Fuzzy Annihilator Ideals of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 11.6374 11.6718" width="11.6374pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g></svg>-</span>Algebra

Norahun, Wondwosen Zemene; Alemayehu, Teferi Getachew; Addis, Gezahagne Mulat

doi:https://doi.org/10.1155/2021/7481960

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Research Article | Open Access

Volume 2021 | Article ID 7481960 | https://doi.org/10.1155/2021/7481960

Fuzzy Annihilator Ideals of -Algebra

Wondwosen Zemene Norahun,¹Teferi Getachew Alemayehu,²and Gezahagne Mulat Addis¹

Academic Editor: Ferdinando Di Martino

Received25 May 2021

Revised09 Aug 2021

Accepted14 Aug 2021

Published10 Sept 2021

Abstract

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.

1. 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 [1–4]). 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 [6–8]), fuzzy ideals of lattices and MS-algebra (see [9–16]), fuzzy ideals of C-algebras (see [17, 18]), and intuitionistic fuzzy ideals of BCK-algebra, BG-algebra, and BCI-algebra (see [19–21]).

On the contrary, Guzman and Squier, in [22], introduced the variety of -algebras as the variety generated by the three-element algebra with the operations of type , which is the algebraic form of the three-valued conditional logic. They proved that the two-element Boolean algebras and are the only subdirectly irreducible -algebras and that the variety of -algebras is a minimal cover of the variety of Boolean algebras. In [23], U. M. Swamy et al. studied the center of a -algebra and proved that the center of a -algebra is a Boolean algebra. In [24], Rao and Sundarayya studied the concept of C-algebra as a poset. In a series of papers (see [25–28]), 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 C-algebras [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.

2. Preliminaries

In this section, we recall some definitions and basic results on algebras.

Definition 1 (see [22]). An algebra of type is called a c-algebra if it satisfies the following axioms:(1)(2)(3)(4)(5)(6)(7), for all

Example 1. The three-element algebra with the operations given by by the following tables is a -algebra.

Note: the identities 2.1 (a) and 2.1 (b) imply that the variety of -algebras satisfies all the dual statements of 2.1 (2) to 2.1 (7) in this view.

Lemma 1 (see [22]). Every algebra satisfies the following identities:(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

The dual statements of the above identities are also valid in a -algebra.

Definition 2 (see [22]). An element of a -algebra is called a left zero for if , for all .

Definition 3 (see [26]). A nonempty subset of a -algebra is called an ideal of if(1)(2), for each .It can also be observed that , for all and all . For any subset , the smallest ideal of containing is called the ideal of generated by and is denoted by . Note that
If , then we write for . In this case, . Moreover, it is observed in [26] that the set is the smallest ideal in .

Definition 4. Let and be two -algebras. Then, a mapping is called a homomorphism if it satisfies the following conditions:(1)(2)(3), for all Here, and .
and are the smallest ideals of the -algebras and , respectively. The kernel of the homomorphism is defined as .
Remember that, for any set , a function is called a fuzzy subset of A. For each , the set,is called the level subset of at [30]. For numbers and in , we write for and for .

Definition 5 (see [17]). A fuzzy subset of is called a fuzzy ideal of if(1), for all (2)(3), for all We denote the class of all fuzzy ideals of by .

Lemma 2 (see [17]). Let be a fuzzy ideal of . Then, the following hold, for all :(1)(2)(3), for each (4); hence, (5)If , then

Let be a fuzzy subset of . Then, the fuzzy ideal generated by is denoted by .

Theorem 1 (see [17]). If and are fuzzy ideals of a C-algebra, then their supremum is given by

We define the binary operations “” and “” on the set of all fuzzy subsets of as

If and are fuzzy ideals of , then is a fuzzy ideal and . However, in a general case, is not a fuzzy ideal.

Definition 6 (see [5]). Let be a function from to , be a fuzzy subset of , and be a fuzzy subset of .(1)The image of under , denoted by , is a fuzzy subset of defined, for each , by(2)The preimage of under , denoted by , is a fuzzy subset of defined, for each , by

Theorem 2 (see [31]). Let be a function from to . Then, the following assertions hold:(1)For all fuzzy subset of , , so .(2)For all fuzzy subset of , , and therefore, .(3). In particular, if is an injection, then , for all fuzzy subset of .(4). In particular, if is a surjection, then , for all fuzzy subset of .(5), for all fuzzy subsets and of and , respectively.

The class of fuzzy ideals of a C-algebra is denoted by .

Note: throughout the rest of this paper, stands for a C-algebra.

3. 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 of and a fuzzy ideal , we defineA fuzzy subset is called fuzzy annihilator of relative to .
For any ,For simplicity, we write

Lemma 3. For any two fuzzy subsets and of a C-algebra , we have

Now, we prove the following lemma.

Lemma 4. For any fuzzy subset of and a fuzzy ideal , we have

Proof. Clearly, . Since , we can easily show that the other inclusion holds. Thus,

Theorem 3. For any fuzzy subset of and a fuzzy ideal , is a fuzzy ideal of .

Proof. Since and is a fuzzy ideal, we get , for all left zero element for .
Let . Then,Since , , and , we get that and . Then,Thus, .
On the contrary, let . Then,Similarly, . So, . Hence, is a fuzzy ideal of .
In the following theorem, we characterize relative fuzzy annihilators in terms of fuzzy points.

Theorem 4. Let be a fuzzy subset of and be a fuzzy ideal. Then, for each ,

Proof. For each , let us define two sets and as follows:Since , then both and are nonempty subsets of . Now, we proceed to show that . Let . Then, for some fuzzy subset of satisfying . If , then we can find such that . On the contrary, suppose that . Then, is a fuzzy point of such that , which implies and . Thus, . So, .
To show , let . Then, is a fuzzy point of such that . This shows that . Thus, . So, . So, . Hence, the result is obtained.

Example 2. Let , and define , , and on as follows.

Now, consider the C-algebra , where . Then, the set of left zero for is .
If we define two fuzzy subsets and of asthen is a fuzzy ideal of and , and . Thus, is a fuzzy ideal of .
In the following lemma, some basic properties of relative fuzzy annihilators can be observed.

Lemma 5. Let and be fuzzy subsets and , and be fuzzy ideals of . Then,(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

Proof. The proof of (3) and (4) is straightforward. Now, we proceed to prove the following.(1)Let . To show , assume that . Then, there is such that . This implies that , for each such that . Thus, is an upper bound of . This shows that . Thus, ; it is a contradiction. So, . The converse part is trivial.(2)Since , we get that . Thus, .(3)(4)Since for every , we can easily verified that .(5)By property (3), we have that . On the contrary, Since and , there exists a fuzzy ideal of contained in and such that and . This implies that . This shows that(6)Since , by (6), we get . Thus,(7)Since , by (8), we get . On the contrary, let for some fuzzy ideal of . Since , we get that . Thus, . Since , by (4), we have . Hence, . So, .(8)If , then, by (6) and by the definition of relative fuzzy annihilator, . Conversely, suppose . Since is a fuzzy ideal, we can express as follows:Let be a fuzzy point of such that . Since , we get . Thus, . Now,Thus, .

Theorem 5. Let be a fuzzy ideal of . If is a class of fuzzy ideals of , then

Proof. We know that for each . Thus, by Lemma 5 (3), we get for each . Thus,On the contrary, put . Then, , for each . By Lemma 5 (10), we have , for each . This impliesSo, by Lemma 5 (10), we have . Thus,So,In the following theorem, we prove that is a relative pseudocomplement of and in the class of .

Theorem 6. Let be a fuzzy subset and and fuzzy ideals of . Then,(1) is the largest fuzzy ideal such that (2) is the largest fuzzy ideal such that

Proof. First, we have to show that . For any ,Thus, .
Now, we show that is the largest fuzzy ideal satisfying . Suppose not. Then, there exists a fuzzy ideal properly containing such that . Then, by Lemma 5 (10), we get that , which is a contradiction. Therefore, is the largest fuzzy ideal, satisfying .
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. The set of all fuzzy ideals of is the Heyting algebra.

Proof. We know that the set of all fuzzy ideals of is a complete distributive lattice. For any fuzzy ideals and of , by Theorem 6, is the largest fuzzy ideal of . Thus,So, is the Heyting algebra.

4. 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 of , the fuzzy subset is a fuzzy ideal denoted by and is called a fuzzy annihilator of .

Lemma 6. Let be a fuzzy subset of . Then,(1)(2)(3), whenever , where (4)

Proof. Here, it is enough to prove property (3). Let be any fuzzy subset of and . Then,This shows that . If , for , then and .

Lemma 7. Let and be fuzzy subsets of . Then,(1)(2)(3), whenever for (4)(5)(6)(7)

Lemma 8. For any fuzzy ideals and of , we have(1)(2)(3)(4)

Theorem 8. The set of all fuzzy ideals of is a pseudocomplemented lattice.

Proof. Let be a fuzzy ideal of . Then, it is clear that is a fuzzy ideal of and that . Suppose now such that . Then, by Lemma 7 (2), , and consequently, is the pseudocomplement of .

Lemma 9. If , for every , then

Proof. Let be family of fuzzy subsets of . Since for each , by Lemma 7 (1), we have . Thus,To prove , it is enough to show that . For any ,Thus, by Lemma 7 (2), we get that . So,Now, we define the fuzzy annihilator ideal.

Definition 9. A fuzzy ideal of is called a fuzzy annihilator ideal if , for some fuzzy subset of , or equivalently, if .
We denote the class of all fuzzy annihilator ideals of by .

Example 3. Consider the three-element C-algebra and given in Example 2. If we define a fuzzy subset of asthen is a fuzzy ideal of and . Thus, is a fuzzy annihilator ideal of .

Lemma 10. Let . Then,(1)(2)

The result (2) of the above lemma can be generalized as given in the following.

Corollary 1. If is a family of fuzzy annihilator ideals of , then

Theorem 9. A map defined by is a closure operator on . That is,(1)(2)(3), for any two fuzzy ideals of

Fuzzy annihilator ideals are simply the closed elements with respect to the closure operator.

Lemma 11. If , the supremum of and is given by

Proof. First, we need to show is a fuzzy annihilator ideal. Clearly is a fuzzy ideal of . Since , we get . Similarly, . This implies is an upper bound of and . Suppose that is a fuzzy annihilator ideals of such that and . Then, we get and and . This implies . Hence, is the smallest fuzzy annihilator ideal containing and .

Corollary 2. Let be a family of fuzzy annihilator ideals of . Then, is the smallest fuzzy annihilator ideal containing each .

In the following theorem, we prove that the class of all fuzzy annihilator ideals forms a complete Boolean algebra.

Theorem 10. The set of all fuzzy annihilator ideals of forms a complete Boolean algebra.

Proof. Clearly, is a complete bounded lattice. To show the distributivity, let . Then,Hence, is a complete distributive lattice. For any , we have and . Hence, is the complement of in . Therefore, is a complete Boolean algebra.

Definition 10. A fuzzy ideal of is called dense fuzzy ideal if .

5. 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.

Throughout this section, and denote C-algebras with the smallest ideals and , respectively, and denotes a C-algebra homomorphism.

Lemma 12. In , the following conditions hold:(1)(2)(3) is a fuzzy ideal of

Lemma 13. If is any fuzzy subset of and is a fuzzy ideal of , then

In particular, if , then .

Proof. Let be any fuzzy subset of and be a fuzzy ideal of . For any ,Then, .

Definition 11. For any fuzzy subset of , is said to be a fuzzy annihilator preserving if .
In the following theorem, we give a sufficient condition for a homomorphism to be fuzzy annihilator preserving.

Theorem 11. If and is onto, then is a fuzzy annihilator preserving.

Proof. Let be any fuzzy subset of . Then, . Since and is onto, and , for all . Let . Then,Thus, . So, preserves fuzzy annihilator.

Theorem 12. If and is onto, then preserves the fuzzy annihilator.

Proof. Let be any fuzzy subset of and . Then,Thus, . Similarly, . Therefore, .

Theorem 13. For any C-algebra , the following conditions hold:(1)If is fuzzy annihilator preserving and onto, then is a fuzzy annihilator ideal of , for every fuzzy annihilator ideal of (2)If preserves the annihilator, then is a fuzzy annihilator ideal of , for every fuzzy annihilator ideal of (3) is a fuzzy annihilator ideal of

6. 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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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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