States on hyper MV-algebras

1 Introduction

Algebraic hyperstructures represent a natural extension of classical algebraic structures. They were introduced in 1934 by the French mathematicianF. Marty [18]. Since then, there appeared many components of hyperalgebras such as hypergroups in [11], hyperrings in [22] etc. Nowadays, algebraic hyperstructure theory has a multiplicity of applications to other disciplines such as geometry, graphs and hypergraphs, binary relations, lattices, groups, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, C-algebras, artificial intelligence and probability theory, see [5]. Seems that one of important applications is in logic where the logical operations is not uniquely determined, i.e., they give some (set) possibilities. This provides sufficient motivations for researchers to study hyperstructures of various logical algebras. Recently, Ghorbani et al. [8], applied hyperstructures to MV-algebras and introduced the concept of hyper MV-algebras which are a generalization ofMV-algebras (introduced by Chang [2] in 1958 to prove the completeness theorem for Lukasiewicz calculus). They also, investigated the congruence relations to define quotient hyper MV-algebras. Rasouli and Davvaz [20] studied homomorphisms and fundamental relations on hyper MV-algebras and gave some results about the connections between hyper MV-algebras and fundamental MV-algebras. Moreover, Jun et al. [12, 13] introduced some new types of deductive systems on hyper MV-algebras and investigated their relations.

States on MV-algebras were introduced by Mundici [19] with the intent of measuring the average truth-value of propositions in Łukasiewicz logic, which are an analogue to probability measure and have a very important role in the theory of quantum structures. States on MV-algebras have been deeply investigated and many profound results have been achieved [6, 14]. Considering that random experiments may also follow the rules of other logical systems, the notion of states has been extended to other logical algebras such as BL-algebras [21], MTL-algebras [15, 16], R₀-algebras [17] and residuated lattices [4, 23] and their non-commutative cases. Different approaches to the generalization mainly gave rise to two different notions, namely, Riečan states [21] and Bosbach states [7]. It was proved by Ciungu [4] that in any good non-commutative residuated lattice, Bosbach states and Riečan states coincide, while the converse is not always true. Thus the notion of Riečan states is more general than that of Bosbach states. We can see that the current study of states focus on various classical logical algebras. In order to develop the theories of states on hyper logical algebras, we introduce and study states on hyper MV-algebras in this paper. In particular, using states on a hyper MV-algebra, we will construct and characterize a quotient hyper MV-algebra.

The paper is organized as follows. In Section 2, we recall some basic notions and some results of hyper MV-algebras. In Section 3, we investigate Riečan states and Bosbach states on a hyper MV-algebra and derive some properties of them. In Section 4, using a Bosbach state on a hyper MV-algebra, we construct and discuss a quotient hyper MV-algebra. In Section 5, we introduce the notion of state morphisms on hyper MV-algebras, and dicuss some relationships between state morphisms and Bosbachstates.

2 Preliminaries

In this section, we summarize some basic notions and properties about MV-algebras and hyper MV-algebras, which will be used in the following sections of the paper.

Definition 2.1. [2, 3] An MV-algebra is an algebra (A, ⊕ , ∗ , 0) with a binary operation “ ⊕ ″, a unary operation “ ∗ ″ and a constant 0 satisfying the following conditions: for all x, y, z ∈ A

(MV1) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z),

(MV2) x ⊕ y = y ⊕ x,

(MV3) x ⊕ 0 = x,

(MV4) (x^∗)^∗= x,

(MV5) x ⊕ 0^∗= 0^∗,

(MV6) (x^∗⊕ y)^∗⊕ y = (y^∗⊕ x)^∗⊕ x.

A partial order ≤ introduced on an MV-algebra A by defining x ≤ y if x^∗⊕ y = 0^∗, which makes A into a distributive lattice.

Let L be a nonempty set and P^*(L) be the set of all nonempty subsets of L. A hyperoperation on L is a map ∘ : L × L ⟶ P^*(L), which associates a nonempty subset a ∘ b with any pair (a, b) of elements of L × L. The couple (L, ∘) is called a hypergroupoid.

Let (L, ∘) be a hypergroupoid. If A and B are nonempty subsets of L, we denote: for all a, b, x ∈ L, (1) x ∘ A = {x} ∘ A = ⋃ _a∈Ax ∘ a, A ∘ x = A ∘ {x} = ⋃ _a∈Aa ∘ x. (2) A ∘ B = ⋃ _a∈A,b∈Ba ∘ b.

Definition 2.2. [8] A hyper MV-algebra is a non-empty set M endowed with a hyperoperation “⊕”, a unary operation “∗” and a constant 0 satisfying the following axioms:

(hMV1) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z),

(hMV2) x ⊕ y = y ⊕ x,

(hMV3) (x^∗)^∗= x,

(hMV4) (x^∗⊕ y)^∗⊕ y = (y^∗⊕ x)^∗⊕ x,

(hMV5) 0^∗∈ x ⊕ 0^∗,

(hMV6) 0^∗∈ x ⊕ x^∗,

(hMV7) if x ⪡ y and y ⪡ x, then x = y, for all x, y, z ∈ M, where x ⪡ y is defined by 0^∗∈ x^∗⊕ y and for every A, B ⊆ M, if there exist a ∈ A and b ∈ B such that a ⪡ b, then we define A ⪡ B. We define 0^∗= 1 and for every A ⊆ M, A^∗= {a^∗|a ∈ A}.

In what follows, unless otherwise specified, we denote a hyper MV-algebra (M, ⊕ , ∗ , 0) by M.

In what follows, we describe some basic examples of hyper MV-algebras.

Example 2.3. [20] Let M = [0, 1]. Define a hyperoperation ⊕ on M by x ⊕ y = [0, min {1, x + y}] and a unary operation ∗ on M by x^∗= 1 - x for all x, y ∈ [0, 1]. Then (M, ⊕ , ∗ , 0) is a hyper MV-algebra.

Example 2.4. [20] Let (M, ⊕ , ∗ , 0) be an MV-algebra. Define a hyperoperation ⊕′ on M by x ⊕ ′y = {t ∈ M|0 ≤ t ≤ x ⊕ y} = [0, x ⊕ y] for all x, y ∈ M. Then (M, ⊕ ′, ∗ , 0) is a hyper MV-algebra.

Example 2.5. [20] Let (G, + , - , 0, ≤) be an additive abelian linearly ordered group. Let c be a distinguished positive element of G and G [c] = {x ∈ G|0 ≤ x ≤ c}. Define a hyperoperation ⊕ on G [c] by x ⊕ y = [0, min {c, x + y}] and a unary operation ∗ on G [c] by x^∗= c - x for all x, y ∈ G [c]. Then (G [c] , ⊕ , ∗ , 0) is a hyper MV-algebra.

Proposition 2.6. [8] Let (M, ⊕ , ∗ , 0) be a hyper MV-algebra. Then for all x, y, z ∈ M and for all non-empty subsets A, B and C of M the following hold:

(1) (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C),

(2) 0 ⪡ x,

(3) x ⪡ x,

(4) x ⪡ y implies y^∗⪡ x^∗, and A ⪡ B implies B^∗⪡ A^∗,

(5) x ⪡ 1,

(6) A ⪡ A,

(7) A ⊆ B implies A ⪡ B,

(8) x ⪡ x ⊕ y, A ⪡ A ⊕ B,

(9) (A^∗)^∗= A,

(10) 0 ⊕ 0 =0,

(11) x ∈ x ⊕ 0,

(12) if y ∈ x ⊕ 0, then y ⪡ x,

(13) 0 ∈ 0 ⊙ x,

(14) 0 ∈ x ⊙ x^∗,

(15) x ⊙ y ⪡ x ⪡ x ⊕ y.

In what follows, let’s review quotient hyper MV-algebras.

Definition 2.7. [1] Let ∼ be an equivalence relation on a hyper MV-algebra M and A, B ⊆ M. (1) If there exist a ∈ A and b ∈ B such that a ∼ b, then we mean A ∼ B. (2) If for all a ∈ A there exists b ∈ B such that a ∼ b and for all b ∈ B there exists a ∈ A such that a ∼ b, we mean A ≈ B. (3) If for all a ∈ A and for all b ∈ B such that a ∼ b, we mean A ≊ B.

Definition 2.8. [1] Let ∼ be an equivalence relation on a hyper MV-algebra M and A, B ⊆ M. (1) If x ∼ y and x′ ∼ y′ imply x^∗∼ y^∗ and x ⊕ x′ ≈ y ⊕ y′ for all x, y, x′, y′ ∈ M, we call ∼ to be a hypercongruence (H-congruence) relation on M. (2) If x ∼ y and x′ ∼ y′ imply x^∗∼ y^∗ and x ⊕ x′ ≊ y ⊕ y′ for all x, y, x′, y′ ∈ M, we call ∼ to be a strong hypercongruence (SH-congruence) relation on M.

Proposition 2.9. [9] Let ∼ be an equivalence relation on a hyper MV-algebra M. Then (1) ∼ is a H-congruence relation on M if and only if for all x, y, z ∈ M (i) x ∼ y implies x^∗∼ y^∗, (ii) x ∼ y implies x ⊕ z ≈ y ⊕ z. (2) ∼ is a strong H-congruence relation on M if and only if for all x, y, z ∈ M (i) x ∼ y implies x^∗∼ y^∗, (ii) x ∼ y implies x ⊕ z ≊ y ⊕ z.

Proposition 2.10. [1] Let ∼ be an equivalence relation on a hyper MV-algebra M and A, B ⊆ M.

(1) if A ≈ B and B ≈ C, then A ≈ C,

(2) if A ≈ B and B ∼ C, then A ∼ C,

(3) if A ∼ B and B ≈ C, then A ∼ C,

(4) if A ≊ B and B ≊ C, then A ≊ C.

Definition 2.11. [9] Let ∼ be a H-congruence relation on a hyper MV-algebra M such that x^∗⊕ y ∼ {1} and y^∗⊕ x ∼ {1} imply that x ∼ y. Then ∼ is called a good H-congruence relation on M.

Theorem 2.12. [9] Let ∼ be a good H-congruence relation on a hyper MV-algebra M. Then ( M ∼ , ⊕ ¯ , ∗ ¯ , [ 0 ] ) is a hyper MV-algebra and it is called the quotient hyper MV-algebra of M respect to ∼, where for all [ x ] , [ y ] ∈ M ∼, [ x ] ⊕ ¯ [ y ] = { [ t ] | t ∈ x ⊕ y } and [ x ] ∗ ¯ = [ x ∗ ]. The relation “⪡” on M ∼ defined by [x] ⪡ [y] if and only if [ 1 ] ∈ [ x ] ∗ ¯ ⊕ ¯ [ y ]. In particular, if ∼ is a good and strong H-congruence relation on a hyper MV-algebra M, then ( M ∼ , ⊕ ¯ , ∗ ¯ , [ 0 ] ) is an MV-algebra.

3 States on hyper MV-algebras

In this section, the concepts of Riečan states and Boabach states on a hyper MV-algebra are introduced, and some properties of them are studied.

In a hyper MV-algebra M, we define x ⊙ y = (x^∗⊕ y^∗)^∗, x ⊖ y = x ⊙ y^∗, x ∨ y = (x ⊖ y) ⊕ y = (x ⊙ y^∗) ⊕ y, x ∧ y = x ⊙ (x^∗⊕ y) = x ⊖ (x ⊖ y) for all x, y ∈ M.

Definition 3.1. A function s : M → [0, 1] is called a Riečan state on a hyper MV-algebra M if it satisfies the following conditions, (1) s (1) =1, (2) s (x ⊕ y) = s (x) + s (y) whenever 0 ∈ x ⊙ y, where s (A) = inf {s (t) |t ∈ A} for all non-empty subset A ⊆ M.

Example 3.2. Let M = {0, b, 1}. Define a hyperoperation “ ⊕” on M by the following table:

Define a unary operation ∗ as follows: 0^∗= 1,  1^∗= 0,  b^∗= b. Then (M, ⊕ , ∗ , 0) is a hyper MV-algebra. And we have the following multiplication table:

Now, we define s (0) =0, s (1) =1, s (b) =0. Then s is a Riečan state on M.

Proposition 3.3. Let M be a hyper MV-algebra. Then we have the following: for all x, y ∈ M, (1) x ⪡ y iff 0 ∈ x ⊙ y^∗, (2) x ⊖ y ⪡ x and 0 ⊖ 0 =0, (3) (x ⊕ y) ⊖ y = x ⊖ (x ⊙ y).

Proof. (1) Note that x ⪡ y iff 1 ∈ x^∗⊕ y iff 1^∗∈ (x^∗⊕ y)^∗ iff 0 ∈ x ⊙ y^∗.

(2) We have that 0 ∈ 0 ⊙ y^∗⊆ (x ⊙ x^∗) ⊙ y^∗= (x ⊙ y^∗) ⊙ x^∗= (x ⊖ y) ⊙ x^∗. By (3), we obtain x ⊖ y ⪡ x. It follows that 0 ⊖ 0 ⪡0, which implies 0 ⊖ 0 =0.

(3) By (hMV4), we have (x ⊕ y)^∗⊕ y = (y^∗⊕ x^∗)^∗⊕ x^∗. Hence (x^∗⊙ y^∗) ⊕ y = (x ⊙ y) ⊕ x^∗. It follows that ((x^∗⊙ y^∗)^∗⊙ y^∗)^∗= (x ⊙ (x ⊙ y)^∗)^∗, which implies ((x ⊕ y) ⊙ y^∗)^∗= (x ⊙ (x ⊙ y)^∗)^∗, that is, ((x ⊕ y) ⊖ y)^∗= (x ⊖ (x ⊙ y))^∗. Therefore, (x ⊕ y) ⊖ y = x ⊖ (x ⊙ y).

Definition 3.4. Let M be a hyper MV-algebra. A map s : M → [0, 1] is called regular if s (x^∗) =1 - s (x) for all x ∈ M. Especially, if a Riečan state s on M is regular, we call it a regular Riečan state on M.

Note that the Riečan state s given in Example 3.2 is not regular. The following example shows the existence of regular Riečan states.

Example 3.5. Let M = {0, a, b, 1}. Define a hyperoperation “ ⊕” on M by the following table:

Define a unary operation ∗ as follows: 0^∗= 1,  1^∗= 0,  a^∗= b,  b^∗= a. Then (M, ⊕ , ∗ , 0) is a hyper MV-algebra. We can get the following multiplication table:

We define s (0) =0, s (a) =0, s (1) =1, s (b) =1. Then s is a regular Riečan state on M.

Proposition 3.6. Let s : M → [0, 1] be a regular map. Then s (A^∗) =1 - s^∗(A) for all A ⊆ M, where s^∗(A) = sup {s (x) |x ∈ A}.

Proof. Note that s (A^∗) = inf {s (x^∗) |x ∈ A} = inf{1 - s (x) |x ∈ A} =1 - sup {s (x) |x ∈ A} =1 - s^∗(A).□

In what follows, we give some basic properties of regular Riečan states on hyper MV-algebras.

Proposition 3.7. Let s be a regular Riečan state on M. Then the following hold: for any x, y ∈ M,

(1) s (0) =0,

(2) s (x ⊕ x^∗) =1,

(3) if x ⪡ y, then s^∗(y ⊖ x) = s (y) - s (x), s (x) ≤ s (y) and s (y^∗) ≤ s (x^∗),

(4) if x ⪡ y, then s (x ⊖ y) =0,

(5) s (x ⊖ y) + s (y) ≤ s^∗(x ∨ y) and s (y ⊖ x) + s (x) ≤ s^∗(x ∨ y),

(6) if x ⪡ y, s (y) ≤ s^∗(x ∨ y),

(7)s (x) ≤ s^∗(x ∧ y) + s^∗(x ⊖ y) and s (y) ≤ s^∗(x ∧ y) + s^∗(y ⊖ x).

Proof. (1) Note that s (0) = s (1^∗) =1 - s (1) =0.

(2) Since 0 ∈ x ⊙ x^∗, we have 1 = s (x) + (1 - s (x)) = s (x) + s (x^∗) = s (x ⊕ x^∗) and hence s (x ⊕ x^∗) =1.

(3) Assume that x ⪡ y, and so 0 ∈ x ⊙ y^∗. Then s (x) + s (y^∗) = s (x ⊕ y^∗) = s ((x^∗⊙ y)^∗) =1 - s^∗(x^∗⊙ y) =1 - s^∗(y ⊖ x) and hence s (x) + s (y^∗) =1 - s^∗(y ⊖ x). This shows that s (x) +1 - s (y) =1 - s^∗(y ⊖ x), that is, s (y) - s (x) = s^∗(y ⊖ x). By s (y) - s (x) = s^∗(y ⊖ x) ≥0, we obtain s (x) ≤ s (y). Moreover, s (x) ≤ s (y) implies 1 - s (y) ≤1 - s (x) and hence s (y^∗) ≤ s (x^∗).

(4) It follows from x ⪡ y that 0 ∈ x ⊙ y^∗= x ⊖ y. Therefore, s (x ⊖ y) =0.

(5) By Proposition 3.3, we have 0 ∈ y ⊙ y^∗, which implies that 0 ∈ x ⊙ y^∗⊙ y = (x ⊖ y) ⊙ y.Then there is t ∈ x ⊖ y such that 0 ∈ t ⊙ y. Hence s (x ⊖ y) + s (y) ≤ s (t) + s (y) = s (t ⊕ y) ≤ s^∗((x ⊖ y) ⊕ y) = s^∗(x ∨ y). Similarly, we can prove s (y ⊖ x) + s (x) ≤ s^∗(x ∨ y).

(6) Let x ⪡ y. We have 0 ∈ x ⊙ y^∗. By (5), s^∗(x ∨ y) ≥ s (x ⊖ y) + s (y) = s (x ⊙ y^∗) + s (y) =0 + s (y) = s (y).

(7) By Proposition 3.3, we have x ⊖ y ⪡ x. Then there exists t ∈ x ⊖ y such that t ⪡ x. Thus s^∗(x∧y) = s^∗(x ⊖ (x ⊖ y)) ≥ s^∗(x ⊖ t) = s (x) - s (t) by (3). It follows that s^∗(x ∧ y) ≥ s (x) - s (t) ≥ s (x) - s^∗(x ⊖ y), that is, s (x) ≤ s^∗(x ∧ y) + s^∗(x ⊖ y). Similarly, we can prove s (y) ≤ s^∗(x ∧ y) + s^∗(y ⊖ x).

Note that in an MV-algebra, the notions of regular Riečan states and Riečan states coincide. From the proof of Proposition 3.7, we can get some interesting properties of Riečan states in MV-algebras.

Corollary 3.8. [19] Let M be an MV-algebra and s be a Riečan state on M. Then the following hold: for any x, y ∈ M,

(1) s (0) =0,

(2) s (x^∗) =1 - s (x),

(3) if x ≤ y, then s (x) ≤ s (y),

(4) s (x) + s (y) = s (x ⊕ y) + s (x ⊙ y),

(5) s (x) + s (y) = s (x ∨ y) + s (x ∧ y),

(6) s (x ∨ y) ≤ s (x ⊕ y) ≤ s (x) + s (y).

Proposition 3.9. Let M be an MV-algebra and s be a Riečan state on M. Then the following hold: for any x, y ∈ M,

(1) if x ≤ y, then s (y ⊖ x) = s (y) - s (x),

(2) if x ≤ y, then s (x ⊖ y) =0,

(3) s (x ∨ y) = s (x ⊖ y) + s (y) = s (y ⊖ x) + s (x),

(4) if x ≤ y, then s (x ∨ y) = s (y),

(5) s (x ∧ y) = s (y) - s (y ⊖ x) = s (x) - s (x ⊖ y),

(6) if x ≤ y, then s (x ∧ y) = s (x).

Proof. (1) Note that if M is an MV-algebra and s is a Riečan state on M, then s^∗= s. By Proposition 3.7, we have that x ≤ y implies s (y ⊖ x) = s^∗(y ⊖ x) = s (y) - s (x) for all x, y ∈ L.

(2) Let x ≤ y, then 0 = x ⊙ y^∗= x ⊖ y and hence s (x ⊖ y) =0.

(3) By Proposition 3.3, we have y ⊙ y^∗= 0 and so 0 = x ⊙ y^∗⊙ y = (x ⊖ y) ⊙ y. Then s (x ⊖ y) + s (y) = s ((x ⊖ y) ⊕ y) = s (x ∨ y). Similarly, we can prove s (x ∨ y) = s (y ⊖ x) + s (x).

(4) Let x ≤ y. Then 0 = x ⊙ y^∗. By (3), s (x ∨ y) = s (x ⊖ y) + s (y) = s (x ⊙ y^∗) + s (y) =0+s (y) = s (y).

(5) Since x ⊖ y ≤ x, we have s (x ∧ y) = s (x ⊖ (x ⊖ y)) = s (x) - s (x ⊖ y) by (1). Similarly, we can obtain s (x ∧ y) = s (y) - s (y ⊖ x).

(6) Let x ≤ y. Then 0 = x ⊙ y^∗. By (5), s (x ∧ y) = s (x) - s (x ⊖ y) = s (x) - s (x ⊙ y^∗) = s (x) -0 = s (x).□

The following theorem gives an equivalent characterization of regular Riečan states in hyper MV-algebras.

Theorem 3.10. Let M be a hyper MV-algebra and a map s : M → [0, 1] be regular. Then the following are equivalent: (1) s is a regular Riečan state on M, (2) x ⪡ y implies s^∗(y ⊖ x) = s (y) - s (x) for all x, y ∈ M.

Proof. (1) ⇒ (2) It follows from Proposition 3.7(3).

(2) ⇒ (1) Let (2) hold. By Proposition 3.3(4), we have 0 ⊖ 0 =0 and hence s (0) = s^∗(0) = s^∗(0 ⊖0) = s (0) - s (0) =0. Moreover, s (1) = s (0^∗) =1-s (0) =1 - 0 =1 and so s (1) =1. Let 0 ∈ x ⊙ y. Then x ⪡ y^∗ by Proposition 3.6(3). By (2) we have s^∗(y^∗⊖ x) = s (y^∗) - s (x) and hence s^∗(y^∗⊙ x^∗) = s^∗((x ⊕ y)^∗) =1 - s (x ⊕ y) by Proposition 3.6. It follows that s (y^∗) - s (x) =1 - s (x ⊕ y) and hence 1 - s (y) - s (x) =1 - s (x ⊕ y). This shows s (x ⊕ y) = s (x) + s (y). Combining the above arguments, we get that s is a regular Riečan state on M.

In what follows, we introduce Bosbach states on a hyper MV-algebra and study some properties of them.

Definition 3.11. A function s : M → [0, 1] is called a Bosbach state on M if it satisfies the following conditions, (1) s (1) =1, (2) s (x) + s (y) = s (x ⊕ y) + s (x ⊙ y), for any x, y∈M, where s (A) = inf {s (t) |t ∈ A} for any non-empty subset A ⊆ M.

Note that the Riečan state s given in Example 3.5. is also a Bosbach state.

The following example shows that not every hyper MV-algebra has a Bosbach state.

Example 3.12. Let M = {0, a, b, 1}. Define a hyperoperation “ ⊕″ and a unary operation “ ∗″ on M as follows:

We can obtain a hyperoperation “ ⊙^″ as follows:

Then M is a hyper MV-algebra. Let s (0) =0, s (a) = α, s (b) = β, s (1) =1. Taking x = 1, y = b in s (x) + s (y) = s (x ⊕ y) + s (x ⊙ y), we get 1 + β = 0 + β. It is a contradiction. Therefore, M does not admit any Bosbach state.

Theorem 3.13. Let s be a Bosbach state on M. Then s is a Riečan state on M.

Proof. Let s be a Bosbach state on M, x, y ∈ Mand 0 ∈ x ⊙ y. Then s (x) + s (y) = s (x ⊕ y) + s (x ⊙ y) = s (x ⊕ y) +0 = s (x ⊕ y). Moreover, since0 ∈ 0 ⊙1, we have s (0) + s (1) = s (0 ⊕1) + s (0 ⊙1) = s (0 ⊕1) ≤1 and hence s (0) =0. Therefore, s is a Riečan state on M.

The next example shows that the converse of Theorem 3.13. may not be true.

Example 3.14. Consider the hyper MV-algebra Mand the Riečan state s on M given in Example 3.2. We can see that s is not a Bosbach state. In fact, if s is a Bosbach state, then s (b) + s (1) =1 = s (b ⊕ 1) + s (b ⊙ 1) = s ({0, b, 1}) + s ({b, 1}) =0 + 0 =0, which is a contradiction.

In the following, we give a characterization of regular Bosbach states on M.

Theorem 3.15. Let a map s : M → [0, 1] be regular. Then the following are equivalent: (1) s^∗(y ⊖ x) - s (x ⊖ y) = s (y) - s (x) for all x, y∈M, (2) s is a regular Bosbach state on M.

Proof. (1)⇒(2) Let (1) hold. Taking x = y = 0 in s^∗(y ⊖ x) - s (x ⊖ y) = s (y) - s (x), we get s^∗(0 ⊖0) - s (0 ⊖0) = s (0) - s (0) and hence s^∗(0 ⊖0) -0 = 0. By Proposition 3.3(4), 0 ⊖ 0 =0 and so s^∗(0) =0. Since s (x) ≤ s^∗(0) =0, we have s (0) =0. Note that s is regular, we get s (1) =1. Moreover, we have s^∗(y^∗⊖ x) - s (x ⊖ y^∗) = s (y^∗) - s (x) for all x, y ∈ M by (1). It follows that s^∗(y^∗⊖ x) - s (x ⊙ y) = (1 - s (y)) - s (x). Since s^∗(y^∗⊖ x) = s^∗(y^∗⊙ x^∗) = s^∗((x ⊕ y)^∗) =1 - s (x ⊕ y) by Proposition 3.6., we have 1 - s (x ⊕ y) - s (x ⊙ y) = (1 - s (y)) - s (x), that is, s (x ⊕ y) + s (x ⊙ y) = s (x) + s (y). Combining the above arguments, we have that s is a regular Bosbach state on M.

(2)⇒(1) Let (2) hold. Then s (x) + s (y^∗) = s (x ⊕ y^∗) + s (x ⊙ y^∗) = s ((x^∗⊙ y)^∗) + s (x ⊖ y) =1 - s^∗(y ⊖ x) + s (x ⊖ y) and hence s (x) +1 - s (y) =1 - s^∗(y ⊖ x) + s (x ⊖ y). This shows that s^∗(y ⊖ x) - s (x ⊖ y) = s (y) - s (x).□

4 States and quotient hyper MV-algebras

In this section, using a Bosbach state on a hyper MV-algebra, we construct and study a quotient hyper MV-algebra.

Definition 4.1. Let s be a Bosbach state and ∼ be a good H-congruence relation on M. Then s is called to be ∼ - compatible if x ∼ y ⇔ s (x) = s (y) for all x, y ∈ M.

Let μ be a fuzzy subset of M. μ is said to have sup-property (inf-property), if for all subset A of M, sup {μ (t) |t ∈ A} = α (inf {μ (t) |t ∈ A} = α) implies μ (t) = α for some t ∈ A.

Lemma 4.2. Let ∼ be a good H-congruence relation and s be a ∼ - compatible regular Bosbach state on M. Then in the bounded quotient hyper MV-algebra ( M ∼ , ⊕ ¯ , ∗ ¯ , I 0 , I 1 ), where I _x  = [x] _∼, I x ⊕ ¯ I y = I x ⊕ y and I x ∗ ¯ = I x ∗ the following hold: (1) if I _x  ⪡ I _y , then s (x ⊖ y) =0, (2) if s has sup-property, then s (x ⊖ y) =0 implies I _x  ⪡ I _y , (3) I _x  = I _y implies s (x) = s (y).

Proof. (1) Note that I _x  ⪡ I _y if and only if I₁∈I x ∗ ⊕ ¯ I y, then there exists z ∈ x^∗⊕ y such thatz ∼ 1. By Definition ref 4.1., we get that s (z) = s (1) =1. It follows from Proposition 3.6. that s (x ⊖ y) = s (x ⊙ y^∗) = s ((x^∗⊕ y)^∗) =1 - s^∗(x^∗⊕ y) =1-sup {s (t) |t ∈ x^∗⊕ y} =0.

(2) Suppose that s (x ⊖ y) =0. Then s ((x^∗⊕ y)^∗)= 1 - s^∗(x^∗⊕ y) =1 - sup {s (t) |t ∈ x^∗⊕ y} =0. Thus there is z ∈ x^∗⊕ y such that s (z) =1 and hence z ∼ 1. Moreover, we get I _z  = I₁, i.e., there is I _z  ∈ I _{x ^∗}  ⊕ I _y such that I _z  = I₁. This means that I _x  ⪡ I _y .

(3) It’s clear that I _x  = I _y iff x ∼ y. By Definition 4.1., we have that I _x  = I _y implies s (x) = s (y).□

Theorem 4.3. Let ∼ be a good H-congruence relation and s be a ∼ - compatible regular Bosbach state on M. Define a map s ˆ : M ∼ → [ 0 , 1 ] by s ˆ ( I x ) = s ( x ) and s ˆ ( { I x | x ∈ A } ) = inf { s ˆ ( I t ) | t ∈ A }, for any x ∈ M and A ⊆ M. Then s ˆ is a Bosbach state on M ∼.

Proof. By Lemma 4.2., we have that s ˆ is well-defined. Clearly, s ˆ ( I 0 ) = s ( 0 ) = 0 and s ˆ ( I 1 ) = s ( 1 ) = 1. Since s ˆ ( I x ⊕ ¯ I y ) = inf { s ˆ ( I t ) | t ∈ x ⊕ y } = inf { s ( t ) | t ∈ x⊕y} = s (x ⊕ y) and s ˆ ( I x ⊙ ¯ I y ) = inf { s ˆ ( I t ) | t ∈ x ⊙y} = inf {s (t) |t ∈ x ⊙ y} = s (x ⊙ y), then s ˆ ( I x ⊕ ¯I y ) + s ˆ ( I x ⊙ ¯ I y ) = s ( x ⊕ y ) + s ( x ⊙ y ) = s ( x ) + s ( y )= s (I _x ) + s (I _y ). Therefore, s ˆ is a Bosbach state on M ∼.□

Definition 4.4. [8] A non-empty subset F of M is called a hyper MV-filter of M if it satisfies the following conditions:

(hF1) 1 ∈ F,

(hF2) if F ⪡ x^∗⊕ y and x ∈ F, then y ∈ F for all x, y ∈ M.

A hyper MV-filter F of M is called a prime hyper MV-filter of M if x ⊕ y ⊆ F implies x ∈ F or y ∈ F for all x, y ∈ M.

Theorem 4.5. Let s be a regular Bosbach state on M. Define K = Ker (s) = {a ∈ M|s (a) =1}, which is called the kernel of s. Then K is a hyper MV-filter of M.

Proof. Clearly, 1 ∈ K. Let K ⪡ x^∗⊕ y and x ∈ K. Then s (x) =1 and there is a ∈ K, b ∈ x^∗⊕ y such that a ⪡ b. By Proposition 3.7.(3), s (a) ≤ s (b) andhence 1 = s (a) ≤ s (b) ≤ s^∗(x^∗⊕ y) = s^∗((x ⊙ y^∗)^∗) = s^∗((x ⊖ y)^∗) =1 - s (x ⊖ y) by Proposition 3.6. Hence s (x ⊖ y) ≤0 and thus s (x ⊖ y) =0. Since s is a Bosbach state on M, we have s^∗(x ⊕ y) - s (x ⊖ y) = s (y) - s (x) and hence s^∗(x ⊕ y) = s (y) - s (x). Note that s (x) =1, we have s (y) = s^∗(x ⊕ y) + s (x) ≥1, or s (y) =1, which shows that y ∈ K. Therefore, K is a hyper MV-filter of M.□

Let s be a Bosbach state on M. Denote K = {t ∈ M ∣ s (t) =1}. Define a binary relation ∼ _K as follows x ∼  _K y iff x^∗⊕ y ⊆ K and y^∗⊕ x ⊆ K for all x, y∈M.

Definition 4.6. Let s be a Bosbach state on M. s is called a strongly isotone Bosbach state (briefly si-Bosbach state) if max {s (x) , m (y)} ≤ s (x ⊕ y) for all x, y ∈ M.

Example 4.7. Consider the hyper MV-algebra M and the regular Bosbach state s on M given in Example 3.5. Then s is a regular si-Bosbach state on M.

In what follows, we give some properties of K in hyper MV-algebras.

Proposition 4.8. Let s be a regular Bosbach state on M. Then the following hold: (1) for all k₁, k₂ ∈ K, we have k₁ ⊙ k₂ ⊆ K. Moreover, for all subsets A, B of K, A ⊙ B ⊆ K, (2) for all A ⊆ M and B ⊆ K, A ⊕ B^∗⊆ K implies A ⊆ K, (3) if s is a si-Bosbach state, then for all k ∈ K and x ∈ M, we have k ⊕ x ⊆ K. Moreover, for all subsets A ⊆ K and B ⊆ M, A ⊕ B ⊆ K.

Proof. (1) Let k₁, k₂ ∈ K. By Theorem 3.15., we have s ( k 1 ⊙ k 2 ) = s ( k 1 ⊖ k 2 ∗ ) = s ∗ ( k 2 ∗ ⊖ k 1 ) - s ( k 2 ∗ ) + s ( k 1 ) = s ∗ ( k 2 ∗ ⊖ k 1 ) + 1 and hence s (k₁ ⊙ k₂) =1. This shows that k₁ ⊙ k₂ ⊆ K. The second part is clear.

(2) Let A ⊆ M, B ⊆ K and A ⊕ B^∗⊆ K. Then for any a ∈ A, b ∈ B, a ⊕ b^∗⊆ K and hence s (a ⊕ b^∗) =1. Thus 1 = s ((a^∗⊙ b)^∗) =1 - s^∗(a^∗⊙ b), which implies that s^∗(a^∗⊙ b) = s^∗(b ⊖ a) =0. By Theorem s^∗(b ⊖ a) - s (a ⊖ b) = s (b) - s (a) =1 - s (a). Hence s (a) =1 + s (a ⊖ b). Thus s (a) =1 or a ∈ K. This shows that A ⊆ K.

(3) Let k ∈ K and x ∈ M. Then s (k ⊕ x) ≥ max {s (k) , s (x)} =1 since s is a si-Bosbach state. Hence s (k ⊕ x) =1. This shows that k ⊕ x ⊆ K. The second part is clear.□

Proposition 4.9. If s is a regular sp-Bosbach state on M, then we have the following: (1) ∼ _K is an equivalent relation on M, (2) x ∼  _K y iff x^∗∼  _K y^∗.

Proof. (1) By Proposition 3.7, we have s (x^∗⊕ x) =1 and hence x^∗⊕ x ⊆ K. That is, x ∼  _K x. The symmetry is clear.

Next, we will prove that the transitivity holds. Assume that x ∼  _K y and y ∼  _K z. Then x ⊕ y^∗⊆ K and y ⊕ z^∗⊆ K. By Propositions 3.7(2) and 4.8(3), we obtain y ⊕ (y ⊕ x^∗)^∗⊕ y^∗⊕ (y^∗⊕ z)^∗⊆ K. Using (hMV6), we get y ⊕ (y ⊕ x^∗)^∗⊕ y^∗⊕ (y^∗⊕ z)^∗ = x ⊕ (x ⊕ y^∗)^∗⊕ z^∗⊕ (z^∗⊕ y)^∗ = (x ⊕ z^∗) ⊕ (x ⊕ y^∗)^∗⊕ (y ⊕ z^∗)^∗ = (x ⊕ z^∗) ⊕ ((x ⊕ y^∗) ⊙ (y ⊕ z^∗))^∗⊆ K. By (1) and (2) of Proposition 4.6, we have x ⊕ z^∗⊆ K. Similarly, we can prove z ⊕ x^∗⊆ K. It follows that x ∼  _K z.

(2) Note that x ∼  _K y iff x^∗⊕ y ⊆ K and y^∗⊕ x ⊆ K iff x^∗⊕ (y^∗)^∗⊆ K and y^∗⊕ (x^∗)^∗⊆ K iff x^∗∼  _K y^∗.□

Theorem 4.10. Let s be a regular si-Bosbach state on M. Then we have the following: (1) ∼ _K is a strong H-congruence relation on M, (2) ∼ _K is good, (3) x ∼  _K y iff s^∗(x ⊖ y) = s^∗(y ⊖ x) =0 for all x, y ∈ M, (4) s is ∼ _K -compatible.

Proof. (1) Assume x ∼  _K y. Then x^∗⊕ y ⊆ K and y^∗⊕ x ⊆ K. For all u ∈ x ⊕ z and v ∈ y ⊕ z, we have u^∗⊕ v ⊆ (x ⊕ z)^∗⊕ (y ⊕ z) = (x^∗⊕ y) ⊕ x ⊙ z ⊆ K by Proposition 4.8(3), and hence u^∗⊕ v ⊆ K. Similarly, we can prove v^∗⊕ u ⊆ K. It follows that (x ⊕ z) ≊ (y ⊕ z). Combining Propositions 2.9 and 4.9, we get that s is a strong H-congruence relation on M.

(2) Let x^∗⊕ y ∼  _K  {1} and y^∗⊕ x ∼  _K  {1}. It follows from (1) that (x^∗⊕ y)^∗∼  _K  {0} and (y^∗⊕ x)^∗∼  _K  {0}. Since ∼ _K is a strong H-congruence relation on M, we obtain (x^∗⊕ y)^∗⊕ y ≊  _K 0 ⊕ y and (y^∗⊕ x)^∗⊕ x ≊  _K 0 ⊕ x. By Definition 2.2 (hMV4), we have (x^∗⊕ y)^∗⊕ y = (y^∗⊕ x)^∗⊕ x. It follows from Proposition 2.10 (4) that 0 ⊕ y ≊  _K 0 ⊕ x. Notice y ∈ 0 ⊕ y and x ∈ 0 ⊕ x, we have x ∼  _K y. Therefore, ∼ _K is good.

(3) Let x ∼  _K y. Then x^∗⊕ y ⊆ K and hence s (x^∗⊕ y) =1. Thus 1 = s (x^∗⊕ y) = s ((x ⊖ y)^∗) =1 - s^∗(x ⊖ y) by Proposition 3.6 Hence we get s^∗(x ⊖ y) =0. Similarly, we can get s^∗(y ⊖ x) =0. Conversely, let s^∗(x ⊖ y) = s^∗(y ⊖ x) =0. Then s (x^∗⊕ y) = s ((x ⊖ y)^∗) =1 - s^∗(x ⊖ y) =1. Therefore, x^∗⊕ y ⊆ K. Similarly, we can prove that y^∗⊕ x ⊆ K. This shows that x ∼  _K y.

(4) Let x ∼  _K y. By (3), we have s^∗(x ⊖ y) =s^∗(y ⊖ x) =0. It follows that s (x ⊖ y) =0. More-over, by Theorem 3.15, we have s^∗(y ⊖ x) = s^∗(y ⊖ x) - s (x ⊖ y) = s (y) - s (x) and thus s (y) - s (x) =0. Therefore, s (x) = s (y).□

In this section, up to now, we assume that M is a hyper MV-algebra, s is a regular si-Bosbach state having the sup property, and K = ker (s) unless specially assigned.

Combining Theorem 2.11, Theorem 4.3 and Theorem 4.10, we have the following corollary.

Corollary 4.11. Let s be a regular sp-Bosbach state on M. Then M ∼ K is an MV-algebra and s ¯ : M ∼ K → [ 0 , 1 ] defined by s ¯ ( [ x ] ∼ K ) = s ( x ), is a Bosbach state on M ∼ K.

We denote M ∼ K by M/K, [x] _{∼ K} by K _x and s ¯ by s _K .

Proposition 4.12. In M/K, s _K has the following properties:

(1) s K ( K x ⊕ ¯ K y ) = s ( x ⊕ y ),

(2) s K ( K x * ¯ ) = 1 — s ( x ) ,

(3)s K ( K x ⊖ ¯ K y ) = s ( x ⊖ y ),

(4) s K ( K x ⊙ ¯ K y ) = s ( x ⊙ y ),

(5)s K ( K x ∧ ¯ K y ) = s ( x ∧ y ),

(6) s K ( K x ∨ ¯ K y ) = s ( x ∨ y ).

Proof. It is straightforward.□

Definition 4.13. A non-empty subset I of M is called a hyper MV-ideal of M, if it satisfies the following conditions: (hF1) 0 ∈ I, (hF2) if y ⊖ x ⪡ I and x ∈ I, then y ∈ I for all x, y ∈ M. Moreover, I is called a prime hyper MV-ideal of M if x ⊙ y ⊆ I implies x ∈ I or y ∈ I for all x, y ∈ M.

Proposition 4.14. Let M be a hyper MV-algebra and s be a regular Bosbach state of M. Define I₀ = {t ∈ M ∣ s (t) =0}. Then I₀ is a hyper MV-ideal of M.

Proof. Clearly, 0 ∈ I₀. Let y ⊖ x ⪡ I₀ and x ∈ I₀. Then s (x) =0 and there is t ∈ y ⊖ x and i ∈ I₀ such that t ⪡ i. By Proposition 3.7(3) we have s (t) ≤ s (i) =0 and thus s (t) =0. Hence s (y ⊖ x) =0. By Theorem 3.15, s^∗(x ⊖ y) - s (y ⊖ x) = s (x) - s (y) or s^∗(x ⊖ y) = - s (y). It follows that s^∗(x ⊖ y) = s (y) =0 and hence y ∈ I₀.□

Theorem 4.15. Let M be a hyper MV-algebra and s be a regular si-Bosbach state. Then the following are equivalent: (1) M/K is an MV-algebra containing two elements, that is, M/K = {K₀, K₁}, (2) K is a prime hyper MV-filter of M, (3) I₀ is a prime hyper MV-ideal of M.

Proof. (1)⇒(2) Let x ⊕ y ⊆ K. Then s (x ⊕ y) =1.If x ∉ K, then x ∈ K₀ by (1), that is, s (x) =0. Hence 1 + s (x ⊙ y) = s (x ⊕ y) + s (x ⊙ y) = s (x) + s (y) = s (y). Therefore, s (y) =1 and hence y ∈ K. This shows that K is a prime hyper MV-filter of M.

(2)⇒(3) Let x ⊙ y ⊆ I₀. Then s^∗(x ⊙ y) =0 and hence s^∗((x^∗⊕ y^∗)^∗) =0. Therefore, 1 - s (x^∗⊕ y^∗) =0. This shows that s (x^∗⊕ y^∗) =1. Thus x^∗⊕ y^∗⊆ K. By (2), K is a prime hyper MV-filter and thus x^∗∈ K or y^∗∈ K. It follows that x ∈ I₀ or y ∈ I₀. This proves that (3) is true.

(3)⇒(1) By Proposition 3.7 s (x^∗⊕ x) =1 andhence s^∗(x ⊙ x^∗) = s^∗((x^∗⊕ x)^∗) =1 - s (x^∗⊕ x) =0, that is, x ⊙ x^∗⊆ I₀. By (3), x ∈ I₀ or x^∗∈ I₀. Thus x ∈ I₀ or x ∈ K₁. Note that I₀ = K₀, so x ∈ K₀ or x ∈ K₁. This shows that M/K = {K₀, K₁}.

5 State morphisms on hyper MV-algebras

In the section, we introduce state morphisms on hyper MV-algebras. Moreover, we discuss some relationships between state morphisms and Bosbach states.

The standard MV-algebra is the MV-algebra [0, 1]  _MV  = ([0, 1] , ⊕  _s , ¬  _s , 0), where r ⊕  _s s = min{r + s, 1}, ¬ _s r = 1 - r. And the derived operations are as follows r ⊖ s s = max { r - s , 0 } , r ⊙ s s = max { r + s - 1 , 0 }, r →  _s s = min {1 - r + s, 1}, r ∨  _s s = max {r, s} , r ∧  _s s = min {r, s}.

Definition 5.1. (M, ⊕ , ∗ , 0) be a hyper MV-algebra and m : M → [0, 1] be a map satisfyingm (x ⊕ y) = m (x) ⊕  _s m (y), m (x ⊙ y) = m (x) ⊙  _s m (y) and m (x^∗) = ¬  _s m (x), where m (A) = inf{m (t) ∣ t ∈ A} for all subset A of M. Then m is said to be a state morphism on M.

Proposition 5.2. Let m be a state morphism on a hyper MV-algebra (M, ⊕ , ∗ , 0). Then for all x, y ∈ M, we have (1) m (0) =0, m (1) =1, (2) m ( x ⊖ y ) = m ( x ) ⊖ s m ( y ).

Proof. (1) By Proposition 3.3, 0 ⊖ 0 =0 and so wehave m (0) = m (0 ⊖0) = m (0 ⊙0^∗) = m (0) ⊙  _s m(0^∗) = max {m (0) + m (0^∗) -1, 0} = max {m (0) + (1 - m (0)) -1, 0} =0. Moreover, m (1) = m (0^∗) = ¬  _s m (0) =1 - m (0) =1.

(2) Note that m (x ⊖ y) = m (x ⊙ y^∗) = m (x)⊙ _s m (y^∗) = max {m (x) + m (y^∗) -1, 0} = max {m(x) + (1 - m (y)) -1, 0} = max {m (x) - m (y)) , 0}= m ( x ) ⊖ s m ( y ).

The following theorem gives a relation between state morphisms and Bosbach states on hyper MV-algebras.

Theorem 5.3. Let m be a state morphism on a hyper MV-algebra (M, ⊕ , ∗ , 0). Then m is a regular si-Bosbach state on M.

Proof. By Proposition 5.2, we have m (1) =1. Moreover, m (x ⊕ y) + m (x ⊙ y) = m (x) ⊕  _s m (y)+m (x) ⊙  _s m (y) = min {m (x) + m (y) , 1} + max{m (x) + m (y) -1, 0}. If m (x) + m (y) ≤1 then m (x ⊕ y) + m (x ⊙ y) = min {m (x) + m (y) , 1} + max {m (x) + m (y) -1, 0} = m (x) + m (y) +0 = m (x) + m (y). If m (x) + m (y) >1, then m (x⊕y) + m (x ⊙ y) = min {m (x) + m (y) , 1} + max{m (x) + m (y) -1, 0} =1 + m (x) + m (y) -1 = m (x) + m (y). By Definition 3.11, we have that mis a Bosbach state on M. Moreover, let x, y ∈ M, we have that m (x ⊕ y) = m (x) ⊕  _s m (y) = min {m(x) + m (y) , 1} ≥ max {s (x) , m (y)}, that is, m is strongly preserve order. Clearly, m (x^∗) =1 - m (x). Therefore, m is a regular si-Bosbach state on M.□

Let M be a hyper MV-algebra and m be a state morphism on M. Define K _m  = ker (m) = {a ∈ M ∣ m (a) =1}, which is called the kernel of m, and I _m  = {t ∈ M ∣ m (t) =0}, which is called the cokernel of m.

By Theorem 4.5, Proposition 4.14 and Theorem 5.3, we have the following corollary.

Corollary 5.4. Let M be a hyper MV-algebra and m be a state morphism on M. Then (1) K _m is a hyper MV-filter of M, (2) I _m is a hyper MV-ideal of M.

Let M be a hyper MV-algebra and m be a state morphism on M. Define a binary relation ∼ _m as follows x ∼  _m y iff x^∗⊕ y ⊆ K _m and y^∗⊕ x ⊆ K _m for all x, y ∈ M.

By Corollary 4.11 and Theorem 5.3, we have the following result.

Proposition 5.5. Let M be a hyper MV-algebra and m be a state morphism on M. Then M ∼ m is an MV-algebra and s m : M ∼ m → [ 0 , 1 ] defined by s _m  ([x] _{∼ m} ) = m (x), is a Bosbach state on M ∼ m.

Proposition 5.6. Let M be a hyper MV-algebra and m be a state morphism on M. Then the following are equivalent: (1) x ∼  _m y, (2) m (x^∗⊕ y) =1 and m (y^∗⊕ x) =1, (3) m (x) = m (y).

Proof. (1)⇒(2) Let x ∼  _m y. Then x^∗⊕ y ⊆ K _m and y^∗⊕ x ⊆ K _m . Hence m (x^∗⊕ y) = inf {m (t) ∣ t ∈ x^∗⊕ y} =1. Similarly, we can prove m (y^∗⊕ x) =1.

(2)⇒(3) Let (2) hold. Assume that (3) is not true. Then m (x) ≠ m (y). Without loss of generality, we assume m (x) > m (y). Hence m (x^∗⊕y) = min {m (x^∗) + m (y) , 1} = min {1 - (m (x) - m (y)) , 1)} =1 - (m (x) - m (y)) <1, a contradiction. Therefore, m (x) = m (y).

(3)⇒(1) Let (3) hold. Then m (x^∗⊕ y) = m (x^∗) ⊕  _s m (y) = min {m (x^∗) + m (y) , 1} = min {1-m (x) + m (y) , 1} =1 and hence x^∗⊕ y ⊆ K _m . Similarly, we can prove y^∗⊕ x ⊆ K _m . Therefore, x ∼  _m y.□

Definition 5.7. [9] Let M be an MV-algebra and S be a non-empty subset of M. If S is an MV-algebra respect to the operation “ ⊕” and the unary operation “ ∗” on M, we say that S is a subalgebra of M.

Proposition 5.8. [9] Let (M, ⊕ , ∗ , 0) be an MV-algebra and S be a non-empty subset of M. Then S is a subalgebra of M if and only if 0 ∈ S and x^∗⊕ y ⊆ S for all x, y ∈ S.

Proposition 5.9. Let M be a hyper MV-algebra and m be a state morphism having inf-property on M. Then m (M) is a subalgebra of [0, 1]  _MV , where m (M) = {m (t) ∣ t ∈ M}.

Proof. Clearly, m (M)≠ ∅ and 0 ∈ m (M). Let x, y ∈ m (M). Then x = m (s) , y = m (t) for some s, t ∈ M. Thus ¬ _s x ⊕  _s y = ¬  _s m (s) ⊕  _s m (t) = m (s^∗) ⊕  _s m (t) = m (s^∗⊕ t). Considering that m has inf-property, we have that there exists t₀ ∈ s^∗⊕ t ⊆ M such that m (s^∗⊕ t) = m (t₀). This shows that ¬ _s x ⊕  _s y = m (t₀) , t₀ ∈ M and hence ¬ _s x ⊕  _s y ∈ m (M). By Proposition 5.8, we have that m (M) is a subalgebra of [0, 1]  _MV .□

Definition 5.10. [8] Let (M₁, ⊕ ₁, ∗ ₁, 0₁) and (M₂, ⊕ ₂, ∗ ₂, 0₂) be two MV-algebras. A map f : M₁ → M₂ is said to be a homomorphism if it satisfies: (1) f (0₁) =0₂, (2) f (x ⊕ ₁y) = f (x) ⊕ ₂f (y), (3) f (x^∗₁) = (f (x))^∗₂.

Moreover, if f is one to one (or onto), then f is said to be a monomorphism (or epimorphism). In particular, if f is both one to one and onto, then f is said to be an isomorphism, denoted by M₁ ≅ M₂.

Theorem 5.11. Let (M, ⊕ , ∗ , 0) be a hyper MV-algebra and m be a state morphism having inf-property on M. Then M ∼ m ≅ m ( M ).

Proof. Consider the map f : M ∼ m → m ( M ) defined by f ([x] _{∼ m} ) = m (x). By Proposition 5.5,f is well defined and bijective. Moreover, for all [ x ] ∼ m , [ y ] ∼ m ∈ M ∼ m, f ( [ x ] ∼ m ⊕ ¯ [ y ] ∼ m ) = f ( [ x ⊕y] _{∼ m} ) = m (x ⊕ y) = m (x) ⊕  _s m (y) = f ([x] _{∼ m} ) ⊕  _s f ([y] _{∼ m} ), f ( [ x ] ∼ m ∗ ¯ ) = f ( [ x ∗ ] ∼ m ) = m ( x ∗ ) = ¬ s m(x) = ¬  _s f ([x] _{∼ m} ) and f ([0] _{∼ m} ) = m (0) =0. Therefore, f is an isomorphism, that is, M ∼ m ≅m (M).□

6 Conclusion

In this paper, we investigate Riečan states and Bosbach states on a hyper MV-algebra and give some characterizations of regular Riečan states and regular Bosbach states on a hyper MV-algebra. Moreover, we construct and discuss a quotient hyper MV-algebra via a Bosbach state. In particular, we obtain some equivalent conditions under which a quotient hyper MV-algebra becomes a Boolean algebra. Finally, we study state morphisms on hyper MV-algebras, and obtain some relationships between state morphisms and Bosbach states. Our further work on this topic will focus on state operators on hyperMV-algebras.