Kul Hur

Intuitionistic Smooth Topological Spaces

We introduce the concept of intuitionistic smooth topology in Lowens sense and we prove that the family IST(X) of all intuitionistic smooth topologies on a set is a meet complete lattice with least element and the greatest element[Proposition 3.6]. Also we introduce the notion of level fuzzy topology on a set X with respect to an intuitionistic smooth topology and we obtain the relation between the intuitionistic smooth topology τ and the intuitionistic smooth topology ŋ generated by level fuzzy topologies with respect to τ [Theorem 3.10].

INTERVAL-VALUED FUZZY SUBGROUPS AND LEVEL SUBGROUPS

We introduce the concept of level subgroups of an interval- valued fuzzy subgroup and study some of its properties. These level subgroups in turn play an important role in the characterization of all interval-valued fuzzy subgroup of a prime cyclic group.

INTERVAL-VALUED FUZZY SUBGROUPS

We study the conditions under which a given interval- valued fuzzy subgroup of a given group can or can not be realized as a union of two interval-valued fuzzy proper subgroups. Moreover, we provide a simple necessary and sucient condition for the union of an arbitrary family of interval-valued fuzzy subgroups to be an interval-valued fuzzy subgroup. Also we formulate the concept of interval-valued fuzzy subgroup generated by a given interval-valued fuzzy set by level subgroups. Furthermore we give characterizations of interval-valued fuzzy conjugate subgroups and interval-valued fuzzy characteristic subgroups by their level subgroups. Also we investigate the level subgroups of the homomorphic image of a given interval-valued fuzzy subgroup.

NEIGHBORHOOD STRUCTURES IN ORDINARY SMOOTH TOPOLOGICAL SPACES

We construct a new definition of a base for ordinary smooth topological spaces and introduce the concept of a neighborhood structure in ordinary smooth topological spaces. Then, we state some of their properties which are generalizations of some results in classical topological spaces.

INTERVAL-VALUED FUZZY GROUP CONGRUENCES

We introduce the concepts of interval-valued fuzzy complete inner-unitary subsemigroups and interval-valued fuzzy group congruences on a semigroup. And we investigate some of their properties. Also, we prove that there is a one to one correspondence between the interval-valued fuzzy complete inner-unitary subsemigroups and the interval-valued fuzzy group congruences on a regular semigroups.

Ω-INTERVAL-VALUED FUZZY SUBSEMIGROUPS IN A SEMIGROUP

By using a set Ω, we introduce the concept of Ω-fuzzy subsemigroups and study some of it’s properties. Also, we show that the homomorphic images and preimages of Ω-fuzzy subsemigroups become Ω-fuzzy subsemigroups.

Closures and Interiors Redefined, and Some Types of Compactness in Ordinary Smooth Topological Spaces

We give a new definition of ordinary smooth closure and ordinary smooth interior of an ordinary subset in an ordinary smooth topological space which have almost all the properties of the corresponding operators in a classical topological space. As a consequence of these definitions we reduce the additional hypotheses in the results of [1] and also generalize several properties of the types of compactness in [1].

Some topological structures of ordinary smooth topological spaces

We introduce the notions of ordinary smooth, quasi-ordinary smooth and weak ordinary smooth structure, showing that various properties of an ordinary smooth topological space can be expressed in terms of these structures. In particular, the definitions and results of [2, 4, 5] may be expressed in terms of the ordinary smooth and quasi-ordinary smooth structures. Furthermore, we present the basic concepts relating to the weak ordinary smooth structure of an ordinary smooth topological space and the fundamental properties of the objects in these structures.

Interval-Valued Fuzzy Cosets

First, we prove a number of results about interval-valued fuzzy groups involving the notions of interval-valued fuzzy cosets and interval-valued fuzzy normal subgroups which are analogs of important results from group theory. Also, we introduce analogs of some group-theoretic concepts such as characteristic subgroup, normalizer and abelian groups. Secondly, we prove that if A is an interval-valued fuzzy subgroup of a group G such that the index of A is the smallest prime dividing the order of G, then A is an interval-valued fuzzy normal subgroup. Finally, we show that there is a one-to-one correspondence the interval-valued fuzzy cosets of an interval-valued fuzzy subgroup A of a group G and the cosets of a certain subgroup H of G.

Intuitionistic Interval-Valued Fuzzy Topological Spaces

By using the concept of intuitionistic interval-valued fuzzy sets, we introduce the notion of intuitionistic interval-valued fuzzy topology. And we study some fundamental properties or intuitionistic interval-valued fuzzy topological spaces: First, we obtain analogues[see Theorem :3.11 and :3.121 of neighborhood systems in ordinary topological spaces. Second, we obtain the result[see Theorem 4.9] corresponding to “the 14-set Theorem” ill ordinary topological spaces. Finally, we give the initial structure on intuitionistic interval-valued fuzzy topologies[see Theorem 5.9].