(\(\in ,\in \vee q_{(\lambda ,\mu )}\) )-Fuzzy Completely Semiprime Ideals of Semigroups
We introduce a new kind of generalized fuzzy completely ideal of a semigroup called \((\in ,\in \vee q_{(\lambda ,\mu )})\) -fuzzy completely semiprime ideals. These generalized fuzzy completely semiprime ideals are characterized.
- PDF / 170,636 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 48 Downloads / 181 Views
ct We introduce a new kind of generalized fuzzy completely ideal of a semigroup called (∈, ∈ ∨q(λ,μ) )-fuzzy completely semiprime ideals. These generalized fuzzy completely semiprime ideals are characterized. Keywords Fuzzy algebra · Fuzzy points · (∈, ∈ ∨q(λ,μ) )-fuzzy completely semiprime ideals · Level subsets · Completely semiprime ideals
1 Introduction Fuzzy semigroup theory plays a prominent role in mathematics with ranging applications in many disciplines such as control engineering, information sciences, fuzzy coding theory, fuzzy finite state machines, fuzzy automata, fuzzy languages. Using the notion of a fuzzy set introduced by Zadeh [1] in 1965, which laid the foundation of fuzzy set theory, Rosenfeld [2] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroups. Since then fuzzy algebra came into being. Bhakat and Das gave the concepts of fuzzy subgroups by using the “belongs to” relation (∈) and “quasi-coincident with” relation (q) between a fuzzy point and a fuzzy set, and introduced the concept of a (∈, ∈ ∨q)-fuzzy subgroup [3–6]. It is worth to point out that the ideal of quasi-coincident of a fuzzy point with a fuzzy set, which is mentioned in [7], played a vital role to generate some different types of fuzzy subgroups. In particular, (∈, ∈ ∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup, which provides sufficient motivation to researchers to review various concepts and results from the realm of Z. Liao (B) · Y. Fan · L. Yi School of Science, Jiangnan University, 214122 Wuxi, China e-mail: [email protected] Z. Liao Department of Mathematics, State University of New York at Buffalo, 14260 New York, USA B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_18, © Springer-Verlag Berlin Heidelberg 2014
173
174
Z. Liao et al.
abstract algebra in the broader framework of fuzzy setting. Zhan [8], Jun et al. [9] introduced the notion of (∈, ∈ ∨q)-fuzzy interior ideals of a semigroup. Davvaz [10– 13] defined (∈, ∈ ∨q)-fuzzy subnear-rings and characterized Hν -fuzzy submodules, R-fuzzy semigroups using the relation (∈, ∈ ∨q). Later, the definition of a generalized fuzzy subgroup was introduced by Yuan [13]. Based on it, Liao [14] expanded common “quasi-coincident with” relationship to generalized “quasi-coincident with” relationship, which is the generalization of Rosenfeld’s fuzzy algebra and Bhakat and Das’s fuzzy algebra. And a series results were gotten by using generalized “quasi-coincident with” relationship [15–18]. When λ = 0 and μ = 1 we get common fuzzy algebra by Rosenfeld and When λ = 0 and μ = 0.5 we get the (∈, ∈ ∨q)-fuzzy algebra defined by Bhakat and Das and when λ = 0 and μ = 0.5 we get the (∈, ∈ ∨ q)-fuzzy algebra. The concept of a fuzzy ideal in semigroups was developed by Kuroki. He studied fuzzy ideals, fuzzy bi-ideals and fuzzy semiprime ideals in semigroups [19–21]. Fuz
Data Loading...
