Strong GE-Filters and GE-Ideals of Bordered GE-Algebras

Öztürk, Mehmet Ali; Lee, Jeong-Gon; Bandaru, Ravikumar; Jun, Young Bae

doi:https://doi.org/10.1155/2021/5520023

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5520023 | https://doi.org/10.1155/2021/5520023

Strong GE-Filters and GE-Ideals of Bordered GE-Algebras

Mehmet Ali Öztürk,¹Jeong-Gon Lee,²Ravikumar Bandaru,³and Young Bae Jun⁴

Academic Editor: Elena Guardo

Received16 Jan 2021

Accepted22 Apr 2021

Published19 May 2021

Abstract

The notion of strong GE-filters and GE-ideals (generated) is introduced, and the related properties are investigated. The intersection of strong GE-filters (resp., GE-ideals) is proved to be a strong GE-filter (resp., GE-ideal), and the union of strong GE-filters (resp., GE-ideals) is generally not a strong GE-filters (resp., GE-ideal) by example. Conditions for a subset of a bordered GE-algebra to be a strong GE-filter are provided, and a characterization of a strong GE-filter is considered. In order to do so, irreducible GE-filter is defined first and its properties are examined. Conditions for a GE-filter to be irreducible are discussed. Given a GE-filter, and a subset in a bordered GE-algebra, the existence of an irreducible GE-filter, which contains the given GE-filter and is disjoint to the given subset, is considered. Conditions under which any subset of a bordered GE-algebra can be a GE-ideal are provided, and GE-ideal that is generated from a subset in a bordered GE-algebra is discussed. Also, what element it is formed into is stated. Finally, the smallest GE-ideal which contains a given GE-ideal and an element in a bordered GE-algebra is established.

1. Introduction

In 1966, Imai and Iséki introduced BCK-algebras (see [1]) as the algebraic semantics for a non classical logic possessing only implication. Since then, the generalized concepts of BCK-algebras have been studied by various scholars. Kim and Kim introduced the notion of a BE-algebra as a generalization of a dual BCK-algebra (see [2]). Hilbert algebras were introduced by Henkin and Skolem in the fifties for investigations in intuitionistic and other non classical logics. Diego proved that Hilbert algebras form a variety which is locally finite (see [3]). Rezaei et al. discussed relations between Hilbert algebras and BE-algebras (see [4]). The generalization process in the study of algebraic structures is also an important area of study. As a generalization of Hilbert algebras, Bandaru et al. introduced the notion of GE-algebras and investigated several properties (see [5, 6]).

In this article, we introduce the notions of strong GE-filters and GE-ideals (generated) and investigate the related properties in bordered GE-algebras. In Section 3, we introduce the concept of strong GE-filters and investigate its properties. We show that the intersection of strong GE-filters is a strong GE-filters, but the union of strong GE-filters is not a strong GE-filter by example. We provide conditions for a subset of a bordered GE-algebra to be a strong GE-filter. We consider a characterization of a strong GE-filter. In order to do so, we define irreducible GE-filter first and examine its properties. We provide conditions for a GE-filter to be irreducible. Given a GE-filter and a subset in a bordered GE-algebra, we consider the existence of an irreducible GE-filter which contains the given GE-filter and is disjoint to the given subset. In Section 4, we introduce the notion of GE-ideals in bordered GE-algebras and study its properties. We show that the intersection of GE-ideals is a GE-ideal, but the union of GE-ideals is not a GE-ideal by example. We suggest conditions under which any subset of a bordered GE-algebra can be a GE-ideal. We discuss GE-ideal that is generated from a subset in a bordered GE-algebra and investigate what element it is formed into. We construct the smallest GE-ideal which contains a given GE-ideal and an element in a bordered GE-algebra.

2. Preliminaries

Definition 1. (see [5]). A GE-algebra is a nonempty set with a constant 1 and a binary operation , satisfying the following axioms:(GE1)(GE2)(GE3) for all

In a GE-algebra , a binary relation “” is defined by

Definition 2. (see [5]). A GE-algebra is said to be(1)Transitive if it satisfies(2)Antisymmetric if the binary relation “” is antisymmetric.

Definition 3. (see [6]). If a GE-algebra has a special element, say , which satisfies for all , we call the bordered GE-algebra.

Definition 4. (see [6]). If a bordered GE-algebra satisfies condition (2), we say that is a transitive bordered GE-algebra.

Definition 5. (see [6]). A bordered GE-algebra is said to be antisymmetric if the binary operation “” is antisymmetric.

Proposition 1 (see [5]). Every GE-algebrasatisfies the following items:If is transitive, then

Lemma 1 (see [5]). In a GE-algebra, the following facts are equivalent to each other:

Definition 6. (see [5]). A subset of a GE-algebra is called a GE-filter of if it satisfies

Lemma 2 (see [5]). In a GE-algebra, every GE-filterofsatisfies

Proposition 2 (see [6]). In a bordered GE-algebra, the following assertions are valid:If is a transitive bordered GE-algebra, thenIf is an antisymmetric bordered GE-algebra, thenIf is a transitive and antisymmetric bordered GE-algebra, then

Definition 7. (see [6]). By a duplex bordered element in a bordered GE-algebra , we mean an element of which satisfies .

The set of all duplex bordered elements of a bordered GE-algebra is denoted by and is called the duplex bordered set of . It is clear that .

Definition 8. (see [6]). A bordered GE-algebra is said to be duplex if every element of is a duplex bordered element, that is, .

3. Strong GE-Filters

Given a subset of a bordered GE-algebra , we define the set

Definition 9. A GE-filter of a bordered GE-algebra is said to be strong if it satisfiesi.e., whenever .

It is clear that every GE-filter is strong in a duplex bordered GE-algebra.

Example 1. Let be a set with a binary operation given in the following table:Then, is a bordered GE-algebra, and it is easy to check that is a strong GE-filter of .

Example 2. Let be a set with a binary operation given in the following table:Then, is a bordered GE-algebra, and , , , , and are all strong GE-filters of .

Theorem 1. Ifandare strong GE-filters of a bordered GE-algebra, then so is.

Proof. Let and be strong GE-filters of a bordered GE-algebra . Then, is a GE-filter of (see [5]). Let . Then, , so and , that is, and . Thus, , which shows that . Therefore, is a strong GE-filter of .
The following example shows that the union of strong GE-filters may not be a GE-filter.

Example 3. Consider a bordered GE-algebra with a binary operation given in the following table:It is routine to verify that and are strong GE-filters of . However, is not a GE-filter of since and but , so it is not a strong GE-filter of .

Lemma 3. In a transitive and antisymmetric bordered GE-algebra, we have the next properties:where .

Proof. Let be a transitive and antisymmetric bordered GE-algebra and . Equation (30) is easily obtained. Since , it follows from (11) that and . Hence, by (4) and (10), so . Using (GE3) and (3), we obtainHence, . Similarly, we haveby (GE3), (3), and (30). Hence, . We know that which implies that by (11). Therefore, by (11) and (25). Hence, . Now, we have by (12) and (24), which implies from (22) and (25) that . Therefore, , which proves (33). We have (34) by the following calculation:Using the above facts, we obtainThis completes the proof.
For every subset of a bordered GE-algebra , consider the following:We find subsets of a bordered GE-algebra satisfying condition (40).

Example 4. Using the bordered GE-algebra in Example 2, we find subsets that satisfy (40) as follows: , , , , , , , , , , , , , , , , , for , and for .

We provide conditions for a subset of a bordered GE-algebra to be a strong GE-filter.

Theorem 2. Letbe a transitive and antisymmetric bordered GE-algebra. If a nonempty subsetofsatisfies (15), (26), and (40), thenis a strong GE-filter of.

Proof. Let be a nonempty subset of satisfying (15), (26), and (40). Since , there exists . As , we have by (15). Let be such that and . Then, and by (9) and (15), which inducesby (25) and (40). It follows from (24) and (25) that . Since by (2), we have by (22), and hence, by (15). Thus, by (26), and thus, is a GE-filter of . Therefore, is a strong GE-filter of .

Definition 10. Let be a GE-algebra. A GE-filter of is said to be irreducible if, for every GEfilters and of ,

Example 5. (1)Let be a set with a binary operation given as follows: Then, is a GE-algebra. Clearly, , , , , , , and are all GE-filters of , and is an irreducible GE-filter of .(2)Let be a set with a binary operation given as follows:Then, is a GE-algebra. We know that , , and are GE-filters of , and is an irreducible GE-filter of .

We provide conditions for a GE-filter to be irreducible.

Lemma 4. Letbe a GE-filter of a GE-algebra. For every, if there existssuch thatand, thenis irreducible.

Proof. Let and be GE-filters of such that . Assume that and . Then, there exist and which imply from the hypothesis that there exists such that and . Hence, and . It follows that , which is a contradiction. Thus, or , and therefore, is an irreducible GE-filter of .
Let be a GE-filter of a transitive GE-algebra , and let be a subset of such thatIf and are disjoint, then there exists an irreducible GE-filter of such that and . In fact, let denote the set of GE-filter such that and , i.e.,where is the set of all GE-filters of . Then, , so is nonempty. It is obvious that the union of a chain of elements of is also contained in , so has a maximal element, say , by Zorn’s lemma. For any , consider the GE-filter and . Then, and . Hence, , so there exists such that and . It follows from (45) that and for some . Hence, and by (11), so and . Therefore, is an irreducible GE-filter of by Lemma 4. Consequently, we have the following theorem.

Theorem 3. Letbe a GE-filter of a transitive GE-algebraand letbe a subset ofsatisfying (44) and (45). If, then there exists an irreducible GE-filterofsuch thatand.

Corollary 1. Letbe an irreducible GE-filter of a transitive GE-algebra. For every,if and only if there exists an irreducible GE-filtersuch that,, and.

Proposition 3. In a transitive bordered GE-algebra, every GE-filtersatisfies (40).

Proof. Suppose that there exists such that . Then, there exists an irreducible GE-filter such that by Theorem 3. It follows from Corollary 1 that there exists an irreducible GE-filter such that and . Since , we have , so . This is a contradiction, and therefore, , that is, satisfies (40).
The above mentioned induces the characterization of a strong GE-filter as follows.

Theorem 4. Letbe a transitive and antisymmetric bordered GE-algebra. A nonempty subsetofis a strong GE-filter ofif and only if it satisfies (15), (26), and (40).

4. GE-Ideals

Definition 11. Let be a bordered GE-algebra. A subset of is called a GE-ideal of if it satisfies

It is clear that is a GE-ideal of a bordered GE-algebra .

Example 6. Let be a set with a binary operation given as follows:Then, is a bordered GE-algebra, and it is a routine to verify that is a GE-ideal of .

Theorem 5. Ifandare GE-ideals of a bordered GE-algebra, then so is.

Proof. It is clear that . Let be such that and . Then, , , , and . It follows from (48) that and . Hence, . This completes the proof.

The following example shows that the union of GE-ideals may not be a GE-ideal.

Example 7. In Example 6, we can observe that and are GE-ideals of . However, is not a GE-ideal of since and , but .

Proposition 4. Letbe a bordered GE-algebra. Every GE-idealofsatisfies

Proof. Let and be such that . Then, , so . It follows from (48) that .
The following corollary is induced directly by (22) and Proposition 4.

Corollary 2. Every GE-idealof a transitive bordered GE-algebrasatisfies

Proposition 5. Every GE-idealof a transitive bordered GE-algebrasatisfies

Proof. Let be a GE-ideal of a transitive bordered GE-algebra and assume that and for all . Using (22) and (23), we have for all . Hence, by Corollary 2. Since , it follows from (48) that .
We suggest the conditions under which any subset of a bordered GE-algebra can be a GE-ideal of .

Theorem 6. In an antisymmetric and duplex bordered GE-algebra, every subsetofsatisfying (47) and (52) is a GE-ideal of.

Proof. Let and be such that . Since is antisymmetric and duplex, it follows from (24) that . Hence, by (52), and therefore, is a GE-ideal of .

Lemma 5 (see [6]). Given an antisymmetric bordered GE-algebra, the following are equivalent:(i) is duplex.(ii) satisfies(iii) satisfies(iv) satisfies

The combination of Theorem 6 and Lemma 5 induces the next corollary.

Corollary 3. If an antisymmetric bordered GE-algebrasatisfies either condition (53), (54), or (55), then every subsetofsatisfying (47) and (52) is a GE-ideal of.

Theorem 7. In a transitive and antisymmetric bordered GE-algebra, every subsetofsatisfying (47) and (52) is a GE-ideal of.

Proof. Let and be such that . Using (24), (25), and (35) inducesIt follows from (52) that . Therefore, is a GE-ideal of .

Definition 12. Let be a subset of a bordered GE-algebra . The GE-ideal generated by is defined to be the intersection of all GE-ideals containing , and it is denoted by .

Example 8. Let be a set with a binary operation given as follows:Then, is a bordered GE-algebra. If we take a subset of , then the GE-ideal of generated by is .

Let be a GE-algebra. For any natural number and , is recursively defined as follows:

Let be a transitive GE-algebra. By (6) and induction, it is easy to prove that

If we put for all in (59), thenProof. It is straightforward by (9).

Proposition 6. Every GE-algebrasatisfiesfor all .

Theorem 8. Letbe a subset of a bordered GE-algebra. Then,(i) and (ii) and (iii)For every subsets and of , if , then (iv)(v)If is a GE-ideal of , then

Proof. Straightforward.
In the next theorem, we discuss the description of the elements of .

Theorem 9. Letbe a nonempty subset of a transitive and antisymmetric bordered GE-algebra. Then,

Proof. Let . Since is nonempty, there exists and . Hence, . Let be such that and . Then, there are such thatUsing (25) and (35) induces . If follows from (11), (59), and (63) thatHence, by (7), soIf we multiply both sides of (66) by from the left, thenIf we repeat this process times, thenso . Hence, , and therefore, is a GE-ideal of . It is clear that . Let be a GE-ideal of containing . If , then for some . On the contrary, we haveby using (6), (9), and (11) repeatedly, which implies from (7) thatthat is, . The combination of this and (61) inducesIf we take and for in (71), thenSince and is a GE-ideal of , we obtainRepeating this process, we obtain . Since by (17), it follows from Corollary 2 that . Hence, , and therefore, . This completes the proof.
If we take for all in Theorem 9 and use (4), then we have the next corollary.

Corollary 4. For every elementin a transitive and antisymmetric bordered GE-algebra, we havethat is, .

Given a GE-ideal and an element in a bordered GE-algebra, we construct the smallest GE-ideal containing and .

Theorem 10. Letbe a GE-ideal of a transitive and antisymmetric bordered GE-algebraand letbe any element of. Then,

Proof. Let . Since , contains the element . For any , we have by (5) and (17). It follows from Proposition 4 that . Thus, , which shows that . Hence, . Since , we get . Let be such that and . Then,for some natural numbers and . Using (4), (25), (30), and (35), we haveso by (76). The combination of (4) and (77) induces . Since is a GE-ideal of , it follows that . Hence, , and thus, is a GE-ideal of . Now, we prove that is the smallest GE-ideal containing . Let be a GE-ideal of containing and . Then, and . Hence, since is GE-ideal of . Therefore, . Hence, .

5. Conclusions

We introduced the notion of strong GE-filters and GE-ideals (generated) and investigated the related properties. We have shown that the intersection of strong GE-filters (resp., GE-ideals) is a strong GE-filter (resp., GE-ideal), but the union of strong GE-filters (resp., GE-ideals) is not a strong GE-filter (resp., GE-ideal) by example. We provided conditions for a subset of a bordered GE-algebra to be a strong GE-filter. We considered a characterization of a strong GE-filter. In order to do so, we defined irreducible GE-filter first and examined its properties. We provided conditions for a GE-filter to be irreducible. Given a GE-filter and a subset in a bordered GE-algebra, we considered the existence of an irreducible GE-filter which contains the given GE-filter and is disjoint to the given subset. We suggested conditions under which any subset of a bordered GE-algebra can be a GE-ideal, discussed GE-ideal that is generated from a subset in a bordered GE-algebra, and investigated what element it is formed into. We constructed the smallest GE-ideal which contains a given GE-ideal and an element in a bordered GE-algebra. It can be seen that the study of cubic structures on BCK/BCI-algebras, etc., has been actively conducted recently (see [7, 8]). In the future, we will study the cubic structure based on the ideas and results of this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07049321).

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Copyright © 2021 Mehmet Ali Öztürk 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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