Prominent interior GE-filters of GE-algebras

Abstract: The concept of a prominent interior GE-filter (of type 1 and type 2) is introduced, and their properties are investigated. The relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter are discussed. Examples to show that any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter are provided. Conditions for an interior GE-filter to be a prominent interior GE-filter are given. Also, conditions under which an internal GEfilter larger than a given internal GE filter can become a prominent internal GE-filter are considered, and an example describing it is given. The relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1 is discussed.


Introduction
Henkin and Skolem introduced Hilbert algebras in the fifties for investigations in intuitionistic and other non-classical logics. Diego [4] proved that Hilbert algebras form a variety which is locally finite. Bandaru et al. introduced the notion of GE-algebras which is a generalization of Hilbert algebras, and investigated several properties (see [1,2,[7][8][9]). The notion of interior operator is introduced by Vorster [12] in an arbitrary category, and it is used in [3] to study the notions of connectedness and disconnectedness in topology. Interior algebras are a certain type of algebraic structure that encodes 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 [6] studied interior operators on bounded residuated lattices, and Svrcek [11] studied multiplicative interior operators on GMV-algebras. Lee et al. [5] applied the interior operator theory to GE-algebras, and they introduced the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigated their relations and properties. Later, Song et al. [10] introduced the notions of an interior GE-filter, a weak interior GE-filter and a belligerent interior GE-filter, and investigate their relations and properties. They provided relations between a belligerent interior GE-filter and an interior GE-filter and conditions for an interior GE-filter to be a belligerent interior GE-filter is considered. Given a subset and an element, they established an interior GE-filter, and they considered conditions for a subset to be a belligerent interior GE-filter. They studied the extensibility of the belligerent interior GE-filter and established relationships between weak interior GE-filter and belligerent interior GE-filter of type 1, type 2 and type 3. Rezaei et al. [7] studied prominent GE-filters in GE-algebras. The purpose of this paper is to study by applying interior operator theory to prominent GE-filters in GE-algebras. We introduce the concept of a prominent interior GE-filter, and investigate their properties. We discuss the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We find and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We provide conditions for an interior GE-filter to be a prominent interior GE-filter. We provide conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and give an example describing it. We also introduce the concept of a prominent interior GE-filter of type 1 and type 2, and investigate their properties. We discuss the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We give examples to show that A and B are independent of each other, where A and B are: (1) A: prominent interior GE-filter of type 1. B: prominent interior GE-filter of type 2.
[1] By a GE-algebra we mean a non-empty set X with a constant 1 and a binary operation * satisfying the following axioms: In a GE-algebra X, a binary relation "≤" is defined by (∀x, y ∈ X) (x ≤ y ⇔ x * y = 1) . (2.1) Every GE-algebra X satisfies the following items: In a GE-algebra X, the following facts are equivalent each other.
[1] A subset F of a GE-algebra X is called a GE-filter of X if it satisfies: In a GE-algebra X, every filter F of X satisfies: Definition 2.7. [7] A subset F of a GE-algebra X is called a prominent GE-filter of X if it satisfies (2.15) and (∀x, y, z ∈ X)(x * (y * z) ∈ F, x ∈ F ⇒ ((z * y) * y) * z ∈ F). (2.18) Note that every prominent GE-filter is a GE-filter in a GE-algebra (see [7]).
Definition 2.8. [5] By an interior GE-algebra we mean a pair (X, f ) in which X is a GE-algebra and f : X → X is a mapping such that Definition 2.9. [10] Let (X, f ) be an interior GE-algebra. A GE-filter F of X is said to be interior if it satisfies: Then X is a GE-algebra. If we define a mapping f on X as follows:

Prominent interior GE-filters
then (X, f ) is an interior GE-algebra and F = {1, 4, 5} is a prominent interior GE-filter in (X, f ).
It is clear that every prominent interior GE-filter is a prominent GE-filter. But any prominent GEfilter may not be a prominent interior GE-filter in an interior GE-algebra as seen in the following example.  Table 2, Table 2. Cayley table for the binary operation " * ". * and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and We discuss relationship between interior GE-filter and prominent interior GE-filter.
Theorem 3.4. In an interior GE-algebra, every prominent interior GE-filter is an interior GE-filter.
Proof. It is straightforward because every prominent GE-filter is a GE-filter in a GE-algebra. □ In the next example, we can see that any interior GE-filter is not a prominent interior GE-filter in general.
Example 3.5. Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in Table 3. Then X is a GE-algebra. If we define a mapping f on X as follows: Proposition 3.6. Every prominent interior GE-filter F in an interior GE-algebra (X, f ) satisfies: Proof. Let F be a prominent interior GE-filter in (X, f ). Let x, y ∈ X be such that f (x * y) ∈ F. Then x * y ∈ F by (2.22), and so 1 * (x * y) = x * y ∈ F by (GE2). Since 1 ∈ F, it follows from (2.18) that ((y * x) * x) * y ∈ F. □ Corollary 3.7. Every prominent interior GE-filter F in an interior GE-algebra (X, f ) satisfies: Proof. Let F be a prominent interior GE-filter in (X, f ). Then F is an interior GE-filter in (X, f ) by Theorem 3.4. Let x, y ∈ X be such that x * y ∈ F. Since x * y ≤ f (x * y) by (2.19), it follows from Lemma 2.6 that f (x * y) ∈ F. Hence ((y * x) * x) * y ∈ F by Proposition 3.6. □ Corollary 3.8. Every prominent interior GE-filter F in an interior GE-algebra (X, f ) satisfies: Proof. Straightforward. □ Corollary 3.9. Every prominent interior GE-filter F in an interior GE-algebra (X, f ) satisfies: Proof. Straightforward. □ In the following example, we can see that any interior GE-filter F in an interior GE-algebra (X, f ) does not satisfy the conditions (3.1) and (3.2).
We provide conditions for an interior GE-filter to be a prominent interior GE-filter.
Theorem 3.11. If an interior GE-filter F in an interior GE-algebra (X, f ) satisfies the condition (3.1), then F is a prominent interior GE-filter in (X, f ).
Proof. Let F be an interior GE-filter in (X, f ) that satisfies the condition (3.1). Let x, y, z ∈ X be such that x * (y * z) ∈ F and x ∈ F. Then y * z ∈ F. Since y * z ≤ f (y * z) by (2.19), it follows from Lemma 2.6 that f (y * z) ∈ F. Hence ((z * y) * y) * z ∈ F by (3.1), and therefore F is a prominent interior GE-filter in (X, f ). □ Lemma 3.12. [10] In an interior GE-algebra, the intersection of interior GE-filters is also an interior GE-filter.
Theorem 3.13. In an interior GE-algebra, the intersection of prominent interior GE-filters is also a prominent interior GE-filter.
Then f (x * y) ∈ F i for all i ∈ Λ. It follows from Proposition 3.6 that ((y * x) * x) * y ∈ F i for all i ∈ Λ. Hence ((y * x) * x) * y ∈ ∩{F i | i ∈ Λ} and therefore ∩{F i | i ∈ Λ} is a prominent interior GE-filter in (X, f ) by Theorem 3.11. □ Theorem 3.14. If an interior GE-filter F in an interior GE-algebra (X, f ) satisfies the condition (3.2), then F is a prominent interior GE-filter in (X, f ).
Proof. Let F be an interior GE-filter in (X, f ) that satisfies the condition (3.2). Let x, y, z ∈ X be such that x * (y * z) ∈ F and x ∈ F. Then y * z ∈ F and thus ((z * y) * y) * z ∈ F. Therefore F is a prominent interior GE-filter in (X, f ). □ Given an interior GE-filter F in an interior GE-algebra (X, f ), we consider an interior GE-filter G which is greater than F in (X, f ), that is, we take two interior GE-filters F and G such that F ⊆ G in an interior GE-algebra (X, f ). We are now trying to find the condition that G can be a prominent interior GE-filter in (X, f ).
Theorem 3.15. Let (X, f ) be an interior GE-algebra in which X is transitive. Let F and G be interior GE-filters in (X, f ). If F ⊆ G and F is a prominent interior GE-filter in (X, f ), then G is also a prominent interior GE-filter in (X, f ).
Thus ((y * x) * x) * y ∈ G by Lemma 2.6. Therefore G is a prominent interior GE-filter in (X, f ). by Theorem 3.11. □ The following example describes Theorem 3.15.
Example 3.16. Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in Table 4, Table 4. Cayley table for the binary operation " * ". * and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra in which X is transitive, and F := {1} and G := {1, 4, 5} are interior GE-filters in (X, f ) with F ⊆ G. Also we can observe that F and G are prominent interior GE-filters in (X, f ).
In Theorem 3.15, if F is an interior GE-filter which is not prominent, then G is also not a prominent interior GE-filter in (X, f ) as shown in the next example.
In Theorem 3.15, if X is not transitive, then Theorem 3.15 is false as seen in the following example.
Example 3.18. Let X = {1, 2, 3, 4, 5, 6} be a set with the Cayley table which is given in Table 6. Table 6. Cayley table for the binary operation " * ". * If we define a mapping f on X as follows: then (X, f ) is an interior GE-algebra in which X is not transitive. Let F := {1} and G := {1, 5, 6}. Then F is a prominent interior GE-filter in (X, f ) and G is an interior GE-filter in (X, f ) with F ⊆ G. But G is not prominent interior GE-filter since 5 * (3 * 4) = 5 * 5 = 1 ∈ G and 5 ∈ G but ((4 * 3) Definition 3.19. Let (X, f ) be an interior GE-algebra and let F be a subset of X which satisfies (2.15). Then F is called: • A prominent interior GE-filter of type 2 in (X, f ) if it satisfies: Example 3.20. (1). Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in Table 7, Table 7. Cayley table for the binary operation " * ". * and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1, 3} is a prominent interior GE-filter of type 1 in (X, f ). (2). Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in Table 8, and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1, 3} is a prominent interior GE-filter of type 2 in (X, f ).
Theorem 3.21. In an interior GE-algebra, every prominent interior GE-filter is of type 1.
Proof. Let F be a prominent interior GE-filter in an interior GE-algebra (X, f ). Let x, y, z ∈ X be such that x * (y * f (z)) ∈ F and f (x) ∈ F. Then x ∈ F by (2.22). It follows from (2.18) that (( f (z) * y) * y) * f (z) ∈ F. Hence F is a prominent interior GE-filter of type 1 in (X, f ). □ The following example shows that the converse of Theorem 3.21 may not be true.
The following example shows that prominent interior GE-filter and prominent interior GE-filter of type 2 are independent of each other, i.e., prominent interior GE-filter is not prominent interior GEfilter of type 2 and neither is the inverse.  Cayley table which is given in the following  Table 10, Table 10. Cayley table for the binary operation " * ". * and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1} F is a prominent interior GE-filter in (X, f ). But it is not a prominent interior GE-filter of type 2 since 1 * (5 * f (2)) = 5 * 5 = 1 ∈ F and f (1) . Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in the following Table 11, Table 11. Cayley table for the binary operation " * ". * and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1} is a prominent interior GE-filter of type2 in (X, f ). But it is not a prominent interior GE-filter in (X, f ) since 1 * (2 * 3) = 1 * 1 = 1 ∈ F and 1 ∈ F but ((3 * 2) * 2) * 3 = (2 * 2) * 3 = 1 * 3 = 3 F.
The following example shows that prominent interior GE-filter of type 1 and prominent interior GE-filter of type 2 are independent of each other.
The following example shows that interior GE-filter and prominent interior GE-filter of type 1 are independent of each other.  Cayley table which is given in the following  Table 14, and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1} is an interior GE-filter in (X, f ). But F is not prominent interior GE-filter of type 1 since 1 * (5 * f (2)) = 1 * (5 * 2) = 1 * 1 = 1 ∈ F and f (1) . Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in the following Table 15, and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1, 2} is a prominent interior GE-filter of type 1 in (X, f ). But it is not an interior GE-filter in (X, f ) since 2 * 4 = 1 and 2 ∈ F but 4 F.
The following example shows that interior GE-filter and prominent interior GE-filter of type 2 are independent of each other.  Cayley table which is given in the following  Table 16, and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1, 4} is an interior GE-filter in (X, f ). But F is not prominent interior GE-filter of type 2 since 4 * (2 * f (3) . Let X = {1, 2, 3, 4, 5} be a set with the Cayley table which is given in the following Table 17, and define a mapping f on X as follows: Then (X, f ) is an interior GE-algebra and F := {1, 2, 5} is a prominent interior GE-filter of type 2 in (X, f ). But it is not an interior GE-filter in (X, f ) since 5 * 4 = 1 and 5 ∈ F but 4 F.
Before we conclude this paper, we raise the following question.
Question. Let (X, f ) be an interior GE-algebra. Let F and G be interior GE-filters in (X, f ). If F ⊆ G and F is a prominent interior GE-filter of type 1 (resp., type 2) in (X, f ), then is G also a prominent interior GE-filter of type 1 (resp., type 2) in (X, f )?

Conclusions
We have introduced the concept of a prominent interior GE-filter (of type 1 and type 2), and have investigated their properties. We have discussed the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We have found and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We have provided conditions for an interior GE-filter to be a prominent interior GE-filter. We have given conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and have provided an example describing it. We have considered the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We have found and provide examples to verify that a prominent interior GE-filter of type 1 and a prominent interior GE-filter of type 2, a prominent interior GE-filter and a prominent interior GE-filter of type 2, an interior GE-filter and a prominent interior GE-filter of type 1, and an interior GE-filter and a prominent interior GE-filter of type 2 are independent each other. In future, we will study the prime and maximal prominent interior GE-filters and their topological properties. Moreover, based on the ideas and results obtained in this paper, we will study the interior operator theory in related algebraic systems such as MV-algebra, BL-algebra, EQ-algebra, etc. It will also be used for pseudo algebra systems and further to conduct research related to the very true operator theory.