Imploring GE-Filters of GE-Algebras

Song, Seok-Zun; Bandaru, Ravikumar; Jun, Young Bae

doi:https://doi.org/10.1155/2021/6651531

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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 6651531 | https://doi.org/10.1155/2021/6651531

Imploring GE-Filters of GE-Algebras

Seok-Zun Song,¹Ravikumar Bandaru,²and Young Bae Jun³

Academic Editor: Efthymios G. Tsionas

Received22 Nov 2020

Revised06 Jan 2021

Accepted11 Feb 2021

Published23 Feb 2021

Abstract

Relations between a transitive GE-algebra, a belligerent GE-algebra, an antisymmetric GE-algebra, and a left exchangeable GE-algebra are displayed. A new substructure, so called imploring GE-filter, is introduced, and its properties are investigated. The relationship between a GE-filter, an imploring GE-filter, a belligerent GE-filter, and a prominent GE-filter are considered. Conditions for an imploring GE-filter to be a belligerent GE-filter are given, and the conditions necessary for a (belligerent) GE-filter to be an imploring GE-filter are found. Relations between a prominent GE-filter and an imploring GE-filter are discussed, and a condition for an imploring GE-filter to be a prominent GE-filter is provided. Examples to show that a belligerent GE-filter and a prominent GE-filer are independent concepts are given. The extension property of the imploring GE-filter is established.

1. Introduction

Hilbert algebras were introduced by Henkin and Skolem in the middle of the last century. Since then, several scholars have participated in the study of Hilbert algebras, for example, the reader can refer to [1–12]. It is well known that these algebras are the algebraic counterpart of the fragment of the intuitionistic propositional calculus. The study of generalization of one known algebraic structure is also an important research task. Bandaru et al. introduced GE-algebra as a generalization of Hilbert algebra and studied its properties (see [13]). As a follow-up study to [13], Bandaru et al. introduced two kinds of GE-filters called voluntary GE-filter and belligerent GE-filters in GE-algebras and investigated related properties (see [14, 15]). Rezaei et al. [16] introduced prominent GE-filters in GE-algebras and studied its properties. As a continuing follow-up to the papers [13–16], we introduce a new substructure called imploring GE-filter and study its properties. We first discuss relations between a transitive GE-algebra, a belligerent GE-algebra, an antisymmetric GE-algebra, and a left exchangeable GE-algebra. We establish the relationship between a GE-filter, an imploring GE-filter, a belligerent GE-filter, and a prominent GE-filter. We discuss conditions for an imploring GE-filter to be a belligerent GE-filter and look at the conditions necessary for a (belligerent) GE-filter to be an imploring GE-filter. We consider relations between a prominent GE-filter and an imploring GE-filter and provide a condition for an imploring GE-filter to be a prominent GE-filter. We give examples to show that a belligerent GE-filter and a prominent GE-filer are independent concepts. We finally establish the extension property of the imploring GE-filter.

2. Preliminaries

Definition 1. (see [13]). A GE-algebra is a nonempty set with a constant 1 and a binary operation , satisfying the following axioms:(i)(GE1) ,(ii)(GE2) ,(iii)(GE3) , for all .In a GE-algebra , a binary relation “” is defined by

Definition 2. (see [13–15]). A GE-algebra is said to be(i)Transitive if it satisfies(ii)Commutative if it satisfies(iii)Left exchangeable if it satisfies(iv)Belligerent if it satisfies(v)Antisymmetric if the binary relation “” is antisymmetric.

Proposition 1. (see [13]). Every GE-algebra satisfies the following items:If is transitive, then

Lemma 1. (see [13]). In a GE-algebra , the following facts are equivalent to each other:

Definition 3. (see [13]). A subset of a GE-algebra is called a GE-filter of if it satisfies

Lemma 2. (see [13]). In a GE-algebra , every GE-filter of satisfies

Definition 4. (see [14]). A subset of a GE-algebra is called a belligerent GE-filter of if it satisfies (17) and

Lemma 3. (see [14]). If a GE-filter of a GE-algebra satisfiesthen is a belligerent GE-filter of .

Definition 5. (see [16]). A subset of a GE-algebra is called a prominent GE-filter of if it satisfies (17) and

Lemma 4. (see [16]). Let be a GE-filter of a GE-algebra . Then, is a prominent GE-filter of if and only if it satisfies

3. Relationship between GE-Algebras in Different Forms

We discuss relations between a transitive GE-algebra, a belligerent GE-algebra, an antisymmetric GE-algebra, and a left exchangeable GE-algebra.

Every belligerent GE-algebra is a transitive GE-algebra, but the converse is not true (see [16]).

Theorem 1. Every antisymmetric GE-algebra is a left exchangeable GE-algebra.

Proof. It is straightforward by (9) and the antisymmetricity of .
The converse of Theorem 1 may not be true as seen in the following example.

Example 1. Let be a set with a binary operation “” given in Table 1.
Then, is a left exchangeable GE-algebra which is not antisymmetric, since and , but .
In the following example, we know that any transitive GE-algebra is neither a belligerent GE-algebra nor a left exchangeable GE-algebra.

Example 2. Let be a set with a binary operation given in Table 2.
Clearly is a transitive GE-algebra. But it is not a left exchangeable GE-algebra sinceAlso, is not a belligerent GE-algebra sinceThe following example shows that any antisymmetric GE-algebra may not be a belligerent GE-algebra.

Example 3. Let be a set with a binary operation given in Table 3.
Then, is an antisymmetric GE-algebra which is not belligerent sinceWe look at the necessary conditions for a transitive GE-algebra or an antisymmetric GE-algebra to change to a belligerent GE-algebra.

Lemma 5. Every transitive GE-algebra satisfies

Proof. For every , we haveBy (2), (7), (9), and (13), it follows from (10) that , i.e., .

Corollary 1. Every belligerent GE-algebra satisfies (27).

Theorem 2. Every antisymmetric and transitive GE-algebra is a belligerent GE-algebra.

Proof. Let be an antisymmetric and transitive GE-algebra. Then, is left exchangeable GE-algebra by Theorem 1 and for all by Lemma 5. Using (GE2), (4), (8), and Lemma 1, we havethat is, . Since is antisymmetric, it follows that for all . Therefore, is a belligerent GE-algebra.

4. Imploring GE-Filters

Definition 6. A subset of a GE-algebra is called an imploring GE-filter of if it satisfies (17) and

Example 4. Let be a set with a binary operation given in Table 4.
Then, is a GE-algebra which is not transitive, not antisymmetric, not left exchangeable, not commutative, and not belligerent. It is easy to check that is an imploring GE-filter of .
Given a subset of a GE-algebra , consider the following assertion:In general, any GE-filter of a GE-algebra does not satisfy condition (31) as seen in the following example.

Example 5. In Example 4, the set is a GE-filter of which does not satisfy condition (31) since , but .
The following example shows that any imploring GE-filter of a GE-algebra may not satisfy condition (31).

Example 6. Let be a set with a binary operation given in Table 5.
Then, is a GE-algebra which is not transitive, not left exchangeable, and not belligerent. It is easy to verify that is an imploring GE-filter of . But it does not satisfy (31) since and .

Proposition 2. Every imploring GE-filter of a transitive GE-algebra satisfies condition (31).

Proof. Let be a transitive GE-algebra. Suppose is an imploring GE-filter of . Then, is a GE-filter of . Let be such that . Combining (11) and (13) inducesIt follows from (9), (13), and (16) thatHence, and so .

Corollary 2. Every imploring GE-filter of a belligerent GE-algebra satisfies condition (31).
We establish the relationship between a GE-filter, an imploring GE-filter, a belligerent GE-filter, and a prominent GE-filter.

Theorem 3. Every imploring GE-filter is a GE-filter.

Proof. Let be an imploring GE-filter of a GE-algebra . Let be such that and . If we put in (30) and use (GE1) and (GE2), then and so by (30). Therefore, is a GE-filter of .
In the following example, we know that the converse of Theorem 3 is not true.

Example 7. If we consider the GE-algebra in Example 4, then the set is a GE-filter of . But it is not an imploring GE-filter of since and , but .
We list the conditions necessary for the converse of Theorem 3 to be established.

Theorem 4. Let be a GE-filter of a GE-algebra . Then, is an imploring GE-filter of if and only if the following implication is valid:

Proof. Assume that is an imploring GE-filter of . Then, (34) is induced by taking in (30).
Conversely, let be a GE-filter of which satisfies (34). Let be such that and . Then, by (18). It follows from (34) that . Therefore, is an imploring GE-filter of .
The following example shows that a belligerent GE-filter and an imploring GE-filter are independent concepts in a GE-algebra, i.e., any belligerent GE-filter is not an imploring GE-filter, and any imploring GE-filter is not a belligerent GE-filter.

Example 8. (i)In Example 6, the imploring GE-filter is not a belligerent GE-filter of since and , but .(ii)Let be a set with a binary operation given in Table 6.Then, is a GE-algebra which is not transitive, not left exchangeable, and not belligerent. The set is a belligerent GE-filter of . But, it is not an imploring GE-filter of since and , but .
We discuss conditions for an imploring GE-filter to be a belligerent GE-filter.
Note that in GE-algebras, every belligerent GE-filter is a GE-filter, but not converse (see [14]).

Lemma 6. In a transitive GE-algebra, every GE-filter is a belligerent GE-filter.

Proof. Let be a GE-filter of a transitive GE-algebra . Let be such that and . Using Lemma 2 and Lemma 5 induces which implies from (18) that . Therefore, is a belligerent GE-filter of .

Corollary 3. In a belligerent GE-algebra, every GE-filter is a belligerent GE-filter.
The following corollaries are direct results of Theorem 3, Lemma 6, and Corollary 3.

Corollary 4. In a transitive GE-algebra, every imploring GE-filter is a belligerent GE-filter.

Corollary 5. In a belligerent GE-algebra, every imploring GE-filter is a belligerent GE-filter.
The converse of Corollary 4 may not be true as seen in the following example.

Example 9. Consider a transitive GE-algebra where and the binary operation is given by Table 7.
Then, is a belligerent GE-filter of but not an imploring GE-filter of since and , but .
The following example shows that the converse of Corollary 5 may not be true.

Example 10. Let be a set with a binary operation given in Table 8.
Then, is a belligerent GE-algebra and is a belligerent GE-filter of . But is not an imploring GE-filter of since and , but .
We now consider relations between a prominent GE-filter and an imploring GE-filter.

Theorem 5. In a GE-algebra, every prominent GE-filter is an imploring GE-filter.

Proof. Suppose is a prominent GE-filter of . Then, is a GE-filter of and . Assume that and . Then, , which implies from (GE1), (GE2), (7), and Lemma 4 thatTherefore, is an imploring GE-filter of .

Corollary 6. In a transitive GE-algebra, every prominent GE-filter is a belligerent GE-filter.
The following example shows that any imploring GE-filter is not a prominent GE-filter in general.

Example 11. Consider a GE-algebra where and the binary operation is given by Table 9.
Then, is an imploring GE-filter of but not a prominent GE-filter of since , but .
We provide a condition for an imploring GE-filter to be a prominent GE-filter.

Theorem 6. In a transitive GE-algebra, every imploring GE-filter is a prominent GE-filter.

Proof. Let be an imploring GE-filter of a transitive GE-algebra . Then, is a GE-filter of . Let be such that . Since by (8), we have by (13). If we put , then by (8). It follows from (15), (9), and (13) thatUsing Lemma 2, we get and so by Theorem 4. Therefore, is a prominent GE-filter of by Lemma 4.

Corollary 7. In a belligerent GE-algebra, every imploring GE-filter is a prominent GE-filter.
We reveal that a belligerent GE-filter and a prominent GE-filer are independent concepts in next examples. In other words, any belligerent GE-filter is not a prominent GE-filer and any prominent GE-filer is not a belligerent GE-filter.

Example 12. (i)Consider a GE-algebra where and the binary operation is given by Table 10. Then, is a belligerent GE-filter of . But, it is not a prominent GE-filter of since , but .(ii)Let be a set with the binary operation given by Table 11.Then, is a GE-algebra which is not transitive, not left exchangeable, and not belligerent. The set is a prominent GE-filter of . But it is not a belligerent GE-filter of since and , but .
The following example shows that in a transitive GE-algebra, any GE-filter may not be an imploring GE-filter.

Example 13. Let and the binary operation is given by Table 12.
Then, is a transitive GE-algebra. Clearly, is a GE-filter of . But, is not an imploring GE-filter of since and , but .
As conditions are strengthened, a GE-filter can be an imploring GE-filter in a transitive GE-algebra.

Theorem 7. Let be a GE-filter of a transitive GE-algebra , satisfying (31). Then, is an imploring GE-filter of .

Proof. Suppose is a GE-filter of a transitive GE-algebra , satisfying (31). Let be such that . Since by (12), it follows from (13) that . Hence, by (7) and Lemma 2, and so by (31). Since by (8), we get by (13) which implies from Lemma 2 that . Hence, , and therefore, is an imploring GE-filter of by Theorem 4.

Corollary 8. Let be a GE-filter of a belligerent GE-algebra , satisfying (31). Then, is an imploring GE-filter of .
The following example shows that in a transitive GE-algebra, a belligerent GE-filter is neither an imploring GE-filter nor a prominent GE-filter, and a GE-filter is not a prominent GE-filter.

Example 14. In Example 13, the set is a belligerent GE-filter and so a GE-filter of . But, it is not an imploring GE-filter of since and , but . Also, since , but , is not a prominent GE-filter of . Since and , we know that is not a prominent GE-filter of .
We finally establish the extension property of the imploring GE-filter.

Lemma 7. In a transitive GE-algebra , every GE-filter satisfies

Proof. Assume that for all . Using (7), (9), (13), and (15), we haveIt follows from Lemma 2 that .

Theorem 8. If is an imploring GE-filter of a transitive GE-algebra , then every GE-filter containing is an imploring GE-filter of .

Proof. Let be an imploring GE-filter of a transitive GE-algebra and let be a GE-filter of such that . Assume that for all . Then, , which implies from Lemma 7 that . Since by (9), we get by (13). Hence, by Lemma 2. It follows from Proposition 2 that . Note thatHence, , and so . Therefore, is an imploring GE-filter of by Theorem 7.

Corollary 9. If is an imploring GE-filter of a belligerent GE-algebra , then every GE-filter containing is an imploring GE-filter of .

5. Conclusions

As a generalization of Hilbert algebras, Bandaru et al. introduced the notion of GE-algebras. After that, Bandaru, Rezaei, Borumand Saeid, and Jun studied belligerent GE-filter, prominent GE-filter, and voluntary GE-filter as a study on the substructure of GE-algebras. As a follow-up to these studies, we studied a new substructure called imploring GE-filter. First of all, we discussed relations between a transitive GE-algebra, a belligerent GE-algebra, an antisymmetric GE-algebra, and a left exchangeable GE-algebra. We established the relationship between GE-filter, imploring GE-filter, belligerent GE-filter, and prominent GE-filter. We considered conditions for the imploring GE-filter to be the belligerent GE-filter and looked at the conditions necessary for the (belligerent) GE-filter to be the imploring GE-filter. We discussed relations between the prominent GE-filter and imploring GE-filter and gave a condition for the imploring GE-filter to be the prominent GE-filter. We cited examples to show that a belligerent GE-filter and a prominent GE-filer are independent concepts. We established the extended properties of the imploring GE-filter. The purpose of our research in the future is to continue to think about these things and define and study new substructures in this algebraic structure.

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 (no. 2016R1D1A1B02006812) and Basic Science Research Program to Research Institute for Basic Sciences (RIBS) of Jeju National University through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2019R1A6A1A10072987).

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Copyright © 2021 Seok-Zun Song 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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