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
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, is a -algebra.
5. Conclusion
We have presented the notion of -algebras and provided some of its initial properties. We showed that, in a -algebra, a neighborhood base of the constant element generates a unique -topology, making a -algebra provided that the given -algebra is commutative. Lastly, we provided some properties of (normal) subalgebras of -algebras and investigated the notion of quotient -algebras. As a consequence of this study, it is best recommended to explore the properties of -algebras with respect to some known topological invariants (e.g., separability and compactness).
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The researcher gratefully acknowledges the support from the Research Unit of Bukidnon State University, Malaybalay City 8700, Philippines, last year 2018.