Jeong Gon Lee

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.

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.

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.

Interval-Valued H-Fuzzy Sets

We introduce the category IVSet(H) of interval-valued H-fuzzy sets and show that IVSet(H) satisfies all the conditions of a topological universe except the terminal separator property. And we study some relations among IVSet(H), ISet(H) and Set(H).

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].