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L-Fuzzy Ideals of Residuated Lattices

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L-Fuzzy Ideals of Residuated Lattices Discussiones Mathematicae General Algebra and Applications 39 (2019) 181–201 doi:10.7151/dmgaa.1313 L-FUZZY IDEALS OF RESIDUATED LATTICES Pierre Carole Kengne 1, Blaise Bleriot´ Koguep 1∗ Daniel Akume 2 and Celestin Lele 1 1Department of Mathematics and Computer Science University of Dschang, Cameroon 2Computer Science Department, HTTTC Kumba University of Buea, Cameroon e-mail: [email protected] [email protected] d−[email protected] [email protected] Abstract This paper mainly focuses on building the L-fuzzy ideals theory of resid- uated lattices. Firstly, we introduce the notion of L-fuzzy ideals of a resid- uated lattice and obtain their properties and equivalent characterizations. Also, we introduce the notion of prime fuzzy ideal, fuzzy prime ideal and fuzzy prime ideal of the second kind of a residuated lattice and establish existing relationships between these types of fuzzy ideals. Finally, we in- vestigate the notions of fuzzy maximal ideal and maximal fuzzy ideal of a residuated lattice and present some characterizations. Keywords: fuzzy ideal, fuzzy prime ideal, prime fuzzy ideal, fuzzy maximal ideal and maximal fuzzy ideal. 2010 Mathematics Subject Classification: no numbers. Introduction Dealing with certain and uncertain information is an important task of the arti- ficial intelligence, in order to make computer simulate human being. To handle such information, in 1965, Zadeh [15] introduced the notion of fuzzy subset of a non empty set X as a function µ : X → I, where I = [0, 1] is the unit in- terval of real numbers. Since then, many authors have been using the above ∗Corresponding author. 182 P.C. Kengne, B.B. Koguep, D. Akume and C. Lele original definition to setup fuzzy mathematical structures. The notion of fuzzy ideal has been studied in several algebraic structures such as rings [13], lattices [1, 7, 12], MV-algebras [4], BL-algebras [9] and residuated lattices [5, 8, 11] and even in hyperstructures, for example in hyperlattices [6]. In 1988, Swany [13], introduced the notion of prime fuzzy ideal and showed the difference with the notion of fuzzy prime ideal given by Attallah [1]. The study of fuzzy maximal ideals was done in rings [10], lattices [14] and MV-algebras [4], but this has not been done in residuated lattice. Many authors have investigated fuzzy algebriac notions taking the linearly ordered set [0, 1] to be the set of degrees of member- ship. However, as Goguen [3] pointed out, in some situations, the structure of a complete bounded lattice L can be a suitable set of truth values. Tonga in [14], gives a new definition of a fuzzy set of a non-empty set by replacing the closed unit interval [0, 1] by a complete bounded lattice. This new definition of fuzzy set was used by Kadji et al. [5] to study the notion of Fuzzy prime and maximal filters of residuated lattices. But so far, these authors mostly focus of fuzzy filters in a residuated lattice. In this work, we will also replace the linearly ordered set [0; 1] by a complete bounded lattice L, but we will focus on fuzzy id eal of residuated lattices. We first recall some basic notions on residuated lattice and fix the different notations. Then, we define the notion of fuzzy ideal in residuated lattice and give some characterizations. By an example, we prove that the number of L-fuzzy ideals of a residuated lattice A is not always equal to the number of L-fuzzy filters of A, we also prove that in a residuated lattice, if µ is a L-fuzzy ideal then η = 1 − µ is not always a L-fuzzy filter. Also, if θ is a L-fuzzy filter then ρ = 1 − θ is not always a L-fuzzy ideal. Next, we present the notions of fuzzy prime ideal, fuzzy prime ideal of the second kind and prime fuzzy ideal. Moreover, we present the notions of maximal fuzzy ideal and fuzzy maximal ideal of a residuated lattice. 1. Preliminaries and notations In this preliminary section we recall some essential facts about ideal and prime ideal of residuated lattices; the reader is expected to refer to [2, 5, 8] or [11] for details. Definition 1.1 [11]. A residuated lattice (RL for short) is an algebraic structure A = (A, ∧, ∨, ⊗, →, 0, 1) of type (2, 2, 2, 2, 0, 0) satisfying the following axioms: (R1) (A, ∧, ∨, 0, 1) is a bounded lattice; (R2) (A, ⊗, 1) is a commutative monoid; (R3) For all x,y,z ∈ A, z ≤ x → y if and only if x ⊗ z ≤ y. L-fuzzy ideals of residuated lattices 183 Definition 1.2 [2]. A residuated lattice A which satisfy the prelinearity condition (i.e., (x → y) ∨ (y → x) = 1, for any x,y ∈ A) is called an MTL-algebra. There are RL that are not MTL-algebras as the follows examples show. Example 1.3. Let A = {0, a, b, c, d, 1} such that 0 < a, b < c < 1 and 0 <b< d< 1. Define → and ⊗ as follows: → 0abcd1 ⊗ 0abcd1 0 111111 0 000000 a d1d1d1 a 0a0a0a b cc1111 b 0000bb c bcd1d1 c 0a0abc d aacc11 d 00bbdd 1 0abcd1 1 0abcd1 (A, ∧, ∨, ⊗, →, 0, 1) is an MTL-algebra. Example 1.4. Let A = {0, a, b, c, 1} such that 0 <a<c< 1 and 0 <b<c. Define → and ⊗ as follows: Table 1. Table of operations ⊗ and −→ defined on A → 0abc1 ⊗ 0abc 1 0 11111 0 00000 a b1b11 a 0a0aa b aa111 b 00bbb c 0ab11 c 0abc c 1 0abc1 1 0abc 1 (A, ∧, ∨, ⊗, →, 0, 1) is a residuated lattice. We have (a → b) ∨ (b → a)= b ∨ a = c = 1, then (A, ∧, ∨, ⊗, →, 0, 1) is not an MTL-algebra. Let us give the following notations in a residuated lattice A. • ∀x ∈ A, x′ := x → 0; • ∀x ∈ A, ∀n ∈ N, x0 := 1 and xn+1 := xn ⊗ x; • ∀x,y ∈ A, x ⊘ y := x′ → y. Theorem 1.5 [8]. For any residuated lattice A = (A, ∧, ∨, ⊗, →, 0, 1), the fol- lowing properties hold for every x,y,z ∈ A: (P1) (x ⊗ y) → z = x → (y → z); (P2) x ⊗ (x → y) ≤ y; 184 P.C. Kengne, B.B. Koguep, D. Akume and C. Lele (P3) (x → y) ⊗ x ≤ x; (P4) x → y ≤ x ⊗ z → y ⊗ z; (P5) x ⊗ y ≤ x ∧ y; (P6) (x ∨ y) ⊗ z = (x ⊗ z) ∨ (y ⊗ z); (P7) If x ≤ y then y′ ≤ x′; (P8) y → z ≤ (x → y) → (x → z); (P9) (x ⊗ y)′ = x → y′; (P10) xm ≤ xn for any n,m ∈ N, m ≥ n; (P11) 1 → x = x, x → x = 1; (P12) x → (y → z)= y → (x → z); (P13) x ≤ y ⇔ x → y = 1; (P14) 0′ = 1, 1′ = 0,x′ = x′′′,x ≤ x′′; (P15) x → y ≤ (x ⊗ y′)′ = y′ → x′; (P16) y → x ≤ (x → z) → (y → z); (P17) x → (y ∧ z) = (x → y) ∧ (x → z); (P18) (x ∨ y) → z = (x → z) ∧ (y → z); (P19) (x ∨ y)′ = x′ ∧ y′ and (x ∧ y)′ ≥ x′ ∨ y′. Definition 1.6 [8]. Let A = (A, ∧, ∨, ⊗, →, 0, 1) be a residuated lattice and I a non empty subset of A. We say that I is an ideal of A if it satisfies the following conditions (I1) For every x,y ∈ A, x ⊘ y ∈ I; (I2) For every x,y ∈ A, if x ≤ y and y ∈ I, then x ∈ I. An ideal I of a residuated lattice A is said to be proper if and only if I = A. The following result is a characterization of ideals. Theorem 1.7 [8]. In any residuated lattice A and I ⊆ A, the following conditions are equivalent. (1) I is an ideal. (2) 0 ∈ I and for every x,y ∈ A, if x′ ⊗ y ∈ I and x ∈ I, then y ∈ I. (3) 0 ∈ I and for every x,y ∈ A, if x ∈ I and (x′ → y′)′ ∈ I, then y ∈ I. There are two types of prime ideal in any residuated lattice. Definition 1.8 [8]. Let I be a proper ideal of a residuated lattice A. L-fuzzy ideals of residuated lattices 185 (i) I is said to be a prime ideal, if for any x,y ∈ A, x ∧ y ∈ I implies x ∈ I or y ∈ I. (ii) I is said to be a prime ideal of second kind, if for any x,y ∈ A, (x → y)′ ∈ I or (y → x)′ ∈ I. Theorem 1.9 [8]. Let A be a residuated lattice. Every prime ideal of the second kind of A is also a prime ideal. If A is an MTL-algebra, then prime ideal of A and prime ideal of second kind of A are equivalent. Let us recall the notion of maximal ideal. Definition 1.10. Let I be a proper ideal of a residuated lattice A. I is said to be a maximal ideal, if for any ideal J of A, I ⊆ J implies J = I or J = A. 2. Fuzzy ideals Let A = (A, ∧, ∨, ⊗, →, 0, 1) be a residuated lattice and L = (L, ⊓, ⊔, ⊥, ⊤) be a complete bounded lattice. We will denote by ≤ and ⊑ the induced order relation defined on A and L respectively. Definition 2.1 [14]. (i) A map µ : A → L is called a L -fuzzy subset of A. (ii) A L-fuzzy subset µ of A is called proper if it is not a constant map. (iii) If µ : A → L is a L-fuzzy subset of A and α ∈ L, then µα = {a ∈ A; µ(a) ⊒ α} is called the α-cut set of µ. Let µ and η be two fuzzy subsets of A. µ ∧ η and µ ∨ η are the fuzzy subsets of A defined by (µ ∧ η)(x) = µ(x) ⊓ η(x) and (µ ∨ η)(x) = µ(x) ⊔ η(x), for any x ∈ A.
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