Interval-Valued Semiprime Fuzzy Ideals of Semigroups

We introduce the notion of (i-v) semiprime (irreducible) fuzzy ideals of semigroups and investigate its different algebraic properties. We study the interrelation among (i-v) prime fuzzy ideals, (i-v) semiprime fuzzy ideals, and (i-v) irreducible fuzzy ideals and characterize regular semigroups by using these (i-v) fuzzy ideals. 1. Introduction Zadeh [1] first introduced the concept of fuzzy sets in 1965. After that it has become an important research tool in mathematics as well as in other fields. It has many applications in many areas like artificial intelligence, coding theory, computer science, control engineering, logic, information sciences, operations research, robotics, and others. Likewise, an idea of connecting the fuzzy sets and algebraic structures came first in Rosenfeld’s mind. He first introduced the notion of fuzzy subgroup [2] in 1971 and studied many results related to groups. After that fuzzification of any algebraic structures has become a new area of research for the researchers. Some of fuzzy algebraic structures are mentioned in [3–9]. During the progress of the research on fuzzy sets, several types of extensions of fuzzy subsets were introduced. Interval-valued (in short, (i-v)) fuzzy subset is one of such extensions. In 1975, the concept of interval-valued fuzzy subset was introduced by Zadeh [10]. In this concept, the degree of membership of each element is a closed subinterval in [0,1]. Using such concept, it is possible to describe an object in a more precise way. There are many applications of (i-v) fuzzy subsets in different areas: Davvaz [11] on near rings, Hedayati [12] on semirings, Gorza？czany [13] on approximate reasoning, Turksen [14] on multivalued logic, Mendel [15] on intelligent control, Roy and Biswas [16] on medical diagnosis, and so on. Similar to fuzzy set theory, (i-v) fuzzy set theory gradually developed on different algebraic structures. Biswas [17] defined the (i-v) fuzzy subgroups of Rosenfeld’s nature and investigated some elementary properties. Narayanan and Manikantan [18] introduced the notions of (i-v) fuzzy subsemigroup and various (i-v) fuzzy ideals in semigroups. In [19], Kar et al. introduced the concept of (i-v) prime (completely prime) fuzzy ideal of semigroups and studied their properties. Khan et al. [20] introduced the concept of a quotient semigroup by an interval-valued fuzzy congruence relation on a semigroup. In [21], Thillaigovindan and Chinnadurai introduced the notion of (i-v) fuzzy interior (quasi, bi) ideals of semigroup and studied their properties. However, the concept of (i-v)