International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 8683965 | https://doi.org/10.1155/2019/8683965

Narciso C. Gonzaga, "Analyzing Some Structural Properties of Topological -Algebras", International Journal of Mathematics and Mathematical Sciences, vol. 2019, Article ID 8683965, 7 pages, 2019. https://doi.org/10.1155/2019/8683965

Analyzing Some Structural Properties of Topological -Algebras

Academic Editor: Susana Montes
Received14 Feb 2019
Accepted16 Jun 2019
Published15 Jul 2019

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 [48] 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 haveTherefore, . 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

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.




Remark 16. Let be a -algebra. Then for any nonempty subsets and of .

On the other hand, the following result provides a necessary and sufficient condition for a -algebra.

Theorem 17. Let be a -algebra and a -topology on the set . Then is a -algebra if and only if, for all and for every nbd of , there are nbds and of and , respectively, such that .

Proof. Suppose that “” is a continuous map. Let and a nbd of . Then is a nbd of in . By the definition of Cartesian product topology, there exist nbds and of and , respectively, such that . By Remark 16, . Hence, there are nbds and of and , respectively, such that .
Conversely, let and a nbd of . Now, by the assumption, there exist nbds and of and , respectively, such that . By the definition of Cartesian product topology, it follows that is a nbd of . Now, by Remark 16, and so there is a nbd of such that . Therefore, “” is a continuous function, completing the proof of the theorem.

Theorem 18. Let be a -algebra and a fixed element of the -algebra . Then the following functions are homeomorphisms:(i) defined by .(ii) defined by .(iii) defined by .

Proof. Let be a fixed element of . Consider the function defined by . Let and a nbd of . Then, by the definition of Cartesian product topology, there are -open sets and in such that . Thus, there is a nbd of such that . This shows that is continuous. Similarly, the function defined by is continuous. Now, observe that and , with “” being a continuous map by Definition 13. Thus, both and are compositions of two continuous functions. Hence, and are continuous.
Next, consider defined by . By the previous argument, is also continuous. By Definition 1 and Lemma 2, we haveAlso, by Definition 1, , Lemma 3, , and , it follows thatHence, we have and . Consequently, is a homeomorphism, proving .
On the other hand, for and a nbd of , there are nbds and of and , respectively, such that . Observe that, by Definition 8 and Remark 9, . Thus, there is a nbd of such that . Therefore, is continuous. Next, by Lemma 3, we have , and so . Hence, is a homeomorphism, proving .
Lastly, by Lemma 4, for all . Thus, ; that is, is a composition of two homeomorphisms. Consequently, is a homeomorphism, proving .

Corollary 19. Let be a -algebra and . Then the following statements are equivalent:(i) is -open in .(ii) is -open in .(iii) is -open in for all .(iv) is -open in for all .

Meanwhile, the following corollary is immediate from the fact that, in a homeomorphism of topological spaces, the closure of the image of a subset of the domain is equal to the image of the closure of the said subset.

Corollary 20. Let be a -algebra. Then the following statements hold:(i) for all .(ii) for all and for all .(iii) for all and for all .

We shall now use Theorem 18 in proving some results pertaining to neighborhoods in -algebras. Let be a -algebra and . Denote the neighborhood filter of by . Also, from Remark 9, we have , where .

Theorem 21. Let be a neighborhood base of the constant element of a -algebra . Then the following conditions hold:(i)For every , there is a such that .(ii)For every , there is a such that .(iii)For every and , there is a such that .

Proof. Let be an arbitrary element of .(i)We have . By the continuity of “” and by Theorem 17, there are members and of such that . Thus, there exists such that . Since is a neighborhood base of 0, there is a such that . Therefore, by Remark 9 and Remark 16,(ii)Similar to , but use the continuity of in Theorem 18.(iii)Fix and let be defined by . Then , and so, in view of Theorem 18, is continuous. Now, by and . This means that if , then . Meanwhile, implies . Thus, we can choose such that . Therefore, we have .

On the other hand, we shall construct a -topology on a -algebra which is generated by a neighborhood base of the constant element, making the space a -algebra provided that the -algebra is commutative.

Theorem 22. Let be a -algebra and a filterbase of which satisfies the statements (i), (ii), and (iii) in Theorem 21. Then there is a unique -topology such that is a neighborhood base of the constant element of in the topology . Moreover, if is commutative, then is a -algebra.

Proof. Firstly, we claim that every member of contains the constant element . Let . Then, by the hypothesis, there exists such that . Let . Then, by , . This proves the claim.
Next, letWe shall show that is a -topology furnished from . Clearly, , and so is satisfied. Let be a subclass of and set . Let . Then for some . Thus, there is a such that . Hence, . Further, let be a finite subclass of and . Let . Then for all . Thus, for each , there is a such that . Now, a filterbase means that there is a such that for all . Therefore, there is a such that . This shows that . Consequently, is a -topology on .
We will now show that is a neighborhood base of . We claim first that every member of contains a member of . Let . Then we have with . From the definition of , Remark 9, and by the fact that , there is a such that . This proves the claim. Next, we claim that . It suffices to show that every member of is a nbd of 0. Let . Then by the previous argument, . Now, setThen since . By construction, . Let . Then there is a such that . Also, there is a such that . Let . Then for some . Thus by and by the fact that , . Hence, by Remark 9, . This means that , and so there is a such that . We have shown that , and so is -open. This proves the second claim; hence, is a neighborhood base of . Consequently, since a neighborhood base generates one and only one topology on , is unique.
Lastly, suppose is commutative. We will then prove the continuity of  “”. First, we claim that if and , then . Let . Then for some . By and Lemma 2, , and so there is a such that by the definition of . By Theorem 6, Proposition 11, Lemma 3, and Remark 9, we haveThis proves the claim. Next, let and a nbd of . Then we have for some . Thus, by the definition of and Lemma 3, for some . Also, implies for some . So, we have . Further, implies there is a such that for all . But since is commutative, it follows from Proposition 12 that . Also, implies there is a such that . Now, since, by Lemma 3 and Definition 1, we have and . Similarly, . Since , it follows that and are nbds of and , respectively. By Proposition 12, Lemma 5, and Remark 9Therefore, by Theorem 17, is a -algebra.

4. Subalgebras and Quotients of -Algebras

In this section, we provide some properties of -algebras in terms of its (normal) subalgebras. Also, we consider the construction of quotient -algebras determined by a normal subalgebra of a -algebra.

Proposition 23. Let be a -algebra. Then is Hausdorff if and only if the trivial subalgebra is a closed subset of .

Proof. Suppose is Hausdorff. Let such that and the subclass of containing all the open neighborhoods of . Also, setThenis an open subset of which does not contain . Hence, is closed in .
Conversely, assume that is a closed subset of . Observe that the binary operation “” is continuous. Thus, is closed in . But by and Lemma 2. Therefore, is Hausdorff.

Proposition 24. Let be a -algebra and a subalgebra of .(i)The closure of is a subalgebra of .(ii)If is a normal subalgebra of , then so does .(iii)If is Hausdorff and is commutative, then is commutative.

Proof. Let be a subalgebra of .(i)We will first prove that, for every nonempty subsets and of , . Indeed, since by Remark 16 and by the continuity of “”, . Now, suppose . Observe that by Proposition 11, . Hence, we see that . Therefore, is a subalgebra of .(ii)Suppose is normal in . Then, by Corollary 10, for all . By Corollary 20, for all . Hence, by Corollary 10, is a normal subalgebra of .(iii)Consider the function defined by . Then is continuous since “” is a continuous map. Now, being a Hausdorff space implies is closed in by Proposition 23. Hence, is closed in . Since is a commutative -algebra, it follows that for all by Theorem 6 and . Thus, ; whence . Note that . Therefore, , and so . This means that for all . Now, by Lemma 2 and Lemma 4,Thus, by Lemma 2, for all . Consequently, by Theorem 6, is commutative.

We shall now consider the construction of quotient -algebras by a normal subalgebra of the given -algebra. Recall first that a function where and are -algebras is said to be a -homomorphism [9] if for all , with “” and “” being the binary operations on and , respectively. In addition, is a -epimorphism if is surjective.

Let be a -algebra and a subalgebra of . Denote as the collection of all left cosets of by . Such members of are distinct, since these are the equivalence classes of an equivalence relation on ([6], Theorem 3). Clearly, the mapping is surjective. If is normal in , then coincides with the quotient -algebra . Moreover, is a -epimorphism [9] and is called the canonical projection of -algebras.

Proposition 25. Let be a -algebra, a subalgebra of , and a nonempty subset of . Then .

Proof. Suppose is a nonempty subset of . Let . ThenThis means that there is an such that . Thus, there are such that . Since is a subalgebra of , it follows that and for every . Now, by Lemma 2 and ,This shows that . Next, suppose . Then there exist and such that . By Proposition 11, we have . Hence, by Definition 7,Therefore, . Consequently, .

Theorem 26. Let be a -algebra and a subalgebra of . For the class , define byThen is a topology on .

Proof. It is straightforward.

Theorem 27. Let be a -algebra and a subalgebra of . Then, with respect to the topological space , the surjective map is continuous and open.

Proof. The continuity of follows directly from the definition of . It remains to show that is an open map. From the definition of , it suffices to show that whenever . So let be -open in . By Proposition 25 and Definition 7,By Corollary 19, for each , is -open in , and so is a union of some -open sets in . Therefore, .

Finally, we introduce the notion of quotient -algebras of -algebras which is stipulated in the result below.

Theorem 28. Let be a -algebra. If is a normal subalgebra of , thenis a -algebra.

Proof. Let and such that . Now, a -epimorphism implies there exist such that and . Moreover, , and so . By Theorem 26, . Since is a -algebra, there are such that , , and by virtue of Theorem 17. Also, we have since is a -homomorphism. Note that and . Now,Since is an open map by Theorem 27, . Hence, there are -open sets and in such that , , and . Therefore, by Theorem 17, <