Journal of Intelligent & Fuzzy Systems 32 (2017) 579–587 DOI:10.3233/JIFS-152523 IOS Press 579 New approaches on some fuzzy algebraic structures Rabah Kellil ∗ University of Majmaah, College of Science at Al-Zulfi, Al-Zulfi, Saudi Arabia Abstract. Starting from the study of the papers [1, 3, 4, 5, 13, 14], we have seen that we can change some concepts and give some new definitions on fuzzy algebraic structures. Our goal by introducing these definitions is to get the classical and well known ones when the fuzzy set is just the characteristic function. As a second purpose we were tempted to generalize as long as it is possible some results known in the classical set theory. We get many interesting results which concern fuzzy relations, fuzzy subgroups of a given group, fuzzy ideals of a ring. We also define ideal generated by an element in the commutative case, prime and maximal ideals. Some relationships have been made and many other still open and which will be of our interest in next studies. Keywords: Fuzzy sets, fuzzy subgroup, fuzzy operation, fuzzy equivalence relation, fuzzy ideal, fuzzy prime ideal, fuzzy maximal ideal MSC [2010]: 03E72, 08A72, 20N25 1. Preliminaries This section contains some definitions and prop- erties related to fuzzy subgroups, perfect T -vague groups and fuzzy equivalence relations that will be needed later. Some properties of fuzzy maps and T -vague operations are also stated so that perfect T -vague groups could be set in this context. Definition 1. Let I = [0, 1], a mapping T : I × I -→ I is a t-norm if, 1. for all a ∈ I, T (1,a) = a, 2. for all a, b ∈ I, T (a, b) = T (b, a), 3. for all a, b, c ∈ I, T (T (a, b),c) = T (a, T (b, c)), 4. for all a, b, c, d ∈ I, T (a, b) ≤ T (c, d ) if a ≤ c and b ≤ d . Throughout the paper T will denote a given t -norm. ∗ Corresponding author. Rabah Kellil, University of Maj- maah, College of Science at Al-Zulfi, Al-Zulfi, Saudi Arabia. Tel.: +00966564070512; Fax: +00966 164044044; E-mail: r.kellil@mu.edu.sa. Definition 2. Let (G, ⋆) be a group, e its identity. A fuzzy subset μ of G is a fuzzy subgroup of G if and only if 1. μ(e) = 1, 2. μ(a⋆b) ≥ min{μ(a),μ(b)}, ∀a, b ∈ G, 3. ∀a ∈ G, μ(a) = μ(a -1 ). It is said normal if in addition μ(a⋆b⋆a -1 ) = μ(b). Definition 3. Let (G, ⋆) be a group. A fuzzy subset μ of G is a T -fuzzy subgroup of G if and only if T (μ(a),μ(b)) ≤ μ(a⋆b -1 ), ∀a, b ∈ G. Definition 4. A fuzzy binary relation or just a fuzzy relation on a set X is a mapping R : X × X -→ [0, 1]. Definition 5. A fuzzy relation R on a set X is a fuzzy equivalence relation if and only if for all a, b, c of X the following properties are satisfied, 1064-1246/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved