Abstract

We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a regular congruence relation and a inf-Bosbach state on . By inducing an inf-Bosbach state on the quotient structure /, we show that / is a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states/measure morphisms/state morphisms) on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebra / by a reflexive hyper BCK-ideal . Further, we prove that / is a bounded commutative BCK-algebra.

1. Introduction

The theory of hyper structures (also called multialgebras) was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians. Then several researchers have worked on this new field and developed it. Corsini studied the theory of Hypergroups; see [2, 3]. Krasner [4] introduced the notion of hyperrings and hyperfields. Massouros [5] introduced the theory of hypercompositional structures into the theory of automata. Jun et al. [6] introduced the concept of hyper BCK-algebras which is a generalization of BCK-algebras and studied some properties of them. They also introduced the notions of hyper BCK-ideals, weak/strong hyper BCK-ideals, and reflexive hyper BCK-ideals and discussed the relations among these notions. From then on, a lot of literatures about hyper BCK/BCI-algebras appear; see [7ā11].

MV-algebras entered mathematics just 50 years ago due to Chang [12], but the notion of states for MV-algebras was introduced by Mundici [13] in 1995 as averaging of the truth-value in ukasiewicz logic. BL-algebras were introduced in the 1990s by HĆ”jek as the equivalent algebraic semantics for its basic fuzzy logic. Ciungu et al. [14] defined a state-operator and a strong state-operator for a BL-algebra and proved some basic properties of them. Liu [15] studied the existence of Bosbach states and Riecan states on finite monoidal -norm based algebras (MTL-algebra for short) and gave some examples to show that there exist MTL-algebras having no Bosbach states and Riecan states.

DvureÄenskij [16] introduced measures and states on BCK-algebras and showed that the set of elements of measure 0 is an ideal and the corresponding quotient BCK-algebra is commutative with a lifted original measure. Corina Ciungu and DvureÄenskij [17] extended the notions of measures and states, which were presented in the paper of DvureÄenskij and PulmannovĆ” [18] to the case of pseudo-BCK-algebras. They also studied similar properties and proved that the notion of states in the sense of DvureÄenskij and PulmannovĆ” [18] coincides with the Bosbach state.

At present, the state theories were set up in various algebraic structures. So far, we have not found research literatures about the state theory on hyper structures. In this paper, we mainly introduce and study the state theory on hyper BCK-algebras.

The paper is organized as follows. In Section 2, we recall some basic notions and some results of hyper BCK-algebras. Then we induce two new operations āā and āāā by the operation āā on hyper BCK-algebras and investigate some properties of them. We also present a relation between hyper BCK-algebras and MV-algebras. In Section 3, we define a Bosbach state and an inf-Bosbach state on a bounded hyper BCK-algebra and discuss some of their basic properties. In Section 4, we study inf-Bosbach states on quotient hyper BCK-algebras. In Section 5, we define hyper measure, hyper states, hyper measure-morphisms, and hyper state-morphisms on hyper BCK-algebras and obtain some interesting results.

2. Preliminaries

In this section, we gather some basic notions and properties relevant to hyper BCK-algebras which we need in the sequel.

Let be a nonempty set with a hyperoperation ā.ā For any two subsets and of , by we mean the set . Hereafter we denote instead of , , or .

Definition 1 (see [10]). Let be a nonempty set endowed with a hyperoperation āā and a constant 0. If satisfies the following axioms: for all ,(HK1),(HK2),(HK3),(HK4) and imply ,then is called a hyper BCK-algebra, where is defined by and for any nonempty subsets of , is defined by for all ; there exists such that .

Example 2 (see [10]). We define an operation āā on by then is a hyper BCK-algebra.

Proposition 3 (see [10]). In a hyper BCK-algebra , the condition (HK3) is equivalent to the following condition: for all , .

Proposition 4 (see [10]). In a hyper BCK-algebra , the following hold.(1)For all , , , and .(2)For any nonempty subsets and , , , .

Proposition 5 (see [7, 10]). In any hyper BCK-algebra , the following properties hold: for all , and for any nonempty subsets ,(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13), ,(14),(15),(16),(17) and .

Definition 6 (see [10]). Let be a nonempty subset of a hyper BCK-algebra . Then is said to be a hyper BCK-ideal of if(1),(2) and imply for all .

Note that if is a hyper BCK-ideal of hyper BCK-algebra , then implies .

Definition 7 (see [19]). A hyper BCK-algebra is called bounded if there is an element such that for all . This element is called the unit of , and we denote a bounded hyper BCK-algebra simply by .

Example 8. Let . Define a hyperoperation āā on as follows: then is a bounded hyper BCK-algebra, where .

In a hyper BCK-algebra , we define a hyperoperation āā by for all . For all , . In general, .

Proposition 9. Let be a hyper BCK-algebra. Then for any ,(1),(2),(3).

Proof. (1)By (HK3) of Definition 1, we have . Therefore .(2)Suppose that ; then , . So by Proposition 5 (10). Hence which implies ; that is, . Also by (1).(3)By Proposition 5 (12), ; that is, .

Let be a bounded hyper BCK-algebra. Then we define for any .

Proposition 10. Let be a bounded hyper BCK-algebra; the following hold:(1),(2),(3).

Proof. (1) Assume that ; it is clear by Proposition 5.
(2) By (HK2), .
(3) By (2) and (HK2), it is easy to prove (3).

An MV-algebra is an algebra of type such that (1) is commutative and associative, (2) , (3) , (4) , and (5) . In [20], we know that MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Now we discuss the relation between a bounded hyper BCK-algebra and an MV-algebra.

Define . Then we get the following results.

Lemma 11 (see [6]). Every hyper BCK-algebra is a BCK-algebra if and only if .

Theorem 12. Let be a hyper BCK-algebra. If satisfies the condition , for all , then(1), for all ,(2), for all ,(3), for all ,(4).

Proof. (1) Suppose that , for all . Let ; then we have by Proposition 5, and so .
(2) Let for every . Since , we get . On the other hand, and thus . Hence , and we conclude that . Consequently .
(3) For all , we get by (1) and (2).
(4) By (3) and Lemma 11 we have .

From Theorem 12, we obtain the relation between hyper BCK-algebras and MV-algebras.

Corollary 13. Let be a bounded hyper BCK-algebra with the condition , for all . Then is a bounded commutative BCK-algebra. We define for all and for all . Then is an MV-algebra.

Now, let us review the structure of quotient hyper BCK-algebras on which we consider inf-Bosbach states in Section 4.

Definition 14 (see [19]). Let be an equivalence relation on a hyper BCK-algebra and . Then,(1) means that there exist and such that ;(2) means that for all there exists such that and for all there exists such that ;(3) is called a congruence relation on ; if and then , for all ;(4) is called a regular relation on ; if and , then for all .

Lemma 15 (see [19]). Let be an equivalence relation on and . If and , then .

Lemma 16 (see [19]). Let be an equivalence relation on . Then the following statements are equivalent:(1) is a congruence relation on ;(2)if , then and , for all .

Theorem 17 (see [19]). Let and be two regular congruence relations on such that . Then .

Lemma 18 (see [19]). Let be a congruence relation on . Then is a strong hyper BCK-ideal of .

Theorem 19 (see [19]). Let be a regular congruence relation on , and , where for all . Then is a hyper BCK-algebra, which is called a quotient hyper BCK-algebra, where āā and āā are defined as follows: and .

3. States on Bounded Hyper BCK-Algebras

In this section, the concepts of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra are defined, and its properties are studied.

In what follows in the paper, we denote a bounded hyper BCK-algebra by or , unless otherwise specified.

Definition 20. A function is called a Bosbach state on if it satisfies the following conditions:(1), ,(2), for any .

Example 21. Let be defined in Example 8. We define , , , , , , and . Then is a Bosbach state on .

Definition 22. A function is called an inf-Bosbach state on if it satisfies the following conditions:(1), ,(2), for any ,
where is an abbreviation of , and is defined by for any .

Example 23. Let be defined in Example 8. Assume that is an inf-Bosbach state on . Then we have and . Assume . Since , we have and hence . Therefore . It follows that is the unique inf-Bosbach state on .

The following example shows that not every bounded hyper BCK-algebra has an inf-Bosbach state.

Example 24. Let . Define a hyperoperation āā on as follows: Then is a bounded hyper BCK-algebra, where . Let , , , and . From , taking , , we get . Taking , , we get . Taking , , we get . It is a contradiction. Hence, does not admit any inf-Bosbach state.

Lemma 25. Let be an inf-Bosbach state on . Then is a Bosbach state on .

Then we give some basic properties of inf-Bosbach states on hyper BCK-algebras.

Proposition 26. Let be an inf-Bosbach state on . Then the following hold:(1),(2),(3).

Proof. (1) and (2) are trivial. By Proposition 3 and (1), we get that . So (3) holds.

Proposition 27. Let be an inf-Bosbach state on . Then,(1),(2),(3), ,(4), ,(5).

Proof. (1) Note that by Proposition 3, so we have .
(2) Combining (1) and Definition 22, we get . Thus .
(3) Since , then . Moreover .
(4) By Proposition 10, we get . So and .
(5) Suppose ; then we have . So .

The following theorem gives an equivalent characterization of inf-Bosbach states.

Theorem 28. Let satisfy . Then the following are equivalent:(1) is an inf-Bosbach state on ;(2) and .

Proof. (1) (2) It follows from Proposition 27 (2) and Proposition 26 (1).
(2) āā. Since , we obtain ; that is, .

Theorem 29. Let be an inf-Bosbach state on . Define which is called the kernel of the inf-Bosbach state . Then is a hyper BCK-ideal of .

Proof. Clearly, . Let and . So . Since , then for all , there is such that . Since is order-preserving, we have . Hence ; that is, . Also note that , so . This shows that . We obtain by the definition of inf-Bosbach state . Therefore, we have .

4. States on Quotient Hyper BCK-Algebras

In this section, we study the inf-Bosbach states on quotient hyper BCK-algebras.

Definition 30. Let be an inf-Bosbach state and let be a congruence relation on . Then is called compatibled if and only if for all .

Lemma 31. Let be a regular congruence relation and let be a inf-Bosbach state on . Define . Then in the bounded quotient hyper BCK-algebra , where , and , the following hold:(1) if and only if ,(2) if and only if .

Proof. (1) Since implies , then there exists such that . By Definition 30, we get that . Then . On the other hand, suppose that . Then there is such that . Moreover we get ; that is, there is such that . This means that .
(2) It is clear that if and only if if and only if .

Theorem 32. Let be a regular congruence relation and let be a inf-Bosbach state on . Take . Define a map by and , for any and . Then is an inf-Bosbach state on .

Proof. By Lemma 31, the definition of is well defined. Clearly, and . Since , then . Therefore, is an inf-Bosbach state on .

Definition 33. Let be a regular congruence relation on . Then is called compatibled if there is such that for all .

Lemma 34. Let be a compatibled regular congruence relation on . Then there is such that for all .

Proof. Since is compatibled, so there exists such that for all . Hence . For any , we have . Since is a congruence relation, then by Lemma 16. Since is compatibled, there exists such that and . This shows that for any , is contained in the same equivalent class. Hence . It follows that .

Lemma 35. Let be a compatibled regular congruence relation on . Then for all .

Proof. By Lemma 34, there exists such that for all . Similarly, there exists such that . By Proposition 27, , and so for some and . Since is compatibled, then . Hence . Therefore, and , which implies .

Lemma 36. Let be a regular congruence relation on and . Then for any , in .

Proof. Note that .

Lemma 37. Let be a regular congruence relation on and let be a inf-Bosbach state on . Then the bounded quotient hyper BCK-algebra is a bounded commutative BCK-algebra.

Proof. Note that . Since is compatibled, then there is such that . This shows that for all . It follows that is a BCK-algebra. Since is compatibled, then by Lemma 34, there is such that . Hence . Note that is a BCK-algebra and by Lemma 36, we get . By Lemma 35, . Therefore, .

Summarizing the above conclusions, we get the following result.

Theorem 38. Let be a regular congruence relation and let be a inf-Bosbach state on . Take . Define for all . Then is an MV-algebra. Moreover, the map defined as Theorem 32 is an inf-Bosbach state on and the following hold:(1),(2),(3),(4).

5. Hyper Measures on Hyper BCK-Algebras

In this section, we study the hyper measures on hyper BCK-algebras.

Define āā on the real interval as follows: , for all . Then is a BCK-algebra.

Definition 39. Let be a hyper BCK-algebra. A map such that, for all ,(1) wherever is said to be a hyper measure;(2)if is bounded, is the unit of , and is a hyper measure with , then is said to be a hyper state;(3) is said to be a hyper measure-morphism;(4)if is bounded, is the unit of , and is a hyper measure-morphism with , then is said to be a hyper state-morphism.

Obviously any hyper measure-morphism on a hyper BCK-algebra is a hyper measure.

Proposition 40. Let be a hyper measure on hyper BCK-algebra . Then for all , one has the following:(1),(2) implies ,(3) implies ,(4).

Proof. (1) Clearly we have .
(2) Since implies , then .
(3) Note that implies .
(4) By Definition 39, .

Theorem 41. Let be a hyper state-morphism on . Then is a Bosbach state on .

Proof. Clearly, and . By Proposition 40, we have for all . Moreover, . Therefore is a Bosbach state on .

Proposition 42. Let be a hyper state-morphism on . Then for all ,(1),(2),(3), ,(4).

Proof. (1) Since , then .
(2) Observe that by Theorem 41. So, we obtain .
(3) Since , then and .
(4) Since , then .

Definition 43 (see [10]). A hyper BCK-ideal of a hyper BCK-algebra is said to be reflexive if for all .

Theorem 44 (see [19]). Let I be a reflexive hyper BCK-ideal of a hyper BCK-algebra . A relation on is defined as follows: for all , Then is a regular congruence relation on and . Moreover, is a BCK-algebra.

Lemma 45. Let be a hyper state-morphism on . Then is a reflexive hyper BCK-ideal.

Proof. Clearly, . Assume that and . For all , there exists , such that . By Proposition 40, we have . Hence . Since , then for all , such that . By Proposition 40, . Hence . By Theorem 41, is a Bosbach state, and so we have . Thus . Therefore, is a hyper BCK-ideal. Since , then . So, is a reflexive hyper BCK-ideal.

Theorem 46. Let be a hyper state-morphism on . A relation on is defined as follows: Then is a bounded commutative BCK-algebra, where and for all . And is defined by .
Moreover, define a map by , . Then,(1) if and only if if and only if ;(2) if and only if if and only if ;(3) is a state-morphism on .

Proof. By Proposition 42, . Then for all there exists such that and for all there exists such that . Therefore, , which implies . So (i.e., ) and (i.e., ). In , . So, we have . By Theorem 44, is a BCK-algebra. Combining the above arguments, we get is a bounded commutative BCK-algebra.
In the following, we prove the second part of the theorem.(1)Note that if and only if if and only if , which implies for all we have . Then we get , which implies . Thus . By , we have ; that is, . Conversely, suppose that ; we have . It follows from the definition of hyper state-morphisms that we have . This means that for all we have , which implies , or . That is, .(2)Similar to (1), we can prove (2).(3)By (2), for all , if and only if if and only if . Therefore, the definition of is well defined. It is obvious that and . Note that ā. Therefore, is a state-morphism on .

Corollary 47. In as Theorem 46, define , and for all . Then is an MV-algebra. The map defined as Theorem 46 is a state-morphism on MV-algebra , and(1),(2),(3),(4).

6. Conclusions

In this paper, we mainly study the state theory on hyper structures and introduce a notion of states on hyper BCK-algebras. In order to adapt a state to hyper operation, we define the state on a subset by . Using the definitions of compatibled regular congruence relations and compatibled inf-Bosbach states on hyper BCK-algebras, we prove that the quotient structure of a bounded hyper BCK-algebra is an MV-algebra. Moreover, we define hyper measures on hyper BCK-algebras; then we introduce hyper states, hyper measure-morphisms, and hyper state-morphisms. We prove that a hyper state-morphism on hyper BCK-algebra is a Bosbach state. In the further work, we will solve the problem of how to define a state on a bounded hyper BCK-algebra to make the quotient structure form a hyper MV-algebra.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are extremely grateful to the editor and the referees for their valuable comments and helpful suggestions which help to improve the presentation of this paper. This research is partially supported by a Grant from Graduate Higher Achievement Foundation of Northwest University (YC13055) and Graduate Independent Innovation Foundation of Northwest University (YZZ12061).