Theory Of Fuzzy Ordered Semigroups And Fuzzy Rough Sets Of Lattices

As a combination of algebraic structures and partially ordered structures,the con?cept of ordered algebras can also be seen as a natural generalization of classical algebras.And fuzzy ordered algebras are obtained by applying fuzzy orderings to ordered alge-braic structures.Recently,with the extensive investigations of fuzzy orders,the study on fuzzy ordered algebras received various attention.However,in the theoretical study of fuzzy ordered algebras,much attention has been paid to fuzzy orders,while little work has been done on extensions to the algebraic structures.On the other hand,extensive research has also been carried out to construct the rough approximation operators on different classes of ordered algebras.Even so we could barely find their fuzzy counter-part in the existing literature.Based on these observations,the main purposes of this thesis are twofold.One is to study some problems in algebraic aspect of fuzzy ordered semigroup theory.The other is to introduce and investigate a new class of fuzzy rough approximation operators based on a triangular norm on a lattice.The detailed contents are listed below:In Chapter 1,a general survey of fuzzy algebraic structures,ordered semigroup theory,rough set theory and fuzzy rough sets are given.In Chapter 2,some concepts and results on fuzzy sets,rough sets,ordered semi-groups and Quantales which will be used in this thesis are given.In addition,two approaches are developed to generate a fuzzy ideal by an arbitrary fuzzy subset of a quantale that improves a result presented in[96].In Chapter 3,we mainly study the fuzzy ideals of fuzzy ordered semigroups and investigate the relations between fuzzy ordered semigroups and fuzzy topological semi-groups.The concepts of fuzzy ideals,fuzzy bi-ideals,fuzzy quasi-ideals and fuzzy interior ideals of a fuzzy ordered semigroup are introduced and the relationships among them are obtained.Lattice structures and fuzzy lattice structures of these fuzzy ideals are discussed.The relationships between fuzzy ordered semigroups and fuzzy topological semigroups are investigated,and then,some categorical isomorphic relations between these two categories are established.As more general types of fuzzy ideals on a fuzzy ordered semigroup,the concepts of(?,? ?qk)-fuzzy ideals and(?,? ?qk)-fuzzy interior ideals of a fuzzy ordered semigroup are introduced and several characterizations of them are developed.In Chapter 4,we investigate the characterizations of fuzzy ordered semigroups in the framework of fuzzy setting.The concepts of regular fuzzy ordered semigroups and duo fuzzy ordered semigroups are proposed.Then their characterizations are derivedin terms of fuzzy ideals,fuzzy bi-ideals and fuzzy quasi-ideals of fuzzy ordered semi-groups.Also,the generalized semisimple fuzzy ordered semigroups are introduced and characterized by their(?,? ?qk)-fuzzy ideals and(?,? ?qk)-fuzzy interior ideals,re-spectively.In Chapter 5,we study the(fuzzy)rough approximation operators on a lattice.Firstly,ideals on a distributive join-semilattice are characterized in terms of upper and lower rough approximation operators.Then,the concepts of TL-fuzzy lower and upper rough approximation operators induced by an L-fuzzy set on a lattice are introduced,and their basic properties are investigated.Particularly,some characterizations of TL-fuzzy ideals on a distributive lattice are developed in terms of the TL-fuzzy rough approxi-mation operators.In addition,we define TL-fuzzy quasi-rough ideals induced by an L-fuzzy set on a lattice,and discuss the relationships among TL-fuzzy ideals,TL-fuzzy rough ideals and TL-fuzzy quasi-rough ideals on a given lattice.