Fuzzy Annihilator Ideals of C-Algebra

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.


Introduction
Fuzzy set theory was guided by the assumption that classical sets were not natural, appropriate, or useful notions in describing the real-life problems because every object encountered in this real physical world carries some degree of fuzziness. A lot of work on fuzzy sets has come into being with many applications to various fields such as computer science, artificial intelligence, expert systems, control systems, decision making, medical diagnosis, management science, operations research, pattern recognition, neural network, and others (see [1][2][3][4]). Many papers on fuzzy algebras have been published since Rosenfeld [5] introduced the concept of fuzzy group in 1971. In particular, fuzzy subgroups of a group (see [6][7][8]), fuzzy ideals of lattices and MS-algebra (see [9][10][11][12][13][14][15][16]), fuzzy ideals of C-algebras (see [17,18]), and intuitionistic fuzzy ideals of BCK-algebra, BG-algebra, and BCI-algebra (see [19][20][21]).
On the contrary, Guzman and Squier, in [22], introduced the variety of C-algebras as the variety generated by the three-element algebra C � T, F, U { } with the operations ∧ , ∨ , ′ of type (2, 2, 1), which is the algebraic form of the three-valued conditional logic. ey proved that the two-element Boolean algebras B and C are the only subdirectly irreducible C-algebras and that the variety of C-algebras is a minimal cover of the variety of Boolean algebras. In [23], U. M. Swamy et al. studied the center B(A) of a C-algebra A and proved that the center of a C-algebra is a Boolean algebra. In [24], Rao and Sundarayya studied the concept of Calgebra as a poset. In a series of papers (see [25][26][27][28]), Vali et al. studied the concept of ideals, principal ideals, and prime ideals of C-algebras as well as the concept of prime spectrum, ideal congruences, and annihilators of C-algebras. Later, Rao carried out a study on annihilator ideals of Calgebras [29].
In this paper, we study the concept of relative fuzzy annihilator ideals in C-algebras. We characterize relative fuzzy annihilators in terms of fuzzy points. Using the concept of the relative fuzzy annihilator, we prove that the class of fuzzy ideals of C-algebras forms the Heything algebra. We also study fuzzy annihilator ideals. Basic properties of fuzzy annihilator ideals are also studied. It is shown that the class of all fuzzy annihilator ideals forms a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism and derived a sufficient condition for a homomorphism to be a fuzzy annihilator preserving. Finally, we prove that the image and preimage of fuzzy annihilator ideals are again fuzzy annihilator ideals.

Preliminaries
In this section, we recall some definitions and basic results on c−algebras.
Definition 1 (see [22]). An algebra (A, ∨ , ∧ , ′ ) of type (2, 2, 1) is called a c-algebra if it satisfies the following axioms: e three-element algebra C � T, F, U { } with the operations given by by the following tables is a C-algebra.
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 (2) to 2.1 (7) in this view.
Lemma 1 (see [22]). Every C−algebra satisfies the following identities: e dual statements of the above identities are also valid in a C-algebra.
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.
Definition 3 (see [26]). A nonempty subset I of a C-algebra 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 containing S is called the ideal of A generated by S and is denoted by Here, I 0 � z ∈ A: z is a left zero for ∧ { } and I 0 ′ � y ∈ A ′ : y is a left zero for ∧ .
I 0 and I 0 ′ are the smallest ideals of the C-algebras A and A ′ , respectively. e kernel of the homomorphism is defined as Kerf � x ∈ A: f(x) ∈ I 0 ′ .
Definition 5 (see [17]). A fuzzy subset λ of A is called a fuzzy ideal of A if We denote the class of all fuzzy ideals of A by FI(A).
Lemma 2 (see [17]). Let λ be a fuzzy ideal of A. en, the following hold, for all a, b ∈ A: Let μ be a fuzzy subset of A.
en, the fuzzy ideal generated by μ is denoted by (μ]. Theorem 1 (see [17]). If λ and ] are fuzzy ideals of a Calgebra, then their supremum is given by We define the binary operations "+" and "." on the set of all fuzzy subsets of A as 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 fuzzy ideal.
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.
(2) e preimage of θ under f, denoted by f − 1 (θ), is a fuzzy subset of X defined, for each x ∈ X, by Theorem 2 (see [31]). Let f be a function from X to Y. en, the following assertions hold: and therefore, , for all fuzzy subsets μ and θ of X and Y, respectively.

e class of fuzzy ideals of a C-algebra is denoted by FI(A).
Note: throughout the rest of this paper, A stands for a Calgebra.

Relative Fuzzy Annihilator
In this section, we study the concept of relative fuzzy annihilator 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 prove that the class of fuzzy ideals of a C-algebra forms the Heyting algebra.
Definition 7. For any fuzzy subset λ of A and a fuzzy ideal ], we define A fuzzy subset (λ: ]) is called fuzzy annihilator of λ relative to ].
For any x ∈ A, For simplicity, we write

Lemma 3. For any two fuzzy subsets λ and ] of a C-algebra
Now, we prove the following lemma.

Lemma 4. For any fuzzy subset λ of A and a fuzzy ideal ], we have
Proof.
In the following lemma, some basic properties of relative fuzzy annihilators can be observed.  Proof.
e proof of (3) and (4) is straightforward. Now, we proceed to prove the following.
is shows that λ(x) ≥ 1. us, it is a contradiction. So, η ⊆ λ. e converse part is trivial.
In [18], Alaba and Addis introduced the concept of fuzzy 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 annihilator ideals of a C-algebra, we prove that the class of fuzzy ideals of a C-algebra forms the Heyting algebra. □

Theorem 7. e set FI(A) of all fuzzy ideals of A is the Heyting algebra.
Proof. We know that the set (FI(A)

Fuzzy Annihilator Ideals
In this section, we study fuzzy annihilator ideals in C-algebras. Some basic properties of fuzzy annihilator ideals are also studied. It is proved that the set of all fuzzy annihilator ideals forms a complete Boolean algebra.
Definition 8. For any fuzzy subset λ of A, the fuzzy subset (λ: χ I 0 ) is a fuzzy ideal denoted by λ * and λ * is called a fuzzy annihilator of λ.

Theorem 8. e set FI(A) of all fuzzy ideals of A is a pseudocomplemented lattice.
Proof. Let λ be a fuzzy ideal of A. en, it is clear that λ * is a fuzzy ideal of A and that λ ∩ λ * � χ I 0 . Suppose now θ ∈ FI(A) such that λ ∩ ] � χ I 0 . en, by Lemma 7 (2), ] ⊆ λ * , and consequently, λ * is the pseudocomplement of λ.
Proof. Let λ i : i ∈ I be family of fuzzy subsets of L. Since λ i ⊆ ( ∪ i∈I λ i ) for each i ∈ I, by Lemma 7 (1), we have us, by Lemma 7 (2), we get that Now, we define the fuzzy annihilator ideal.
□ Definition 9. A fuzzy ideal λ of A is called a fuzzy annihilator 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 FI * (A). a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 } given in Example 2. If we define a fuzzy subset λ of C as

Example 3. Consider the three-element C-algebra
then λ is a fuzzy ideal of C and λ � λ * * . us, λ is a fuzzy annihilator ideal of C. Lemma 10. Let λ, ] ∈ FI * (A). en, (2) of the above lemma can be generalized as given in the following.   en, ⊔ i∈I λ i is the smallest fuzzy annihilator ideal containing each λ i .

Fuzzy Annihilator Preserving Homomorphism
In this section, we study some basic properties of fuzzy annihilator preserving homomorphisms. We give a sufficient condition for a homomorphism to be fuzzy annihilator preserving. Finally, we show that the images and inverse images of fuzzy annihilator ideals are again fuzzy annihilator ideals.
roughout this section, A and A ′ denote C-algebras with the smallest ideals I 0 and I 0 ′ , respectively, and f: A ⟶ A ′ denotes a C-algebra homomorphism.

Lemma 12.
In A, the following conditions hold: (40) Proof. Let λ be any fuzzy subset of A and ] be a fuzzy ideal of A. For any y ∈ A ′ , □ Definition 11. For any fuzzy subset λ of L, f is said to be a fuzzy annihilator preserving if f(λ * ) � (f(λ)) * . In the following theorem, we give a sufficient condition for a homomorphism to be fuzzy annihilator preserving.

Conclusion
In this work, we studied the concept of relative fuzzy annihilator ideals of C-algebras. We characterized relative fuzzy annihilators in terms of fuzzy points. We proved that the class of fuzzy ideals of C-algebras forms the Heything algebra. We also studied fuzzy annihilator ideals and investigate some its properties. It is shown that the class of all fuzzy annihilator ideals forms a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism. Our future work will focus on fuzzy congruence relation on C-algebras.

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