P-Fuzzy Ideals and P-Fuzzy Filters in P-Algebras
P-Fuzzy Ideals and P-Fuzzy Filters in P-Algebras
Norahun, Wondwosen Zemene
2021-07-01 00:00:00
Hindawi Advances in Fuzzy Systems Volume 2021, Article ID 4561087, 9 pages https://doi.org/10.1155/2021/4561087 Research Article Wondwosen Zemene Norahun Department of Mathematics, University of Gondar, Gondar, Ethiopia Correspondence should be addressed to Wondwosen Zemene Norahun; wondie1976@gmail.com Received 28 April 2021; Accepted 14 June 2021; Published 1 July 2021 Academic Editor: Antonin Dvora´k Copyright © 2021 Wondwosen Zemene Norahun. -is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce the concept of p-fuzzy ideals and p-fuzzy filters in a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filter. Theorem 1 (see [15]). For any two elements a, b of a p-al- 1. Introduction gebra, we have the following: -e concept of fuzzy sets was firstly introduced by Zadeh [1]. In ∗∗ (1) 0 � 0 1971, Rosenfeld used the notion of a fuzzy subset of a set to (2) a∧ a � 0 introduce the concept of a fuzzy subgroup of a group [2]. ∗ ∗ Rosenfeld’s paper inspired the development of fuzzy abstract (3) a≤ b⇒b ≤ a algebra. Since then, several authors have developed interesting ∗∗ (4) a≤ a results on fuzzy theory (see [3–14]). In this paper, we introduce ∗∗∗ ∗ (5) a � a the concept of p-fuzzy ideals and p-fuzzy filters in p-algebra. ∗ ∗ (6) (a∨b) � a ∧b We provide a set of equivalent conditions for a fuzzy ideal to be ∗∗ ∗∗ ∗∗ a p-fuzzy ideal and a p-algebra to be a Boolean algebra. (7) (a∧b) � a ∧b Moreover, we prove that, for any fuzzy ideal of L, there is the ∗∗ An element x of a p-algebra is called closed, if x � x . smallest p-fuzzy ideal containing it. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we prove that the image and inverse image of a p-fuzzy ideal is a Definition 2 (see [15]). A nonempty subset I of L is called an p-fuzzy ideal under a ∗-epimorphism mapping. Finally, we ideal of L if for any x, y ∈ I, x∨y ∈ I and x ∈ I, y ∈ L, show that there is an isomorphism between the class of p-fuzzy x∧y ∈ I. ideals and p-fuzzy filters. Definition 3 (see [15]). A nonempty subset F of L is called a 2. Preliminaries filter of L if for any x, y ∈ F, x∧y ∈ F and x ∈ F, y ∈ L, x∨ y ∈ I. In this section, we recall some definitions and basic results on p-algebra and fuzzy theory. Definition 4 (see [16]). An ideal I of L is called a p-ideal if ∗∗ Definition 1 (see [15]). An algebra L � (L;∧,∨, , 0, 1) of type any x ∈ I, x ∈ I. (2, 2, 1, 0, 0) is a p-algebra if the following conditions hold: (1) (L;∧,∨, 0, 1) is a bounded lattice Definition 5 (see [16]). A filter F of L is called a p-filter if for ∗ ∗∗ (2) For all a, b ∈ L, a∧b � 0⇔a∧b � a any x ∈ F, x ∈ F. 2 Advances in Fuzzy Systems Definition 6 (see [1]). Let X be any nonempty set. A (2) μ(x∨y)≥ μ(x)∨μ(y) mapping μ: X ⟶ [0, 1] is called a fuzzy subset of X. (3) μ(x∧y)≥ μ(x)∧μ(y) -e unit interval [0, 1] together with the operations min We define the binary operations “+” and “·” on the set of and max forms a complete lattice satisfying the infinite meet all fuzzy subsets of L as distributive law; i.e., (μ + θ)(x) � Supμ(y)∧θ(z): y, z ∈ L, y∨z � x, (4) α∧ ∨ β � ∨ (α∧β), (1) β∈M β∈M (μ · θ)(x) � Supμ(y)∧θ(z): y, z ∈ L, y∧z � x. for all α ∈ [0, 1] and any M⊆[0, 1]. If μ and θ are fuzzy ideals of L, then μ · θ � μ∧θ � μ∩ θ We often write ∧ for minimum or infimum and ∨ for and μ + θ � μ∨θ is a fuzzy ideal generated by μ∪ θ. maximum or supremum. -at is, for all α, β ∈ [0, 1], we have If μ and θ are fuzzy filters of L, then μ + θ � μ∧θ (the α∧β � minα, β and α∨β � maxα, β. pointwise infimum of μ and θ) and μ · θ � μ∨θ (the -e characteristics function of any set A is defined as supremum of μ and θ). 1, if x ∈ A, Theorem 2 (see [18]). Let L be a lattice, x ∈ L and α ∈ [0, 1]. χ (x) � (2) 0, if x ∉ A. Define a fuzzy subset α of L as 1, if y≤ x, Definition 7 (see [2]). Let μ and θ be fuzzy subsets of a set A. α (y) � (5) α, if y≰x, Define the fuzzy subsets μ∪ θ and μ∩ θ of A as follows: for each x ∈ A, (μ∪ θ)(x) � μ(x)∨θ(x) and (μ∩ θ)(x) � which is a fuzzy ideal of L. μ(x)∧θ(x). -en, μ∪ θ and μ∩ θ are called the union and inter- Remark 1 (see [18]). α is called the α-level principal fuzzy section of μ and θ, respectively. ideal corresponding to x. For any collection, μ : i ∈ I of fuzzy subsets of X, where i x Similarly, a fuzzy subset α of L defined as I is a nonempty index set, the least upper bound ∪ μ and i∈I i 1, if x≤ y, the greatest lower bound ∩ μ of the μ ’s are given for each i∈I i i α (y) � (6) x ∈ X, (∪ μ )(x) � ∨ μ (x) and (∩ μ )(x) �∧ μ (x), i∈I i i∈I i i∈I i i∈I i α, if x≰y, respectively. For each t ∈ [0, 1], the set is the α-level principal fuzzy filter corresponding to x. μ � x ∈ A: μ(x)≥ t (3) Remark 2. Let L be a lattice with 0. -en, L is called 0- distributive if for any a, b, c ∈ L with a∧b � 0 � a∧c im- is called the level subset of μ at t [1]. plying a∧(b∨c) � 0. -roughout the rest of this paper, L stands for the 0- Definition 8 (see [2]). Let f be a function from X into Y, μ distributive p-algebra unless otherwise mentioned. be a fuzzy subset of X, and θ be a fuzzy subset of Y. (1) -e image of μ under f, denoted by f(μ), is a fuzzy 3. P-Fuzzy Ideals subset of Y defined by the following: for each y ∈ Y, −1 −1 In this section, we study the concept of p-fuzzy ideals of a Supμ(x): x∈ f (y), if f (y)≠ϕ, f(μ)(y) � p-algebra. We provide a set of equivalent conditions for a fuzzy 0, otherwise − 1 ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean (2) -e preimage of θ under f, denoted by f (θ), is a algebra. Moreover, we prove that, for any fuzzy ideal of L, there fuzzy subset of X defined by for each x ∈ X, − 1 is the smallest p-fuzzy ideal containing it. It is proved that the f (θ)(x) � θ(f(x)) class of p-fuzzy ideals forms a complete distributive lattice. Finally, we show that the image and inverse image of a p-fuzzy Definition 9 (see [17]). A fuzzy subset μ of a bounded lattice ideal is a p-fuzzy ideal under a ∗-epimorphism mapping. L is called a fuzzy ideal of L, if for all x, y ∈ L the following conditions are satisfied: Definition 11. A fuzzy ideal μ of L is called a p-fuzzy ideal if ∗∗ μ(x) � μ(x ) for each x ∈ L. (1) μ(0) � 1 (2) μ(x∨y)≥ μ(x)∧μ(y) Lemma 1. A fuzzy ideal μ of L is a p-fuzzy ideal if and only if ∗ ∗ ∗ (3) μ(x∧y)≥ μ(x)∨μ(y) μ((a ∧b ) )≥ μ(a)∧μ(b) for all a, b ∈ L. Lemma 2. For any x ∈ L, α is a p-fuzzy ideal. Definition 10 (see [17]). A fuzzy subset μ of a bounded lattice L is called a fuzzy filter of L, if for all x, y ∈ L the ∗ ∗∗ ∗ Proof. Let a ∈ L. If α (a) � 1, then a≤ x and a ≤ x . following conditions are satisfied: ∗∗ ∗ ∗∗ ∗ -us, α ∗ (a ) � 1. If α ∗ (a) � α, then a≰x and a ≰x . x x ∗∗ (1) μ(1) � 1 -us, α (a ) � α. Hence, α is a p-fuzzy ideal. □ ∗ ∗ x x Advances in Fuzzy Systems 3 Theorem 3. α is a p-fuzzy ideal if and only if x is a closed Now we proceed to show that p(μ) is the smallest element. p-fuzzy ideal containing μ. Let θ be a p-fuzzy ideal con- ∗∗ ∗∗ taining μ. Let x, a ∈ L such that x≤ a . -en, θ(a )≤ θ(x). ∗∗ Since θ is a p-fuzzy ideal and μ⊆θ, we get that θ(a)≤ θ(x) Proof. Let α be a p-fuzzy ideal. -en, α (x ) � 1 and x x ∗∗ ∗∗ ∗∗ and μ(a)≤ θ(x). -is shows that θ(x) is an upper bound of x ≤ x. Since x≤ x , we have x � x . -us, x is closed. ∗∗ ∗∗ μ(a): x≤ a , a ∈ L. -is implies Sup μ(a): x≤ a , Conversely, suppose that x is a closed element. Let a ∈ L. ∗∗ ∗∗ a ∈ L}≤ θ(x). -us, p(μ)⊆θ. So p(μ) is the smallest p-fuzzy If α (a) � 1, then a≤ x and a ≤ x. -us, α (a ) � 1. If x x ∗∗ ∗∗ ideal containing μ. □ α (a) � α, then a≰x and a ≰x. -us, α (a ) � α. Hence x x α is a p-fuzzy ideal. □ Lemma 3. If μ and θ are fuzzy ideals of L, then μ⊆θ implies Corollary 1. 9e following conditions on L are equivalent: p(μ)⊆ p(θ). (1) Every fuzzy ideal is a p-fuzzy ideal (2) Every level principal fuzzy ideal is a p-fuzzy ideal Lemma 4. For any two fuzzy ideals μ and θ of L, p(μ∩ θ) � p(μ)∩ p(θ). (3) L is Boolean algebra Proof. Let μ and θ be two fuzzy ideals of L. Clearly Proof. -e proofs of 1 ⇒ 2 and 3⇒ 1 are straightforward. p(μ∩ θ)⊆p(μ)∩ p(θ). To show the other inclusion, let To show that 2⇒ 3, suppose that every level principal fuzzy x ∈ L. -en ideal is a p-fuzzy ideal. -en, by the above theorem, every ∗∗ element of L is closed. For any x, y ∈ L, the supremum is (p(μ)∩ p(θ))(x) � Supμ(a): x≤ a , a ∈ L ∗ ∗ given by x⊻y � (x ∧y ) . ∗∗ ∧Supθ(b): x≤ b , b ∈ L To show that L is distributive, it suffices to prove that ∗∗ ∗∗ ≤ Sup μ(a∧b)∧θ(a∧b): x≤ a ∧b x∧(y⊻z)≤ (x∧y)⊻(x∧z), ∀x, y, z ∈ L. (7) ∗∗ ≤ Sup(μ∩ θ)(a∧b): x≤ (a∧b) For this purpose, let t � (x∧y)∨(x∧z). -en, x∧y≤ t � ∗∗ ≤ Sup (μ∩ θ)(c): x≤ c , c ∈ L ∗∗ ∗ ∗ ∗ ∗ ∗ t gives x∧y∧t � 0 and x∧t ≤ y . Similarly, x∧t ≤ z and ∗∗ ∗ ∗ ∗ ∗ ∗ therefore x∧t ≤ y ∧z � (y ∧z ) . It follows from this � p(μ∩ θ)(x). ∗ ∗ ∗ ∗ that x∧t ∧(y ∧z ) � 0 and hence that x∧(y⊻z) � (9) ∗ ∗ ∗∗ x∧(y ∧z ) ≤ t � t � (x∧y)⊻(x∧z). To see that L is also complemented, observe that 1 � 0 -us, p(μ∩ θ) � p(μ)∩ p(θ). □ ∗ ∗ and 0 � 1 . Since every x ∈ L, we have x∧x � 0 and ∗ ∗ ∗∗ ∗ ∗ x∨x � (x ∧x ) � 0 � 1. We see that the complement of Lemma 5. For any fuzzy ideal μ of L, the map μ ⟶ p(μ) is x is x . -us, L is complemented. Hence L is a Boolean a closure operator on FI(L). 9at is, algebra. □ (1) μ⊆p(μ) (2) p(p(μ)) � p(μ) Theorem 4. A fuzzy subset μ of L is a p-fuzzy ideal if and only if every level subset of μ is a p-ideal of L. (3) μ⊆θ⇒p(μ)⊆p(θ), for any two fuzzy ideals μ, θ of L p-fuzzy ideals are simply the closed elements of FI(L) with Corollary 2. A nonempty subset I of L is a p-ideal if and only respect to the closure operator. if χ is a p-fuzzy ideal. Every p-fuzzy ideal is a fuzzy ideal but the converse may In the following result, we prove that, for any fuzzy ideal not be true. For this, we have the following example. of L, there is the smallest p-fuzzy ideal containing it. Example 1. Consider the p-algebra L � 0, a, b, c, 1 whose { } Theorem 5. Let μ be a fuzzy ideal of L. Define Hasse diagram is given below. ∗∗ Define fuzzy subsets μ and θ of L as follows: p(μ)(x) � Supμ(a): x≤ a : a ∈ L. (8) -en, p(μ) is the smallest p-fuzzy ideal containing μ and hence μ is a p-fuzzy ideal if and only if μ � p(μ). Proof. Let μ be a fuzzy ideal of L. -en clearly p(μ) is a fuzzy a b ideal of L. To prove p(μ) is a p-fuzzy ideal, let x ∈ L. Clearly ∗∗ p(μ)(x)≥ p(μ)(x ). On the other hand, p(μ)(x) � ∗∗ ∗∗ ∗∗ ∗∗ Supμ(a): x≤ a ≤ Supμ(a): x ≤ a � p(μ)(x ). -us, p(μ) is a p-fuzzy ideal containing μ. 0 4 Advances in Fuzzy Systems Theorem 6. If λ, η ∈ FI (L), then the supremum of λ and η is μ(0) � 1 � μ(a), given by μ(b) � μ(c) � μ(1) � 0, ∗ ∗ (10) (λ⊻η)(x) � Supλ(a)∧η(b): x≤ a ∧b , a, b ∈ L. θ(0) � θ(b) � 1, (11) θ(a) � θ(c) � θ(1) � 0. -en, it can be easily verified that μ and θ are p-fuzzy Proof. Put c � λ⊻η. Clearly c(0) � 1. For any x, y ∈ L, ideals of L. Moreover, we observe that the fuzzy ideal μ∨θ of L is not a p-fuzzy ideal. -e set of all p-fuzzy ideals of L is denoted by FI (L). We now prove that FI (L) is a lattice. ∗ ∗ c(x)∧c(y) � Supλ