Type-2Fuzzy Sets And Type-2Fuzzy Rough Sets

Type-2fuzzy sets are increasing more and more popular as a generalization of ordinary fuzzy sets. In this dissertation, we study three aspects of type-2fuzzy sets: firstly, type-2fuzzy relations and their compositions; secondly, extended operations and their properties; thirdly, type-2fuzzy rough sets. In particular, our work is listed as follows:Chapter2contains some fundamental concepts and related properties of extension operations on t-norms. Furthermore, we propose extended supremum and extended infimum, and investigate their properties.Chapter3presents type-2fuzzy sets and type-2fuzzy relations, and extends the compositions of type-2fuzzy relations. Moreover, type-2fuzzy quasi-order relations, type-2fuzzy partial order relations and partially ordered type-2fuzzy relations are extended and studied.Chapter4discusses the properties of extended triangular norms and gives the sufficient condition, when extended continuous triangular norms are type-2triangu-lar norms. We investigate the lattice structures of the algebra of fuzzy values and prove that the algebra of fuzzy values is a complete completely distributive lattice. Then fuzzy-valued triangular norms induced by arbitrary triangular norms are pro-posed. Residuated lattices are proposed on the algebra of fuzzy values based on the investigation to fuzzy-valued t-norms induced arbitrary left-continuous t-norms.Chapter5investigates the properties of extended fuzzy implications and lists the sufficient (resp. necessary) conditions, when extended continuous fuzzy implications are type-2fuzzy implications. We propose fuzzy-valued fuzzy implications induced by arbitrary fuzzy implications and discuss the relationship between them and fuzzy-valued fuzzy implications proposed in Chapter4. As an application of fuzzy-valued t-norms and fuzzy-valued fuzzy implications, fuzzy-valued approximate reasoning is extended from the view of type-2fuzzy sets. In the sequel, type-2fuzzy rough sets are studied from the constructive way and axiomatic way. Particularly, two aspects are included:(1) Chapter6investigates strong type-2fuzzy rough sets based on extended con-tinuous fuzzy operators on finite universes, and defines type-2fuzzy closure operator and type-2fuzzy interior operator.(2) Chapter7proposes fuzzy-valued fuzzy rough sets on arbitrary universes, espe-cially infinite universes. Moreover, three different axiomatic characterizations of lower rough approximation operators defined by fuzzy-valued fuzzy implications are proposed on the basis of the properties of fuzzy implications.Fuzzy-valued fuzzy rough sets are investigated from the view of generalized residu-ated lattices in Chapter8. Firstly, we propose fuzzy-valued pseudo-t-norms induced by arbitrary pseudo-t-norms. Secondly, we construct generalized residuated lattices on the algebra of fuzzy values. For a more generalized discussion, we take generalized resid-uated lattices as a basic structure in the following investigation. Generalized L-fuzzy rough sets and generalized L-fuzzy complete lattices are studied. Particularly, there are two aspects:(1) First, a quadruple of generalized L-fuzzy rough approximation operators are defined to suit the situation when generalized residuated lattices are non-commutative. When generalized residuated lattices are complete completely distributive lattices, each generalized L-fuzzy rough set is proven to be consisted of two complete completely distributive rough sets. Second, the topological structures of generalized L-fuzzy ap-proximation spaces are studied. The one-to-one correspondence is constructed between the set of generalized L-fuzzy preorder relations and the set of left (resp. right) Alexan-drov L-topologies on an arbitrary universe.(2) In a generalized L-fuzzy complete lattice, the relationships among tensors, cotensors and generalized L-fuzzy partial order relations are investigated. Generalized L-fuzzy partial order relation and cotensor can be viewed as special left implication and right implication of tensor, respectively. Furthermore, the properties of generalized L-fuzzy powerset operators are studied.