International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 8683965, 7 pages

https://doi.org/10.1155/2019/8683965

## Analyzing Some Structural Properties of Topological -Algebras

Mathematics Department, Bukidnon State University, Malaybalay City 8700, Philippines

Correspondence should be addressed to Narciso C. Gonzaga Jr.; moc.liamg@5270htam.eizran

Received 14 February 2019; Accepted 16 June 2019; Published 15 July 2019

Academic Editor: Susana Montes

Copyright © 2019 Narciso C. Gonzaga Jr. 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.

#### Abstract

In this study, we investigate the topology on -algebras: an algebraic system of propositional logic. We define here the notion of topological -algebras (briefly, -algebras) and some properties are investigated. A characterization of -algebras based on neighborhoods is provided. We also provide a filterbase that generates a unique -topology, making a -algebra in which the filterbase is a neighborhood base of the constant element, provided that the given -algebra is commutative. Finally, we investigate subalgebras of -algebras and introduce the notion of quotient -algebras of the given -algebra.

#### 1. Introduction

J. Neggers and H. S. Kim [1] introduced the concept of -algebras in 2002. It is an algebraic system which is related to /-algebras but seems to have more profound properties without being excessively complicated. The authors in [2, 3] proved that there is a direct correspondence between -algebras and groups. Thus, certain properties and results on -algebras were established via the properties of groups. Readers may refer to [4–8] for some of these “group-alike” properties and results on -algebras.

The theory of topological groups is one of the already well-defined concepts in both algebra and topology. Thus, in connection to the researches previously mentioned, this study initiates the notion of topological -algebras. Some fundamental properties of a topological -algebra that anchors on the basic topological and algebraic concepts will be investigated. This will provide the foundation of future investigations regarding the overall structure of a topological -algebra.

#### 2. Preliminaries

In this section, we recall some elementary concepts in -algebras that are necessary in this paper.

A -algebra [1] is defined as a triple where is a nonempty set with “” being its constant element and “” being a binary operation that satisfies the following conditions for all :

,

,

.

In addition, is said to be a* commutative *-*algebra* if the following condition holds for all :

.

Throughout the remainder of this study, shall conveniently denote the -algebra unless otherwise specified.

Let be a -algebra. A nonempty subset is said to be a* subalgebra* of [9] if for all . In addition, is said to be a* normal subset* of [10] if whenever . Accordingly, every normal subset of is a subalgebra of . Thus, it is more precise to say that is a* normal subalgebra* of if the condition of normality is satisfied.

Some basic notations and properties of (commutative) -algebras are the following.

*Definition 1 (see [7]). *Let be a -algebra. For each , denotes the expression .

Lemma 2 (see [1]). *Let be a -algebra. Then, for all ,*(i)*;*(ii)*;*(iii)* if and only if .*

*Lemma 3 (see [11]). If is a -algebra, then for all .*

*Lemma 4 (see [10]). Let be a -algebra. Then for all .*

*Lemma 5 (see [1]). Let be a commutative -algebra. Then, for every , .*

*Theorem 6 (see [12]). Let be a -algebra. Then is commutative if and only if for all .*

*On the other hand, the following notions appear in [6]. Let be a -algebra and a normal subalgebra of . For , the left coset and the right coset of in by are, respectively, the setsDenote the set of all distinct left cosets of in by and define a binary operation “” on as follows: for all , . Then is a -algebra and is called the quotient -algebra of by . For simplicity of this study, we shall denote the quotient -algebra by only, unless otherwise specified.*

*We finish this section by providing some new classes of sets in -algebras and their properties that will be needed in this study.*

*Definition 7. *Let and be nonempty subsets of a -algebra :(i)For each , the sets and are called the* left* and* right translates* of by , respectively.(ii)The* product* of and , denoted by , is given byWe also recall the following notation which has appeared first in [6].

*Definition 8. *Let be a nonempty subset of a -algebra . The* inverse* of , denoted by , is the set .

*Note that if is a subalgebra of a -algebra , then, as explained earlier, the sets and are left and right cosets of in , respectively.*

*Some obvious results on the notations presented above are the following.*

*Remark 9. *Let and be nonempty subsets of a -algebra :(i) and .(ii)If , then .(iii) if and only if and .

* The following result, which appears in [6], will be considered in this study.*

*Corollary 10. Let be a subalgebra of a -algebra . Then is a normal subalgebra of if and only if for all .*

*Proposition 11. Let be a nonempty subset of a -algebra . Then the following statements hold:(i).(ii)For all , .(iii)If is a subalgebra of , then .(iv)If is a subalgebra of , then , for all and for all .*

*Proof. *Let be a nonempty subset of and :(i)It follows from Lemma 3.(ii)It follows from Lemma 2.(iii)It follows from Remark 9 and from the fact that is a subalgebra of .(iv)Suppose is a subalgebra of . Let and . Then for , there is an such that . By , , with . Hence, and so . Further, if , then there is an such that . By Lemma 3, Definition 1,and Lemma 4, , , and , we have Therefore, . Consequently, .

* Meanwhile, we consider the following properties that are present in a commutative -algebra.*

*Proposition 12. Let and be nonempty subsets of a commutative -algebra . Then the following hold:(i)For all , .(ii)If , for all , then .*

*Proof. *(i) It suffices to show that for all . Now, by , Lemma 5, , Lemma 4, and Theorem 6,(ii) Let . Then, for each , , and so, by the hypothesis, there is a such that . Observe that by , , and ,By Theorem 6, Lemma 2, Lemma 3, , , and Definition 1,Therefore, .

*3. Initial Properties of Topological -Algebras*

*3. Initial Properties of Topological -Algebras**This section presents the definition of a topological -algebra as well as some of its fundamental properties. Also, some characterizations of (commutative) topological -algebras in terms of neighborhood and filters are investigated.*

*Definition 13. *Let be a -algebra. A topology furnished on is called a -*topology* on . In addition, is called a* topological *-*algebra* (or -*algebra*) if is a -topology on and the binary operation is continuous, where the Cartesian product topology on is furnished by .

*Example 14. *Consider the group of real numbers under the usual addition operation “+”. Then it is well known that together with “+” and the Euclidean topology is a topological group. Let “” be a binary operation on defined by for all . Then it is easy to show that is a -algebra. Now, one of the properties of the topological group is that the map is continuous. This implies that is a continuous map. Therefore, is also a -algebra.

*Example 15. *Consider the -algebra with the binary operation “” defined on the Cayley table provided in Table 1 (see [13]). Let . Then is a -topology on . By routine calculations, we see that is a -algebra.

In view of Definition 7, the following remark is straightforward.