Abstract
The concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras are introduced, and their relations and properties are investigated. Many examples are given to support these concepts. A semigroup is formed using the set of interior GE-algebras. An example is given that the set of interior GE-algebras is not a GE-algebra. It is clear that if is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra is transitive (resp., commutative, belligerent, and left exchangeable), but examples are given to show that the converse is not true in general. An interior GE-algebra is constructed using a bordered interior GE-algebra with certain conditions, and an example is given to explain this.
1. Introduction
In 1966, Imai and Iséki introduced BCK-algebras (see [1]) as the algebraic semantics for a nonclassical 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 nonclassical 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 [5–8]).
The notion of the interior operator was introduced by Vorster [9] in an arbitrary category, and it was used in [10] to study the notions of connectedness and disconnectedness in topology. Interior algebras are a certain type of algebraic structures that encode the idea of the topological interior of a set and are a generalization of topological spaces defined by means of topological interior operators. Rachůnek and Svoboda [11] studied interior operators on bounded residuated lattices, and Svrcek [12] studied multiplicative interior operators on GMV-algebras.
In this article, we apply the interior operator theory to GE-algebras. We introduce the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigate their relations and properties. We find and present many examples to illustrate these concepts. We use the set of interior GE-algebras to make up a semigroup. We give examples to show that the set of interior GE-algebras is not a GE-algebra. It is clear that if is a transitive (resp., commutative, belligerent and left exchangeable) GE-algebra, then the interior GE-algebra is transitive (resp., commutative, belligerent and left exchangeable), but we give examples to show that its inverse is not established. We make up the internal GE-algebra using a bordered interior GE-algebra with certain conditions and give examples describing this.
2. Preliminaries
Definition 1. (see [5]). By a GE-algebra, we mean a nonempty set with constant 1 and 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, 6, 8]). 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 [5]). Every GE-algebra satisfies the following items:If is transitive, then
Lemma 1 (see [5]). In a GE-algebra , the following facts are equivalent to each other:
3. Interior GE-Algebras
Definition 3. By an interior GE-algebra, we mean a pair in which is a GE-algebra and is a mapping such that
Example 1. Consider a GE-algebra with the binary operation which is given in the following table:Then, it is routine to verify that is an interior GE-algebra, where
It is clear that if is a GE-algebra, then and are interior GE-algebras, where and .
In the following example, we know that there is a constant map , where , on a GE-algebra such that is not an interior GE-algebra.
Example 2. Consider a GE-algebra , where and is a binary operation on , which is given in the following table:If we take a constant mapping , then . Hence, is not an interior GE-algebra.
Let be the set of all interior GE-algebras. For every , we definewhere and .
Let . The following example shows that the composition of and may not be an interior GE-algebra, and .
Example 3. Let be a set with the binary operation given in the following table:Then, is a GE-algebra. Define two mappings:Then, and are interior GE-algebras, and the composition of and is calculated as follows:Since , the composition of and is not an interior GE-algebra. Also, since
We consider the following condition:for .
Denote by the set of all interior GE-algebras satisfying condition (21).
Theorem 1. If is a GE-algebra, then is a semigroup.
Proof. It is sufficient to show that is closed under . Let . Using (10), we have for all , and so, satisfies condition (10). Also,for all , which shows that satisfies condition (11). For every , if , then , and so, . This shows that is an interior GE-algebra, that is, is closed under . Therefore, is a semigroup.
The following example describes Theorem 1.
Example 4. (1)Consider a GE-algebra , where and is a binary operation on , which is given in the following Cayley table: The set of all interior GE-algebras is , where the self-maps and are given by Table 1. We can check by the following Cayley table: And is a semigroup.(2)Consider a GE-algebra with the binary operation which is given in the following table: The set of all interior GE-algebras consists of in which each , , is given in Table 2. The operation “” in is calculated as follows: in which , which is not contained in , is given as follows: We know that is not closed under the operation “,” and is a semigroup, where . Let . The following example shows that may not be an interior GE-algebra, and .
| 1 | |||||
| 1 | 1 | 1 |
Example 5. Consider in Example 4 (2). Then, the operation “” in is calculated as follows:in which , and they are calculated as follows:
We also know that ; for example, .
| 1 | |||||
| 1 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 1 | 1 | 1 |
Example 5 generally shows that cannot be a GE-algebra. However, the following example shows that becomes a GE-algebra sometimes.
Example 6. Consider in Example 4 (1). Then, the operation “” in is calculated as follows:It is routine to verify that is a GE-algebra.
For every , we define
For every , the setsare called the identity part and the kernel of , respectively.
Proposition 2. Let be an antisymmetric and transitive GE-algebra. For every , we have(i)(ii)
Proof. (i)If , then for all , and so, for all . Also, which implies that for all . Hence, for all , and therefore, . Conversely, assume that . Then, for all . Thus, .(ii)It is clear that if , then . Suppose that . Condition (11) induces , and so, for all . Hence, . Similarly, . Using (10) and (12), we have and . Thus, for all , and therefore, .
Lemma 2 (see [5]). Every GE-algebra satisfies (i) , (ii) , (iii) , (iv) , (v) , (vi) ,for all .
Proposition 3. If is an interior GE-algebra, then(i)(ii)(iii)(iv)(v)(vi)(vii)
Proof. (i) is straightforward, and (ii)–(vii) follow from (12) and Lemma 2.
Question 1. If is an interior GE-algebra, will the next items be established?
The following example shows that the answer to the above question is negative.
Example 7. (1)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is an interior GE-algebra, but does not satisfy (32) and (33) since (2) Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is an interior GE-algebra, but does not satisfy (34) since(3)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is an interior GE-algebra, but does not satisfy (35) since
Definition 4. An interior GE-algebra is said to be transitive (resp., commutative, belligerent, and left exchangeable) if it satisfies (32) (resp., (33), (34), and (35)).
Example 8. (1)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a transitive interior GE-algebra.(2)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a commutative interior GE-algebra.(3)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a belligerent interior GE-algebra.(4)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a left exchangeable interior GE-algebra.
It is clear that if is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra is transitive (resp., commutative, belligerent, and left exchangeable), but the converse is not true in general as seen in the following example.
Example 9. (1)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a transitive interior GE-algebra, but is not transitive GE-algebra since(2)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a commutative interior GE-algebra, but is not commutative GE-algebra since(3)Consider a GE-algebra with the binary operation given in the following Cayley table:Define by Then, is a belligerent interior GE-algebra, but is not belligerent GE-algebra since(4)Consider a GE-algebra with the binary operation given in the following Cayley table: Define by Then, is a left exchangeable interior GE-algebra, but is not left exchangeable GE-algebra since
The following example shows that any interior GE-algebra does not satisfy the following:
Example 10. Consider a GE-algebra with the binary operation given in the following Cayley table:Define byThen, is an interior GE-algebra, but does not satisfy (65) and (66) since
Proposition 4. Every transitive interior GE-algebra satisfies (65) and (66).
Proof. Let . Using (7) and (10) induces which proves (65). Since and , it follows from (7) that . Hence, (66) is valid.
Definition 5. (see [7]). If a GE-algebra has a special element, say 0, which satisfies for all , we call the bordered GE-algebra.
Definition 6. (see [7]). 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 7. (see [7]). A bordered GE-algebra is said to be duplex if every element of is a duplex bordered element, that is, .
Definition 8. By a bordered interior GE-algebra, we mean an interior GE-algebra in which is a bordered GE-algebra.
Example 11. Consider a bordered GE-algebra with the binary operation given in the following Cayley table:Define byIt is routine to verify that is a bordered interior GE-algebra.
Proposition 5. In a bordered interior GE-algebra in which is transitive, we have
Proof. If we take in (66), then . Taking , , and in (8) induces , and so, by (10) and (12).
Lemma 3 (see [7]). The duplex bordered set of a transitive and antisymmetric bordered GE-algebra is closed under the binary operation in , that is, it is a GE-subalgebra of and is also bordered.
Theorem 2. Let be a transitive and antisymmetric bordered interior GE-algebra. Then, is an interior GE-algebra, where
Proof. Let . Then, by (7), andLet be such that . Then, by (12), and thus, . This completes the proof.
The following example describes Theorem 2.
Example 12. Consider a GE-algebra with the binary operation given in the following Cayley table:Define byThen, is a transitive and antisymmetric bordered interior GE-algebra, and is a GE-algebra, where . We know that is calculated as follows:and it is routine to observe that is an interior GE-algebra.
4. Conclusions
We have introduced the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigated their relations and properties. We have found and presented many examples to illustrate these concepts. We have formed a semigroup using the set of interior GE-algebras. We have provided examples to show that the set of interior GE-algebras is not a GE-algebra. It is clear that if is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra is transitive (resp., commutative, belligerent, and left exchangeable), but we have considered examples to show that its inverse is not established. We have provided examples of how to construct and explain interior GE-algebra using a bordered interior GE-algebra under certain conditions. In the future work, we will use the idea and results given in this paper to study other (hyper) algebraic structures, for example, (hyper) hoop, (hyper) BCH-algebra, (hyper) equality algebra, and (hyper) MV-algebra.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Y.B.J. and J.-G.L. created and conceptualized ideas. R.K.B. and Y.B.J. found examples. Y.B.J. developed the methodology. K.H. and R.K.B. reviewed and edited the article. J.-G.L. contributed to funding acquisition. All authors read and agreed to the published version of the manuscript.
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).