Pyung Ki Lim

Interval-Valued Fuzzy Congruences on a Semigroup

We introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: First, for any interval-valued fuzzy congruence R on a group G, the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G. Second, for any interval-valued fuzzy congruence R on a groupoid S, we show that a binary operation * an S=R is well-defined and also we obtain some results related to additional conditions for S. Also we improve that for any two interval-valued fuzzy congruences R and Q on a semigroup S such that R ⊂ Q, there exists a unique semigroup homomorphism g : S/R → S/G.

Interval-valued Fuzzy Normal Subgroups

We study some properties of interval-valued fuzzy normal subgroups of a group. In particular, we obtain two characterizations of interval-valued fuzzy normal subgroups. Moreover, we introduce the concept of an interval-valued fuzzy coset and obtain several results which are analogous of some basic theorems of group theory.

INTUITIONISTIC FUZZY NORMAL SUBGROUP AND INTUITIONISTIC FUZZY ⊙-CONGRUENCES

We unite the two con concepts - normality We unite the two con concepts - normality and congruence - in an intuitionistic fuzzy subgroup setting. In particular, we prove that every intuitionistic fuzzy congruence determines an intuitionistic fuzzy subgroup. Conversely, given an intuitionistic fuzzy normal subgroup, we can associate an intuitionistic fuzzy congruence. The association between intuitionistic fuzzy normal sbgroups and intuitionistic fuzzy congruences is bijective and unigue. This leads to a new concept of co sets and a corresponding concept of guotient.

Closure, Interior and Compactness in Ordinary Smooth Topological Spaces

It presents the concepts of ordinary smooth interior and ordinary smooth closure of an ordinary subset and their structural properties. It also introduces the notion of ordinary smooth (open) preserving mapping and addresses some their properties. In addition, it develops the notions of ordinary smooth compactness, ordinary smooth almost compactness, and ordinary near compactness and discusses them in the general framework of ordinary smooth topological spaces.

Ordinary Smooth Topological Spaces

In this paper, we introduce the concept of ordinary smooth topology on a set X by considering the gradation of openness of ordinary subsets of X. And we obtain the result [Corollary 2.13] : An ordinary smooth topology is fully determined its decomposition in classical topologies. Also we introduce the notion of ordinary smooth [resp. strong and weak] continuity and study some its properties. Also we introduce the concepts of a base and a subbase in an ordinary smooth topological space and study their properties. Finally, we investigate some properties of an ordinary smooth subspace.

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.

Lattices of Interval-Valued Fuzzy Subgroups

We discuss some interesting sublattices of interval-valued fuzzy subgroups. In our main result, we consider the set of all interval-valued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We then prove that this set forms a modular sublattice of the lattice of interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we employ a lattice diagram to exhibit the interrelationship among these sublattices.

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.