we introduce the notion of rough fuzzy sets in ordered semigroups and give some properties related to them. Secondly, we propose notions of (upper, lower) rough fuzzy subordered semigroups (ideals, prime ideals, semiprime ideals, primary ideals), investigate their properties and the links among rough fuzzy subordered semigroups, rough fuzzy ideals, rough fuzzy prime ideals, rough fuzzy semiprime ideals, and rough fuzzy primary ideals. Finally, we prove that:fuzzy subordered semigroups (resp. ideals, prime ideals, semiprime ideals, primary ideals) are (upper, lower) rough fuzzy subor-dered semigroups (resp. ideals, prime ideals, semiprime ideals, primary ideals), the intersection (product) of two fuzzy subordered semigroups is (upper, lower) rough fuzzy subordered semigroups, the intersection (union, product) of two fuzzy ideals (semiprime ideals) is (upper, lower) rough fuzzy ideals (semiprime ideals).