Abstract
In this paper, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given. Further, it is shown that a finite product (union) of belligerent fuzzy GE-filters of GE-algebras is a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras.
1. Introduction
In 1966, Diego [1] introduced the concept of Hilbert algebras. Diego discussed several properties of Hilbert algebras and deductive systems. More studies on the ideas of Hilbert algebras and deductive systems were performed by Busneag [2, 3]. Bandaru discussed a new algebraic structure, called GE-algebra [4] as a generalization of Hilbert algebra. Filters, upper sets, and congruence kernels in GE-algebra are considered, and the congruence kernel of transitive GE-algebra is characterized. The concept of belligerent GE-filter in GE-algebra is introduced by Bandaru [5]. Relationships between a belligerent GE-filter and a GE-filter are established, and the necessary and sufficient conditions on which a GE-filter to be a belligerent GE-filter are presented. Further, different properties of the union and the product of GE-algebras are investigated.
Zadeh, in 1965, studied the notion of fuzzy sets [6]. Following this, Rosenfeld in 1971, used this theory and has formulated the concept of a fuzzy subgroup of a group [7]. Since then, many researchers have been studying several algebraic structures including fuzzy subalgebras, fuzzy subgroups, fuzzy ideals, and fuzzy filters. Jun and Hong in [8] studied the concept of a fuzzy deductive system in Hilbert algebra. Akram and Zham [9] studied sensible fuzzy ideals of BCK-algebras with respect to a t-conorm. Akram and Dar [10] investigated -fuzzy ideals in BCI-algebras. Additionally, Jun, Akram, and Pasha [11] presented intuitionistic fuzzy quasi-associative ideals in BCI-algebras. Recently, Alemayehu and EShetie [12] studied Fuzzy ideals and fuzzy filters of GE-algebras.
Motivated by the above results, in this research, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given. Further, it is shown that a finite product (union) of belligerent fuzzy GE-filters of GE-algebras is a belligerent fuzzy GE-filter of a finite product (union) of GE-algebras.
2. Preliminaries
This section deals on definitions and basic results that we use in the sequel such as the concepts on GE-algebras, GE-filters of GE-algebras, belligerent GE-filters, and fuzzy filters on a given set.
Definition 1 (see [13]). Hilbert algebra is an algebra of type such that the following axioms hold, for all .(1),(2),(3)if , then .
Definition 2 (see [4]). GE-algebra is a non-empty set with a constant 1 and a binary operation satisfying axioms:(1),(2) ,(3), for all .
Definition 3 (see [4]). GE-algebra is said to be transitive if it satisfiesfor all .
Theorem 1 (see [4]). In a transitive GE-algebra , for all , the following conditions hold(1)implies(2)(3)(4)andimplies.
Definition 4 (see [4]). GE-algebra is said to be commutative if it satisfiesfor all .
Theorem 2 (see [4]). Every commutative GE-algebra is a generalized Hilbert algebra.
Definition 5 (see [4]). A subset of GE-algebra is called a GE-filter of if it satisfies the following:(1)(2)if and , then .
Theorem 3 (see [4]). Let be a filter of . If and , then .
Theorem 4 (see [4]). A non-empty subset of GE-algebra is a filter of if and only if it satisfies , and implies that for all .
Definition 6 (see [5]). A subset of GE-algebra is called a belligerent GE-filter of if it satisfies andfor all .
Definition 7 (see [5]). GE-algebra is said to be left exchangeable if for all .
Theorem 5 (see [5]). Let be a GE-filter of a transitive and left exchangeable GE-algebra . If satisfies the condition, implies that for all , then is a belligerent GE-filter of .
Recall that, for any set a function, is called a fuzzy subset of [6], where is a unit interval, and for all . A fuzzy subset of is proper if it is a nonconstant function. A fuzzy subset such that for all is an improper fuzzy subset. Let . For every , the level subset of is .
Definition 8 (see [14, 15]). Let . A fuzzy point of is a fuzzy subset that is defined as
Definition 9 (see [16]). A fuzzy subset of a bounded lattice is said to be a fuzzy ideal of , if for all ,(1)(2)(3).In [16], Swamy and Raju discussed that, a fuzzy subset of a bounded lattice is a fuzzy ideal if and only if and , for all .
Theorem 6 (see [16]). Let be a fuzzy subset of , then is a fuzzy ideal of if and only if, for any , is an ideal of .
Definition 10 (see [17]). A fuzzy subset of a bounded lattice is said to be a fuzzy filter of , if for all ,(1)(2)(3).
3. Belligerent Fuzzy GE-Filters
In this topic, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given.
Some concepts of about fuzzy GE-filters of GE-algebras are recalled.
Definition 11 (see [12]). Let be a fuzzy subset of a GE-algebra , then is called a fuzzy GE-filter on if the following conditions hold for any :(1),(2).
Lemma 1 (see [12]). A fuzzy GE-filter on a GE-algebra satisfies the following if then for all .
Theorem 7 (see [12]). Let be a fuzzy subset of such that and . Then, is a fuzzy GE-filter.
Definition 12. Let be GE-algebra and be a fuzzy subset on , then is said to be a belligerent fuzzy GE-filter on if for any .(1),(2).
Example 1. Consider a set and define a binary operation on as in Table 1.
It is easy to check that is GE-algebra. Now, define a fuzzy subset on by , and . It is easy to verify that is a belligerent fuzzy GE-filter on .
The following results show that the relation between belligerent fuzzy GE-filter and fuzzy GE-filter.
Theorem 8. is a belligerent fuzzy GE-filter of GE-algebra if and only if, , is a belligerent GE-filter.
Proof. Suppose be a belligerent fuzzy GE-filter of GE-algebra , then . This implies . Let such that and , then and , so that ; thus, . Hence is belligerent GE-filter.
Conversely, suppose that be a belligerent GE-filter of GE-algebra . Since for all , . Hence, . Let such that for all . Clearly, and . So, and which shows that . Consequently, . Hence is a belligerent fuzzy GE-filter.
Corollary 1. is a belligerent GE-filter of GE-algebra if and only if is a belligerent fuzzy GE-filter.
Theorem 9. Let be a belligerent fuzzy GE-filter of GE-algebra G, then is a fuzzy GE-filter.
Proof. Since and is a belligerent fuzzy GE-filter of , we have which means . So, is a fuzzy GE-filter. But the converse of this theorem may not be true.
Example 2. Consider set and the binary operation on given in the above examples (3) and (5). Define a fuzzy GE-filter on by , . Since . This implies is not a belligerent fuzzy GE-filter on .
Let be GE-algebra, and be any fuzzy subset of . The following equation is considered asAny fuzzy GE-filter cannot satisfy equation (5), so we look at the following examples:
Example 3. Consider the GE-algebra in Example 1 and fuzzy GE-filters on as , does not satisfies equation (1), since and . Thus .
The next result shows that the condition on which a fuzzy GE-filter to be a belligerent fuzzy GE-filter.
Theorem 10. Let be a fuzzy GE-filter of GE-algebra G such that , then is a belligerent fuzzy GE-filter of G.
Proof. Since is a fuzzy GE-filter of , and . Also, from the assumption we have so that is a belligerent fuzzy GE-filter of .
Corollary 2. In belligerent GE-algebra, the trivial fuzzy GE-filter is a belligerent fuzzy GE-filter.
GE-algebra is said to be left exchangeable and transitive if and for all , respectively [4, 5].
Theorem 11. The trivial fuzzy GE-filter of transitive GE-algebra is a belligerent fuzzy GE-filter.
Proof. Let be the trivial fuzzy GE-filter on transitive GE-algebra . Let , then . Hence, is a belligerent fuzzy GE-filter.
The fuzzy subset given in Theorem 7 is not a belligerent fuzzy GE-filter. For if we consider set and the binary operation on given in the above example and define a fuzzy subset on by , . Since . Hence, is not belligerent fuzzy GE-filter on G.
Theorem 12. Let be a fuzzy GE-filter on such that . If is transitive and left exchangeable, then .
Proof. Assume that is transitive and left exchangeable and be a fuzzy GE-filter on such that . From the transitivity property of , we get and . From the left exchangeable property of , we get . Now, applying the given condition in the theorem and the left exchangeable property of yields .
Theorem 13. Let be a fuzzy GE-filter of a transitive and left exchangeable GE-algebra such that , then is a belligerent fuzzy GE-filter of .
Proof. Suppose the conditions in the theorem hold, then, by Theorem 12 , we get and, hence, from Theorem 10, we conclude that is a belligerent fuzzy GE-filter of .
For a point and a non-empty fuzzy subset of GE-algebra , define a fuzzy subset on as
Lemma 2. If is a fuzzy GE-filter of GE-algebra, then .
If is a fuzzy GE-filter of , then fuzzy subset may not be a fuzzy GE-filter.
Example 4. Let be a GE-algebra of Example 1. A fuzzy subset of , such as and , is a fuzzy GE-filter of but is not a fuzzy GE-filter. Since , , and . This implies . Thus, is not a fuzzy GE-filter of .
However, we give the conditions that is a fuzzy GE-filter of in the following result.
Theorem 14. Let be a belligerent fuzzy GE-filter on , then is a fuzzy GE-filter on .
Proof. Let . Since is a belligerent fuzzy GE-filter on , we obtain that , i.e., . So, is a fuzzy GE-filter on .
We suggest the conditions under which a fuzzy GE-filter can be a belligerent fuzzy GE-filter.
Theorem 15. Let be a fuzzy subset on , and for any , be a fuzzy GE-filter on . Then, is a belligerent fuzzy GE-filter on .
Proof. Since is a fuzzy GE-filter on , for all , we getSo that . Thus, is a belligerent fuzzy GE-filter on .
Corollary 2. Let be a fuzzy GE-filter and be a fuzzy GE-filter of for every , then is a belligerent fuzzy GE-filter on .
Theorem 16. Let be a belligerent fuzzy GE-filter on , then fuzzy subset , where is the smallest fuzzy GE-filter on such that .
Proof. From Theorem 14, is a fuzzy GE-filter on . For any , we obtain that and hence, . Hence . Let be a fuzzy GE-filter on such that and . Hence . Now, for any , . This implies is the least fuzzy GE-filter of .
Theorem 17. Let be a GE-morphism from GE-filters to and be a belligerent fuzzy GE-filter of , then is a belligerent fuzzy GE-filter of .
Proof. Suppose the conditions hold. Since for all and is a morphism, andHence, is a fuzzy GE-filter of . Also,so that is a belligerent fuzzy GE-filter of .
4. Direct Product of Finite Belligerent Fuzzy GE-Filters of GE-Algebras
In this section, it is discussed that the finite product (union) of belligerent fuzzy GE-filters of GE-algebras becomes a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras. Further the sufficient and necessary conditions on which the finite product of a belligerent GE-filter of a finite product of GE-algebras becomes a belligerent fuzzy GE-filter is given.
Definition 13. Let and be fuzzy subsets of GE-algebras and , respectively. Then the direct product of fuzzy subsets and is denoted by and is defined asfor any and .
Definition 14. Let be n fuzzy subsets of GE-algebras , respectively. Then the direct product of finite fuzzy subsets of GE-algebras is denoted by and is defined as for any .
Theorem 18. Let be n (belligerent) fuzzy GE-filters of GE-algebras , respectively. Then is (belligerent) fuzzy GE-filter of .
Proof. Suppose that the assumptions hold and let . Define a binary operation on bywhere are the binary operations on , respectively. Since is fuzzy GE-filter for each , it is trivial to show that . From the given hypothesis is a belligerent fuzzy GE-filter for each . So,After some steps it follows thatTherefore, is a belligerent fuzzy GE-filter of .
Example 5. Let and be GE-algebras where the binary operations and are defined by Tables 2 and 3.
Then, is GE-algebra with point wise operation.
Now, define by and by . Clearly, and are fuzzy GE-filters of and , respectively. Now, we define by , = = = = . Then, clearly, is a fuzzy GE-filter of .
Theorem 19. If and are (belligerent) fuzzy GE-filters of and , respectively, then the union is a (belligerent) fuzzy GE-filter of .
Proof. Clearly, . Recall that for any , define a binary operation on as Now,Hence, is a fuzzy GE-filter of . Next, assume that and be belligerent fuzzy GE-filters of and , respectively. Thenso that is a belligerent fuzzy GE-filter of .
Theorem 20. Let be fuzzy subsets of GE-algebras , respectively, and let, then.
Proof. Let and for any elementHence .
Theorem 21. Let be fuzzy subsets of GE-algebra , respectively. Then is a fuzzy GE-filter of a GE-algebra if and only if for any , is a GE-filter of a GE-algebra .
Proof. Suppose that is a fuzzy GE-filter of GE-algebra . Since . This implies .
If and , then , . This implies . This implies . Thus, . Thus, is a GE-filter of .
Conversely, suppose that is a GE-filter. Clearly . Let . This implies . This implies and . Thus, . This implies .
Thus, is a fuzzy GE-filter.
Theorem 22. Let be fuzzy subsets of GE-algebra , respectively. Then, is a belligerent fuzzy GE-filter of GE-algebra if and only if for any , is belligerent GE-filter of GE-algebra .
Proof. The proof of it is straight forward by Theorems 8 and 18.
5. Conclusion and Future Work
In this paper, the concept of belligerent fuzzy GE-filters is introduced. We investigate the relationships between fuzzy GE-filter and a belligerent fuzzy GE-filter. Additionally prove that that the finite product (union) of belligerent fuzzy GE-filters of GE-algebras becomes a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras. We hope in the future, we study prominent fuzzy GE-filters, imploring fuzzy GE-filters, soft GE-filter, and Soft GE-ideals of GE-algebras.
Data Availability
No data were used.
Conflicts of Interest
The authors declare that they have no conflicts of interest.