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- Hindawi Publishing Corporation
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- Copyright © 2021 Wondwosen Zemene Norahun et al. This 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.
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Abstract
Hindawi Advances in Fuzzy Systems Volume 2021, Article ID 7481960, 10 pages https://doi.org/10.1155/2021/7481960 Research Article Fuzzy Annihilator Ideals of C-Algebra 1 2 Wondwosen Zemene Norahun , Teferi Getachew Alemayehu , and Gezahagne Mulat Addis Department of Mathematics, University of Gondar, Gondar, Ethiopia Department of Mathematics, Debre Berehan University, Debre Berhan, Ethiopia Correspondence should be addressed to Wondwosen Zemene Norahun; wondie1976@gmail.com Received 25 May 2021; Revised 9 August 2021; Accepted 14 August 2021; Published 10 September 2021 Academic Editor: Ferdinando Di Martino Copyright © 2021 Wondwosen Zemene Norahun et al. ,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 relative fuzzy annihilator ideals in C-algebras and investigate some its properties. We characterize relative fuzzy annihilators in terms of fuzzy points. It is proved that the class of fuzzy ideals of C-algebras forms Heything algebra. We observe that the class of all fuzzy annihilator ideals can be made as a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism. We provide a sufficient condition for a homomorphism to be a fuzzy annihilator preserving. element Boolean algebras B and C are the only subdirectly 1. Introduction irreducible C-algebras and that the variety of C-algebras is a Fuzzy set theory was guided by the assumption that minimal cover of the variety of Boolean algebras. In [23], classical sets were not natural, appropriate, or useful U. M. Swamy et al. studied the center B(A) of a C-algebra A notions in describing the real-life problems because every and proved that the center of a C-algebra is a Boolean al- object encountered in this real physical world carries gebra. In [24], Rao and Sundarayya studied the concept of C- some degree of fuzziness. A lot of work on fuzzy sets has algebra as a poset. In a series of papers (see [25–28]), Vali come into being with many applications to various fields et al. studied the concept of ideals, principal ideals, and such as computer science, artificial intelligence, expert prime ideals of C-algebras as well as the concept of prime systems, control systems, decision making, medical di- spectrum, ideal congruences, and annihilators of C-algebras. agnosis, management science, operations research, pat- Later, Rao carried out a study on annihilator ideals of C- tern recognition, neural network, and others (see [1–4]). algebras [29]. Many papers on fuzzy algebras have been published since In this paper, we study the concept of relative fuzzy Rosenfeld [5] introduced the concept of fuzzy group in annihilator ideals in C-algebras. We characterize relative 1971. In particular, fuzzy subgroups of a group (see fuzzy annihilators in terms of fuzzy points. Using the [6–8]), fuzzy ideals of lattices and MS-algebra (see concept of the relative fuzzy annihilator, we prove that [9–16]), fuzzy ideals of C-algebras (see [17, 18]), and the class of fuzzy ideals of C-algebras forms the Heything intuitionistic fuzzy ideals of BCK-algebra, BG-algebra, algebra. We also study fuzzy annihilator ideals. Basic and BCI-algebra (see [19–21]). properties of fuzzy annihilator ideals are also studied. It is On the contrary, Guzman and Squier, in [22], introduced shown that the class of all fuzzy annihilator ideals forms a the variety of C-algebras as the variety generated by the complete Boolean algebra. Moreover, we study the three-element algebra C � {T, F, U} with the operations concept of fuzzy annihilator preserving homomorphism ∧ , ∨ , of type (2, 2, 1), which is the algebraic form of the and derived a sufficient condition for a homomorphism three-valued conditional logic. ,ey proved that the two- to be a fuzzy annihilator preserving. Finally, we prove 2 Advances in Fuzzy Systems that the image and preimage of fuzzy annihilator ideals ,e dual statements of the above identities are also valid are again fuzzy annihilator ideals. in a C-algebra. 2. Preliminaries Definition 2 (see [22]). An element z of a C-algebra A is called a left zero for ∧ if z ∧ x � z, for all x ∈ A. In this section, we recall some definitions and basic results on c−algebras. Definition 3 (see [26]). A nonempty subset I of a C-algebra A is called an ideal of A if Definition 1 (see [22]). An algebra (A, ∨ , ∧ , ) of type (1) a, b ∈ I⇒a ∨ b, a ∧ x ∈ I (2, 2, 1) is called a c-algebra if it satisfies the following axioms: (2) a ∈ I⇒x ∧ a ∈ I, for each x ∈ A. (1) a � b It can also be observed that a ∧ x ∈ I, for all a ∈ I and all x ∈ A. For any subset S ⊆ A, the smallest ideal of A con- ′ ′ ′ (2) (a ∧ b) � a ∨ b taining S is called the ideal of A generated by S and is (3) (a ∧ b) ∧ c � a ∧ (b ∧ c) denoted by ⟨S]. Note that (4) a ∧ (b ∨ c) � (a ∧ b) ∨ (a ∧ c) ⟨S] � ∨ (y ∧ x ): y ∈ A, x ∈ S, i � 1, . . . , n i i i i (5) (a ∨ b) ∧ c � (a ∧ c) ∨ (a ∧ b ∧ c) for some n ∈ Z } If S � {a}, then we write ⟨a] for ⟨S]. In this case, (6) a ∨ (a ∧ b) � a ⟨a] � {x ∧ a: x ∈ A}. Moreover, it is observed in [26] that (7) (a ∧ b) ∨ (b ∧ a) � (b ∧ a) ∨ (a ∧ b), for all a, b, c ∈ A the set I � x ∧ x : x ∈ A is the smallest ideal in A. ′ ′ ′ ′ Definition 4. Let (A, ∨ , ∧ , , I ) and (A , ∨ , ∧ , , I ) be two Example 1. ,e three-element algebra C � {T, F, U} with the 0 0 operations given by by the following tables is a C-algebra. C-algebras. ,en, a mapping f: A ⟶ A is called a ho- momorphism if it satisfies the following conditions: (1) f(a ∨ b) � f(a) ∨ f(b) (2) f(a ∧ b) � f(a) ∧ f(b) ′ ′ (3) f(a ) � f(a) , for all a, b ∈ A Here, I � {z ∈ A: z is a left zero for ∧} and I � 0 0 y ∈ A : y is a left zero for ∧. I and I are the smallest ideals of the C-algebras A and 0 0 A , respectively. ,e kernel of the homomorphism is defined as Kerf � x ∈ A: f(x) ∈ I . Remember that, for any set A, a function μ: A ⟶ [0, 1] is called a fuzzy subset of A. For each t ∈ [0, 1], the set, μ � x ∈ A: μ(x) ≥ t , (1) is called the level subset of μ at t [30]. For numbers α and β in [0, 1], we write α∧β for min α, β and α ∨ β for max α, β . Note: the identities 2.1 (a) and 2.1 (b) imply that the variety of C-algebras satisfies all the dual statements of 2.1 Definition 5 (see [17]). A fuzzy subset λ of A is called a fuzzy (2) to 2.1 (7) in this view. ideal of A if Lemma 1 (see [22]). Every C−algebra satisfies the following (1) λ(z) � 1, for all z ∈ I identities: (2) λ(a ∨ b) ≥ λ(a) ∧ λ(b) (1) x ∧ x � x (3) λ(a ∨ b) ≥ λ(b), for all a, b ∈ A ′ ′ (2) x ∧ x � x ∧ x We denote the class of all fuzzy ideals of A by FI(A). (3) x ∧ y ∧ x � x ∧ y ′ ′ (4) x ∧ x ∧ y � x ∧ x Lemma 2 (see [17]). Let λ be a fuzzy ideal of A. 6en, the following hold, for all a, b ∈ A: (5) x ∧ y � (x ∨ y) ∧ x (6) x ∧ y � x ∧ (y ∨ x ) (1) λ(a ∧ b) ≥ λ(a) (7) x ∧ y � x ∧(x ∨ y) (2) λ(a ∧ b) ≥ λ(b ∧ a) ′ ′ (8) x ∧ y ∧ x � x ∧ y ∧ y (3) λ(a ∧ x ∧ b) ≥ λ(a ∧ b), for each x ∈ A (9) (x ∨ y) ∧ x � x ∨ (y ∧ x) (4) λ(a) ≥ λ(a ∨ b); hence, λ(a) ∧ λ(b) ∧ λ(b ∨ a) ′ ′ ′ (10) x ∧ (x ∨ x) � (x ∨ x) ∧ x � (x ∨ x ) ∧ x (5) If x ∈ (a], then λ(x) ≥ λ(a) Advances in Fuzzy Systems 3 Let μ be a fuzzy subset of A. ,en, the fuzzy ideal ,e class of fuzzy ideals of a C-algebra is denoted by generated by μ is denoted by (μ]. FI(A). Note: throughout the rest of this paper, A stands for a C- Theorem 1 (see [17]). If λ and ] are fuzzy ideals of a C- algebra. algebra, then their supremum is given by 3. Relative Fuzzy Annihilator n n (λ ∨ ])(x) � Sup∧ λ b ∨ ] b : x � ∨ b , b ∈ A. i i i i i�1 i�1 In this section, we study the concept of relative fuzzy an- (2) nihilator ideals in a C-algebra. Basic properties of relative fuzzy annihilator ideals are also studied. We characterize relative fuzzy annihilator in terms of fuzzy points. Finally, we We define the binary operations “+” and “.” on the set of prove that the class of fuzzy ideals of a C-algebra forms the all fuzzy subsets of A as Heyting algebra. (λ + ])(x) � Supλ(y)∧](z): y, z ∈ A, y ∨ z � x, (λ · ])(x) � Supλ(y)∧](z): y, z ∈ A, y∧z � x. Definition 7. For any fuzzy subset λ of A and a fuzzy ideal ], we define (3) (λ: ]) � ⋃η: η ∈ [0, 1] , η · λ ⊆ ]. (6) If λ and ] are fuzzy ideals of A, then λ · ] is a fuzzy ideal and λ · ] � λ ∩ ]. However, in a general case, λ + ] is not a A fuzzy subset (λ: ]) is called fuzzy annihilator of λ fuzzy ideal. relative to ]. For any x ∈ A, Definition 6 (see [5]). Let f be a function from X to Y, μ be a fuzzy subset of X, and θ be a fuzzy subset of Y. (λ: ])(x) � Supη(x): η ∈ [0, 1] , η · λ ⊆ ]. (7) (1) ,e image of μ under f, denoted by f(μ), is a fuzzy For simplicity, we write subset of Y defined, for each y ∈ Y, by (λ: ]) � Supη: η ∈ [0, 1] , η · λ ⊆ ]. (8) − 1 − 1 ⎧ ⎨ Sup μ(x): x ∈ f (y) , if f (y) ≠ ϕ, f(μ)(y) � 0, otherwise. Lemma 3. For any two fuzzy subsets λ and ] of a C-algebra (4) A, we have − 1 (λ · ]] � (λ]∧(]]. (9) (2) ,e preimage of θ under f, denoted by f (θ), is a fuzzy subset of X defined, for each x ∈ X, by Now, we prove the following lemma. − 1 (5) f (θ)(x) � θ(f(x)). Lemma 4. For any fuzzy subset λ of A and a fuzzy ideal ], we have (λ: ]) � ⋃η: η ∈ FI(A), η · λ ⊆ ]. (10) Theorem 2 (see [31]). Let f be a function from X to Y. 6en, the following assertions hold: (1) For all fuzzy subset μ of X, i ∈ I, f( ∪ μ ) � ∪ i i∈I i i∈I Proof. Clearly, ⋃η: η ∈ FI(A), η · λ ⊆ ] ⊆ ⋃{δ: δ ∈ f(μ ), so μ ⊆ μ ⇒f(μ ) ⊆ f(μ ). i 1 2 1 2 [0, 1] , δ · λ ⊆ ]}. Since (η · λ] � (η]∧(λ], we can easily show (2) For all fuzzy subset θ of Y, j ∈ J, that the other inclusion holds. ,us, − 1 − 1 − 1 f ( ∪ θ ) � ∪ f (θ ), f ( ∩ θ ) � j∈J j j∈J j j∈J j (λ: ]) � Sup η: η ∈ FI(A), η · λ ⊆ ] . (11) − 1 − 1 ∩ f (θ ), and therefore, θ ⊆ θ ⇒f j∈J j 1 2 − 1 (θ ) ⊆ f (θ ). 1 2 − 1 (3) μ ⊆ f (f(μ)). In particular, if f is an injection, then Theorem 3. For any fuzzy subset λ of A and a fuzzy ideal ], − 1 μ � f (f(μ)), for all fuzzy subset μ of X. (λ: ]) is a fuzzy ideal of A. − 1 (4) f(f (θ)) ⊆ θ. In particular, if f is a surjection, then − 1 f(f (θ)) � θ, for all fuzzy subset θ of Y. Proof. Since λ · ] ⊆ ] and ] is a fuzzy ideal, we get ](z) � 1, − 1 (5) f(μ) ⊆ θ⇔μ ⊆ f (θ), for all fuzzy subsets μ and θ of for all left zero element z for ∧. X and Y, respectively. Let x, y ∈ A. ,en, 4 Advances in Fuzzy Systems (λ: ])(x)∧(λ: ])(y) � Supη(x): η ∈ FI(A), η · λ ⊆ ] ∧ Supσ(y): σ ∈ FI(A), σ · λ ⊆ ] � Supη(x) ∧ σ(y): η, σ ∈ FI(A), η · λ ⊆ ], σ · λ ⊆ ] (12) ≤ Sup(η ∨ σ)(x) ∧ (η ∨ σ)(y): η, σ ∈ FI(A), η · λ ⊆ ], σ · λ ⊆ ]. Since η, σ ∈ FI(A), η · λ ⊆ ], and σ · λ ⊆ ], we get that η ∨ σ ∈ FI(A) and (η ∨ σ) · λ ⊆ ]. ,en, (λ: ])(x) ∧ (λ: ])(y) ≤ Supc(x)∧c(y): c ∈ FI(A), λ · λ ⊆ ] � Supc(x ∨ y): c ∈ FI(A), c · λ ⊆ ] � (λ: ])(x ∨ y). (13) ,us, (λ: ])(x ∨ y) ≥ (λ: ])(x)∧(λ: ])(y). On the contrary, let x ∈ A. ,en, Now, consider the C-algebra C � A × A � a , a , a , a , 1 2 3 4 (λ: ])(x) � Supη(x): η ∈ FI(A), η · λ ⊆ ] a , a , a , a , a }, where a � (T, U), a � (F, U), 5 6 7 8 9 1 2 ≤ Supη(x∧y): η ∈ FI(A), η · λ ⊆ ] (14) a � (U, T), a � (U, F), a � (U, U), a � (T, T), 3 4 5 6 a � (F, F), a � (T, F), and a � (F, T). ,en, the set of left � (λ: ])(x∧y). 7 8 9 zero for ∧ is I � a , a , a , a . 0 2 4 5 7 Similarly, (λ: ])(y) ≤ (λ: ])(x ∧ y). So, (λ: ])(x∧y) If we define two fuzzy subsets θ and μ of C as ≥ (λ: ])(x) ∨ (λ: ])(y). Hence, (λ: ]) is a fuzzy ideal of A. In the following theorem, we characterize relative fuzzy θ a � θ a � θ a � θ a � 1, 2 4 5 7 annihilators in terms of fuzzy points. □ θ a � θ a � 0.6, 1 8 θ a � θ a � θ a � 0.5, 3 6 9 Theorem 4. Let λ be a fuzzy subset of A and ] be a fuzzy (17) μ a � μ a � μ a � 0.8, ideal. 6en, for each x ∈ A, 1 2 3 μ a � μ a � μ a � 0.7, 4 5 6 (λ: ])(x) � Supα ∈ [0, 1]: x · λ ⊆ ]. (15) μ a � μ a � μ a � 0.4, 7 8 9 Proof. For each x ∈ A, let us define two sets C and B as x x then θ is a fuzzy ideal of C and (μ: θ)(a ) � (μ: θ) follows: (a ) � (μ: θ)(a ) � (μ: θ)(a ) � 1, (μ: θ)(a ) � (μ: θ) 4 5 7 1 A (a ) � 0.6 and (μ: θ)(a ) � (μ: θ)(a ) � (μ: θ)(a ) � 0.5. 8 3 6 9 C � η(x): η ∈ [0, 1] , η · λ ⊆ ] , ,us, (μ: θ) is a fuzzy ideal of C. (16) B � α ∈ [0, 1]: x · λ ⊆ ]. In the following lemma, some basic properties of relative x α fuzzy annihilators can be observed. Since λ · ] ⊆ ], then both C and B are nonempty x x subsets of [0, 1]. Now, we proceed to show that ∨ C � ∨ B . x x Lemma 5. Let η and ξ be fuzzy subsets and λ, ], and c be Let α ∈ C . ,en, α � η(x) for some fuzzy subset η of A fuzzy ideals of A. 6en, satisfying η · λ ⊆ ]. If α � 0, then we can find β ∈ B such that α ≤ β. On the contrary, suppose that α ≠ 0. ,en, x is a fuzzy α (1) (η: λ) � χ ⇔ η ⊆ λ point of A such that x ⊆ η, which implies x · λ ⊆ η · ] and α α (2) ] ⊆ (η: ]) α ∈ B . ,us, C ⊆ B . So, ∨ C ≤ ∨ B . x x x x x (3) η ⊆ ξ⇒(ξ: λ) ⊆ (η: λ) To show ∨ B ≤ ∨ C , let β ∈ B . ,en, x is a fuzzy point x x x β of A such that x · λ ⊆ ]. ,is shows that β ∈ C . ,us, (4) λ ⊆ ]⇒(ξ: λ) ⊆ (ξ: ]) β x B ⊆ C . So, ∨ B ≤ ∨ C . So, ∨ B � ∨ C . Hence, the x x x x x x (5) (η: λ ∩ ]) � (η: λ) ∩ (η: ]) result is obtained. □ (6) ((η]: λ) � (η: λ) (7) (η ∪ ξ: λ) � (η: λ) ∩ (ξ: λ) Example 2. Let A � T, F, U , and define ∨ , ∧, and on A as { } (8) (λ ∨ ]: c) � (λ: c) ∩ (]: c) follows. (9) (λ: ]) � (λ ∨ ]: ]) � (λ: λ∧]) (10) (η] ∩ ] ⊆ λ ⇔ ] ⊆ (η: λ) Proof. ,e proof of (3) and (4) is straightforward. Now, we proceed to prove the following. Advances in Fuzzy Systems 5 (1) Let (η: λ) � χ . To show η ⊆ λ, assume that η⊈λ. λ(x) ≥ η(x); it is a contradiction. So, η ⊆ λ. ,e ,en, there is x ∈ A such that η(x) > μ(x). ,is converse part is trivial. implies that δ(x) ≤ λ(x), for each δ such that (2) Since ]∧(η] ⊆ ], we get that ] · η ⊆ ]. ,us, ] ⊆ (η: ]). δ · η ⊆ λ. ,us, λ(x) is an upper bound of (3) δ(x): δ · η ⊆ λ. ,is shows that λ(x) ≥ 1. ,us, (η: λ) ∩ (η: ]) � Sup c ∈ FI(A), c · η ⊆ λ ∧Sup c ∈ FI(A), c · η ⊆ ] 1 1 2 2 � Supc ∧c : c · η ⊆ λ, c · η ⊆ ] 1 2 1 2 � Sup c: c · η ⊆ λ, c · η ⊆ ] , Where c � c ∧c (18) 1 2 1 2 Supc: c ∈ FI(A), c · η ⊆ λ ∩ ] � (η: λ ∩ ]). (4) Since (c · η] � c ∧ (η], for every c ∈ FI(A), we can (5) By property (3), we have that easily verified that (η: λ) � t((η]: λ). (η ∪ ξ, λ) ⊆ (η, λ) ∩ (ξ, λ). On the contrary, (η: λ) ∩ (ξ: λ) � ((η]: λ) ∩ ((ξ]: λ) � Supc ∈ FI(A): c ∧(η] ⊆ λ∧Supc ∈ FI(A), c ∧(ξ] ⊆ λ (19) 1 1 2 2 � Supc ∧c : c , c ∈ FI(L), c ∧(η] ⊆ λ, c ∧(ξ] ⊆ λ. 1 2 1 2 1 2 (η] ∩ ] � ∨ x ∩ (η] Since c ∧ (η] ⊆ λ and c ∧(ξ] ⊆ λ, there exists a fuzzy α 1 2 x ⊆] ideal c of A contained in c and c such that 1 2 (23) � ∨ x · η by Lemma 3 c∧(η] ⊆ λ and c∧(ξ] ⊆ λ. ,is implies that x ⊆] (c∧((η] ∨ (ξ])) ⊆ λ. ,is shows that ⊆ ]. (η: λ) ∩ (ξ: λ) ⊆ ,us, (η] ∩ ] ⊆ λ. □ ⊆ Sup c: c∧((η] ∨ (ξ]) ⊆ λ (20) ⊆ Supc: c · (η ∪ ξ) ⊆ λ Theorem 5. Let ] be a fuzzy ideal of A. If λ is a class of α α∈Δ � (η ∪ ξ: λ). fuzzy ideals of A, then (6) Since λ ∨ ] � (λ ∪ ]], by (6), we get ∪ λ : ] � ∩ λ : ] . (24) α α α∈Δ α∈Δ (λ ∨ ]: c) � (λ ∪ ]: c). ,us, (λ ∨ ]: c) � (λ: c) ∩ (]: c). (21) Proof. We know that λ ⊆ ∪ λ for each α ∈ Δ. ,us, by α α∈Δ α (7) Since (θ: θ) � χ , by (8), we get (λ ∨ ]: ]) � (λ: ]). A Lemma 5 (3), we get ( ∪ λ : ]) ⊆ (λ : ]) for each α ∈ Δ. α∈Δ α α On the contrary, let c∧λ ⊆ ] for some fuzzy ideal c of ,us, A. Since c ∧ λ ⊆ λ, we get that c∧λ ⊆ λ∧]. ,us, (λ: ]) ⊆ (λ: λ ∧ ]). Since λ∧] ⊆ ], by (4), we have ∪ λ : ] ⊆ ∩ λ : ] . (25) α α α∈Δ α∈Δ (λ: λ∧]) ⊆ (λ: ]). Hence, (λ: ]) � (λ: λ ∧ ]). So, (λ: ]) � (λ ∨ ]: ]) � (λ: λ∧]). On the contrary, put η � ∩ (λ : ]). ,en, η ⊆ (λ : ]), α∈Δ α α for each α ∈ Δ. By Lemma 5 (10), we have λ ∩ η ⊆ ], for each (8) If (η] ∩ ] ⊆ λ, then, by (6) and by the definition of relative fuzzy annihilator, ] ⊆ (η: λ). Conversely, α ∈ Δ. ,is implies suppose ] ⊆ (η: λ). Since ] is a fuzzy ideal, we can express ] as follows: ∨ λ ∩ η � ∨ λ ∩ η ⊆ ]. (26) α α α∈Δ α∈Δ So, by Lemma 5 (10), we have η ⊆ ( ∨ λ : ]). ,us, α∈Δ α ] � ∨ x . α (22) x ⊆] ∩ λ : ] ⊆ ∪ λ : ]. (27) α α α∈Δ α∈Δ Let x be a fuzzy point of A such that x ⊆ ]. Since α α ] ⊆ (η: λ), we get x ⊆ (η: λ). ,us, x · η ⊆ λ. Now, So, α α 6 Advances in Fuzzy Systems Theorem 7. 6e set FI (A) of all fuzzy ideals of A is the ∪ λ : ] � ∩ λ : ] . (28) α α Heyting algebra. α∈Δ α∈Δ In the following theorem, we prove that (λ: ]) is a Proof. We know that the set (FI(A), ∨ , ∩ , χ , χ ) of all relative pseudocomplement of λ and ] in the class of I A fuzzy ideals of A is a complete distributive lattice. For any FI(A). □ fuzzy ideals λ and ] of A, by ,eorem 6, (λ: ]) is the largest fuzzy ideal of c ∈ FI(A): c ∩ λ ⊆ ] . ,us, Theorem 6. Let η be a fuzzy subset and λ and ] fuzzy ideals of A. 6en, λ ⟶ ] � (λ: ]). (30) (1) (η: λ) is the largest fuzzy ideal such that So, (FI(A), ∨ , ∩ , ⟶ , χ , χ ) is the Heyting I A (η] ∩ (η: λ) ⊆ λ algebra. □ (2) (λ: ]) is the largest fuzzy ideal such that λ ∩ (λ: ]) ⊆ ] 4. Fuzzy Annihilator Ideals Proof. First, we have to show that (η] ∩ (η: λ) ⊆ λ. For any x ∈ A, In this section, we study fuzzy annihilator ideals in C-al- gebras. Some basic properties of fuzzy annihilator ideals are ((η] ∩ (η: λ))(x) � (η](x) ∧ Sup c(x): c ∈ FI(A), c ∧ (η] ⊆ λ also studied. It is proved that the set of all fuzzy annihilator � Sup(η](x) ∧ c(x): c ∧ (η] ⊆ λ ideals forms a complete Boolean algebra. � Sup((η] ∩ c)(x): c ∩ (η] ⊆ λ ≤ λ(x). Definition 8. For any fuzzy subset λ of A, the fuzzy subset ∗ ∗ (29) (λ: χ ) is a fuzzy ideal denoted by λ and λ is called a fuzzy annihilator of λ. ,us, (η] ∩ (η: λ) ⊆ λ. Now, we show that (η: λ) is the largest fuzzy ideal Lemma 6. Let λ be a fuzzy subset of A. 6en, satisfying (η] ∩ (η: λ) ⊆ λ. Suppose not. ,en, there exists a fuzzy ideal c properly containing (η: λ) such that (η] ∩ c ⊆ λ. ∗ (1) χ ⊆ λ ,en, by Lemma 5 (10), we get that c ⊆ (η: λ), which is a (2) λ · λ ⊆ χ contradiction. ,erefore, (η: λ) is the largest fuzzy ideal, 0 satisfying (η] ∩ (η: λ) ⊆ λ. (3) λ · λ � χ , whenever λ(z) � 1, where z ∈ I I 0 ∗ ∗∗ In [18], Alaba and Addis introduced the concept of fuzzy (4) λ ∧λ � χ ideals of C-algebra, and they proved that the class of all fuzzy ideals of a C-algebra is a complete distributive lattice. In the following theorem, using the concept of relative fuzzy an- Proof. Here, it is enough to prove property (3). Let λ be any nihilator ideals of a C-algebra, we prove that the class of fuzzy subset of A and x ∈ A. ,en, fuzzy ideals of a C-algebra forms the Heyting algebra. □ ∗ ∗ λ · λ (x) � Supλ(a)∧λ (b): x � a∧b � Supλ(a)∧Supη(b): η ∈ FI(A), η · λ ⊆ χ : x � a∧b � SupSupλ(a)∧η(b): η · λ ⊆ χ : x � a∧b (31) � SupSupλ(a)∧η(b): x � a∧b: η · λ ⊆ χ � Sup (λ · η)(x): η · λ ⊆ χ ≤ χ (x). ∗ ∗∗ ,is shows that λ · λ ⊆ χ . If λ(z) � 1, for z ∈ I , then (4) λ ⊆ λ I 0 ∗ ∗ ∗ ∗∗∗ (λ · λ )(z) � 1 and χ � λ · λ . □ (5) λ � λ ∗ ∗ (6) ((λ]) � λ Lemma 7. Let λ and ] be fuzzy subsets of A. 6en, ∗ ∗ (7) (λ ∪ ]) � λ ∩ ] (1) λ ⊆ ]⇒] ⊆ λ (2) ] · λ ⊆ χ ⇔ ] ⊆ λ Lemma 8. For any fuzzy ideals λ and ] of A, we have (3) ] · λ � χ ⇔ ] ⊆ λ , whenever λ(z) � 1 � ](z) for ∗ ∗ (1) (λ ∨ ]) � λ ∩ ] z ∈ I 0 Advances in Fuzzy Systems 7 ∗ ∗ (2) (λ ∨ λ ) � χ I ∗ ∪ λ � ∩ λ . (32) i i i∈I i∈ I (3) (χ ) � χ I A (4) (χ ) � χ A I Proof. Let λ : i ∈ I be family of fuzzy subsets of L. Since Theorem 8. 6e set FI (A) of all fuzzy ideals of A is a λ ⊆ ( ∪ λ ) for each i ∈ I, by Lemma 7 (1), we have i i∈I i ∗ ∗ pseudocomplemented lattice. ( ∪ λ ) ⊆ λ . ,us, i∈I i Proof. Let λ be a fuzzy ideal of A. ,en, it is clear that λ is a ∪ λ ⊆ ∩ λ . (33) i i ∗ i∈ I i∈I fuzzy ideal of A and that λ ∩ λ � χ . Suppose now ∗ ∗ θ ∈ FI(A) such that λ ∩ ] � χ . ,en, by Lemma 7 (2), ] ⊆ λ , I To prove ∩ λ ⊆ ( ∪ λ ) , it is enough to show that i∈I i i∈I i and consequently, λ is the pseudocomplement of λ. □ ( ∩ λ ) · ( ∪ λ ) ⊆ χ . For any x ∈ A, i∈I i j∈I j I Lemma 9. If λ ∈ [0, 1] , for every i ∈ I, then ∗ ∗ ∩ λ · ∪ λ (x) � Sup ∩ λ (a)∧ ∪ λ (b): a∧b � x i j i j i∈I i∈I j∈I j∈I � Sup∧ λ (a)∧ ∨ λ (b): a∧b � x i∈I j∈I � Sup ∨ ∧ λ (a) ∧λ (b) : a∧b � x i j j∈I i∈I (34) ≤ Sup ∨ λ (a)∧λ (b): a∧b � x j j j∈I ≤ Sup ∨ λ · λ (x): a∧b � x j j j∈I ≤ Sup ∨ χ (x) j∈I � χ (x). ∗ ∗ ∗∗ ,us, by Lemma 7 (2), we get that ( ∩ λ ) ⊆ ( ∪ λ ) . then λ is a fuzzy ideal of C and λ � λ . ,us, λ is a fuzzy i∈I i i∈I i So, annihilator ideal of C. Lemma 10. Let λ, ] ∈ FI (A). 6en, ∩ λ � ∪ λ . (35) i i i∈I i∈ I ∗ ∗ (1) λ ∩ ] � (λ ∨ ] ) ∗∗ Now, we define the fuzzy annihilator ideal. □ (2) λ ∩ ] � (λ ∩ ]) ,e result (2) of the above lemma can be generalized as Definition 9. A fuzzy ideal λ of A is called a fuzzy annihilator given in the following. ideal if λ � ] , for some fuzzy subset ] of A, or equivalently, ∗∗ if λ � λ . We denote the class of all fuzzy annihilator ideals of A by Corollary 1. If λ : i ∈ Δ is a family of fuzzy annihilator FI (A). ideals of A, then ∗∗ ∩ λ � ∩ λ . (37) i i Example 3. Consider the three-element C-algebra i∈Δ i∈Δ A � {T, F, U} and C � A × A � a , a , a , a , a , a , a , 1 2 3 4 5 6 7 a , a } given in Example 2. If we define a fuzzy subset λ of C 8 9 Theorem 9. A map α: FI(A) ⟶ FI(A) defined by α(λ) � as ∗∗ λ , ∀λ ∈ FI(A) is a closure operator on FI(A). 6at is, λ a � λ a � λ a � λ a � λ a � λ a � 1, 1 2 4 5 7 8 (1) λ ⊆ α(λ) λ a � λ a � λ a � 0.4, 3 6 9 (2) α(α(λ)) � α(λ) (36) (3) λ ⊆ ]⇒α(λ) ⊆ α(]), for any two fuzzy ideals λ, ] of A 8 Advances in Fuzzy Systems Fuzzy annihilator ideals are simply the closed elements ,roughout this section, A and A denote C-algebras with respect to the closure operator. with the smallest ideals I and I , respectively, and 0 0 f: A ⟶ A denotes a C-algebra homomorphism. Lemma 11. If λ, ] ∈ FI (L), the supremum of λ and ] is given by Lemma 12. In A, the following conditions hold: ∗ ∗ λ⊔] � λ ∩ ] . (38) (1) χ ⊆ χ I Kerf (2) f(χ ) ⊆ χ I I 0 0 (3) χ is a fuzzy ideal of A Proof. First, we need to show λ⊔] is a fuzzy annihilator Kerf ∗ ∗ ideal. Clearly λ⊔] is a fuzzy ideal of A. Since λ ∩ ] ⊆ λ , we ∗∗ ∗ ∗ get λ � λ ⊆ (] ⊆ λ ) � λ⊔]. Similarly, ] ⊆ λ⊔]. ,is im- Lemma 13. If λ is any fuzzy subset of A and ] is a fuzzy ideal plies λ⊔] is an upper bound of λ and ]. Suppose that c is a of A, then fuzzy annihilator ideals of A such that λ ⊆ c and ] ⊆ c. ,en, ∗ ∗ ∗ ∗ ∗ ∗ we get c ⊆ λ and c ⊆ ] and c ⊆ λ ∩ ] . ,is implies f((λ: ])) ⊆ (f(λ): f(])). (40) ∗ ∗ ∗ ∗∗ λ⊔] � (λ ∩ ] ) ⊆ c � c. Hence, λ⊔] is the smallest fuzzy annihilator ideal containing λ and ]. □ ∗ ∗ In particular, if ] � χ , then f(λ ) ⊆ (f(λ)) . Corollary 2. Let (λ ) be a family of fuzzy annihilator ideals i i∈I of A. 6en, ⊔ λ is the smallest fuzzy annihilator ideal Proof. Let λ be any fuzzy subset of A and ] be a fuzzy ideal of i∈I i containing each λ . A. For any y ∈ A , − 1 f((λ: ]))(y) � Sup(λ: ])(x): x ∈ f (y) In the following theorem, we prove that the class of all −1 fuzzy annihilator ideals forms a complete Boolean algebra. � SupSupη(x): η ∈ FI(A), η · λ ⊆ ]: x ∈ f (y) � Supf(η)(y): η · λ ⊆ ] Theorem 10. 6e set FI (A) of all fuzzy annihilator ideals of ≤ Supc(y): c ∈ FI A , λ · f(λ) ⊆ f(]) A forms a complete Boolean algebra. � (f(λ): f(]))(y). (41) Proof. Clearly, (FI (A), ∩ , ⊔, χ , χ ) is a complete boun- I A ded lattice. To show the distributivity, let λ, ], η ∈ FI (A). ,en, f((λ: ])) ⊆ (f(λ): f(])). □ ,en, ∗ ∗ ∗ λ⊔(] ∩ η) � λ ∩ (] ∩ η) Definition 11. For any fuzzy subset λ of L, f is said to be a ∗ ∗ ∗∗∗ ∗∗ ∗∗ ∗ fuzzy annihilator preserving if f(λ ) � (f(λ)) . � λ ∩ ] ∩ η In the following theorem, we give a sufficient condition ∗∗∗ ∗ ∗ ∗ � λ ∩ ] ∨ η for a homomorphism to be fuzzy annihilator preserving. (39) ∗ ∗ ∗ ∗ ∗ � λ ∩ ] ∨ λ ∩ η ∗ ∗ Theorem 11. If Kerf � I and f is onto, then f is a fuzzy ∗ ∗ ∗ ∗ � λ ∩ ] ∩ λ ∩ η annihilator preserving. � (λ⊔]) ∩ (λ⊔η). ∗ Proof. Let λ be any fuzzy subset of A. ,en, Hence, (FI (A), ∩ , ⊔, χ , χ ) is a complete distributive I A ∗ ∗ − 1 ∗ ∗ f(λ ) ⊆ (f(λ)) . Since Kerf � I and f is onto, f (χ ) � 0 ′ lattice. For any λ ∈ FI (A), we have λ ∩ λ � χ and 0 I A 0 − 1 ∗ ∗ ∗ ∗ χ and ] � f(f (])), for all ] ∈ [0, 1] . Let y ∈ A . ,en, λ∧λ � (λ⊔λ ) � χ . Hence, λ is the complement of λ in ∗ ∗ FI (A). ,erefore, (FI (A), ∩ , ⊔, χ , χ ) is a complete ′ (f(λ)) (y) � Sup ](y): ] ∈ FI A , ] · f(λ) ⊆ χ I A 0 I Boolean algebra. □ − 1 � Sup](y): ff (]) · f(λ) ⊆ χ − 1 � Sup](y): ff (]) · λ ⊆ χ Definition 10. A fuzzy ideal μ of A is called dense fuzzy ideal ′ if μ � χ . − 1 − 1 0 � Sup ](y): f (]) · λ ⊆ f χ − 1 � Sup](f(x)): f (]) · λ ⊆ χ , f(x) � y 5. 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Advances in Fuzzy Systems – Hindawi Publishing Corporation
Published: Sep 10, 2021