The generalized Bosbach states on EQ-algebras

Abstract

The main goal of this paper is to discuss the generalized Bosbach states on EQ-algebras. The notions of generalized Bosbach states of type I and generalized Bosbach states of type II (resp. III, IV, V, VI and VII) are proposed. The equivalent characterizations of these generalized Bosbach states on special EQ-algebras are given. The properties of these generalized Bosbach states and the relationship between them are investigated.

Keywords

EQ-algebra generailized Bosbach state of type I generailized Bosbach state of type II (resp. III,IV,V,VI,VII)

1 Introduction

In [30], Nov $\overset{´}{a}$ k introduced the notion of EQ-algebras aiming to be the algebraric structure of truth values for a higher-order fuzzy logic [29]. There are there binary operations (meet, multiplication and fuzzy equlity) and a top element on EQ-algebras. Its implication is derived from the fuzzy equality and isn’t closely linked with the multiplication and so, EQ-algebras are the generalizations of commutative residuated lattices.

In order to measure the average truth-value of propositions in Łukasiewicz logic, Mundici [28] introduced the notion of states on MV-algebras, which generalizes the probability measures on Boolean algebras. It is possible that random experiments can follow the rules of other logical system. Based on this, the notion of states has been extended to many logical algebras such as BL-algebras [31], R0-algebras [23], MTL-algebras [22], Rℓ-monoids [11], semi-divisible residuated lattices [32], residuated lattices [3, 21], their non-commutative cases [8–10 , 18] and EQ-algebras [1]. These generalization mainly concentrated on the study of Rie $\overset{ˇ}{c}$ an states and Bosbach states.

Totally different from the research direction above, Flaminio and Montagna [17] used the same MV-algrbas as the domain and codomain of a state and presented the notion of internal states on MV-algebras, which preserved the usual properties of states in the original states of Mundici. For detailed investigations, see [6 , 14]. Afterwards, the notions of internal states were introduced into other algebraic structures [2 , 20].

Recently, Georgescu, Mureşan and Ciungu [5, 19] noted that Bosbach states could be defined by the canonical structure of the standard MV-algebras [0, 1], replaced the codomain [0, 1] of Rie $\overset{ˇ}{c}$ an/Bosbach states with an arbitrary residuated lattice and proposed the notions of generalized Rie $\overset{ˇ}{c}$ an/Bosbach states on residuated lattices. They discussed the properties of generalized Bosbach and generalized Rie $\overset{ˇ}{c}$ an states and obtained a lot of meaningful results. They also left some open problems. In [36], Zhou and Zhao sloved there open problems about generalized Bosbach states on residuated lattices. They also introduced the notion of relative negation and discussed many issues such as Glivenko property, semidivisibility, generalized Rie $\overset{ˇ}{c}$ an state of residuated lattices in the context of relative negations. Ma [26] observed that there were some different definition ways of Bosbach states on the standard MV-algebras [0, 1], rewrote the equations of Bosbach states and defined new types of generalized Bosbach states on residuated lattices.

The paper is motivated by the following considerations: (1) Whether these generalized Bosbach states exist on EQ-algebras; (2) which properties do these generalized Bosbach states possess in the case of existence? (3) whether these generailzed Bosbach states preserve the corresponding properties on residuated lattices. Based on this, we put forward the notion of generalized Bosbach states of type I and generalized Bosbach states of type II (resp. III, IV, V, VI, VII) on EQ-algebras and dedicate to the algebraric study of these generalized Bosbach states.

2 Preliminaries

Definition 2.1. [30] An algebra (E, ∧ , ⊗ , ∼ , 1) of type (2,2,2,0) is called an EQ-algebra if it satisfies the following conditions: for all a, b, c, d ∈ E,

(E1) (E, ∧ , 1) is a ∧-semilattice with top element 1;

(E2) (E, ⊗ , 1) is a commutative monoid and ⊗ is isotone in both arguments w.r.t. a ≤ b;

(E3) a ∼ a = 1;

(E4) ((a ∧ b) ∼ c) ⊗ (d ∼ a) ≤ c ∼ (d ∧ b);

(E5) (a ∼ b) ⊗ (c ∼ d) ≤ (a ∼ c) ∼ (b ∼ d);

(E6) (a ∧ b ∧ c) ∼ a ≤ a ∧ b ∼ a;

(E7) a ⊗ b ≤ a ∼ b.

Clearly, (E, ≤) is a partially ordered set, where a ≤ b iff a ∧ b = a. For all a, b ∈ E, we put a → b = (a ∧ b) ∼ a, $\tilde{a} = a \sim 1$ and d_E (a, b) = (a → b) ∧ (b → a).

In what follows, E is an EQ-algebra unless otherwise specified. We appoint that E has the bottom element 0 and denote ¬a = a ∼ 0.

For convenience, for all a ∈ E and n ∈ N, we define if n = 0, then a⁰ = 1; if n ≥ 1, then aⁿ = a^n-1 ⊗ a.

Definition 2.2. [30] Let E be an EQ-algebra. Then E is called

seperated, if for all a, b ∈ E, a ∼ b = 1 implies a = b.

good, if for all $a \in E, \tilde{a} = a$ .

residuated, if for all a, b, c ∈ E, (a ⊗ b) ∧ c = a ⊗ b iff a ∧ ((b ∧ c) ∼ b) = a.

involutive (or an IEQ-algebra), if for all a ∈ E, ¬¬ a = a.

a lattice-order EQ-algebra, if it has a lattice reduct.

a lattice EQ-algebra (or ℓEQ-algebra), if it is a lattice-order EQ-algebra and satisfies: for all a, b, c, d ∈ E, ((a ∨ b) ∼ c) ⊗ (d ∼ a) ≤ c ∼ (d ∨ b).

prelinear, if for all a, b ∈ E, 1 is the unique upper bound of the set {a → b, b → a}.

EQ-algebras have many interesting properties, we list some which are used in the sequel.

Proposition 2.3. [15, 16, 30] If E is an EQ-algebra, then for all a, b, c ∈ E:

(P1) $\neg a = a \to 0, \neg 0 = 1, \neg 1 = \tilde{0}$ .

(P2) a → b = a → a ∧ b.

(P3) If a ≤ b, then a → b = 1, a ∼ b = b → a, c → a ≤ c → b, b → c ≤ a → c.

(P4) a ⊗ b ≤ a ∧ b.

(P5) a → b ≤ (c → a) → (c → b), a → b ≤ (b → c) → (a → c), a → b ≤ ¬ b → ¬ a.

(P6) a → b ≤ a ∧ c → b ∧ c.

(P7) $a \otimes (a \to b) \leq \tilde{b}$ ; If a ≤ b → c, then $a \otimes b \leq \tilde{c}$ .

(P8) b ≤ a → b.

If E is good, then

(P9) a → (b → c) = b → (a → c).

(P10) a ≤ (a → b) → b. Specially a ≤ ¬¬ a.

(P11) a ⊗ (a → b) ≤ a, b.

If E is a ℓEQ-algebra, then

(P12) a → b = a ∨ b → b = (a ∨ b) ∼ b.

(P13) a → b ≤ (a ∨ c) → (b ∨ c).

If E is prelinear and good, then

(P14) a ∼ b = d_E (a, b) = (a ∨ b) → (a ∧ b).

If E is a good ℓEQ-algebra, then

(P15) a ∨ b → c = (a → c) ∧ (b → c). ¬ (a ∨ b) = ¬ a ∧ ¬ b.

If E is residuated, then

(P16) a ≤ b → a ⊗ b.

(P17) (a ⊗ b) → c = a → (b → c).

Remark 2.4. The special EQ-algebras above have the close relationship: Every residuated EQ-algebra is good and every good EQ-algebra is separated; Every prelinear, good EQ-algebra is a prelinear, good ℓEQ-algebra, where the join operation “∨” is defined by a ∨ b = ((a → b) → b) ∧ ((b → a) → a); Every IEQ-algebra is a good ℓEQ-algebra ([15, 16, 30]).

Lemma 2.5. [16, 30] Let E be an EQ-algebra. Then the following assertions are equivalent:

(1) E is separated.

(2) a ≤ b iff a → b = 1 for all a, b ∈ E.

Lemma 2.6. [16, 30] Let E be an EQ-algebra. Then the following assertions are equivalent:

(1) E is good.

(2) 1 → a = a for all a ∈ E.

Lemma 2.7. If E is good, then ((a → b) → b) → b = a → b. Specially ¬¬ ¬ a = ¬ a.

Proof. By Proposition 2.3 (P3), (P10), a → b ≥ ((a → b) → b) → b. On the other hand, by a ≤ (a → b) → b, we have a → b ≤ ((a → b) → b) → b. Thus ((a → b) → b) → b = a → b. Put b = 0, we have ¬¬ ¬ a = ¬ a.

Definition 2.8. [1] A mapping s : E ⟶ [0, 1] is called a Bosbach state on E if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0 and s (1) =1;

(2) s (a) + s (a → b) = s (b) + s (b → a).

Proposition 2.9. [1] Let s : E ⟶ [0, 1] be a mapping such that s (0) =0. Then the following are equivalent, for all a, b ∈ E:

(1) s is a Bosbach state.

(2) b ≤ a implies s (a → b) =1 - s (a) + s (b).

(3) s (a → b) =1 - s (a) + s (a ∧ b).

Proposition 2.10. [1] Let s be a Bosbach state on E. Then, for all a, b ∈ E:

(1) a ≤ b implies s (a) ≤ s (b).

(2) s (¬ a) =1 - s (a).

(3) s (¬¬ a) = s (a).

(4) $s (\tilde{a}) = s (a)$ .

3 Generalized Bosbach states on EQ-algebras

Throughout this section, let E, L be EQ-algebras with the bottom element 0 and s : E ⟶ L an arbitrary mapping.

3.1 Type I states

Definition 3.1. A mapping s : E ⟶ L is called a generalized Bosbach state of type I (briefly, a type I state) if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0;

(2) s (a → b) = s (a) → s (a ∧ b).

Example 3.2. Let E = {0, a, b, c, d, 1} such that 0 < a < b < d < 1, 0 < a < c < d < 1, b and c incomparable. The operations “⊗” and “∼” are defined as:

⊗	a	b	c	d	1
0	0	0	0	0	0
a	0	0	0	0	a
b	0	a	a	a	b
c	0	a	0	a	c
d	0	a	a	a	d
1	a	b	c	d	1

∼	0	a	b	c	d	1
0	1	0	0	0	0	0
a	0	1	d	d	d	d
b	0	d	1	d	d	d
c	0	d	d	1	d	d
d	0	d	d	d	1	1
1	0	d	d	d	1	1

Then E is an EQ-algebra ([30]). The mapping id_E : E ⟶ E is always an order-preserving type I state. Define s : E ⟶ E as s (0) =0, s (a) = s (b) = s (d) = s (1) =1, s (c) = d. It is routine to verify that s is a type I state but not order-preserving.

Theorem 3.3. A mapping s : E ⟶ L with s (0) =0 is a type I state iff b ≤ a implies s (a → b) = s (a) → s (b) for all a, b ∈ E.

Proof. Suppose b ≤ a, then s (a → b) = s (a) → s (a ∧ b) = s (a) → s (b).

Conversely, by a ∧ b ≤ a and Proposition 2.3 (P2), s (a → b) = s (a → a ∧ b) = s (a) → s (a ∧ b).

Theorem 3.4. Let E be a ℓEQ-algebra. Then s : E ⟶ L with s (0) =0 is a type I state iff s (a → b) = s (a ∨ b) → s (b).

Proof. By Proposition 2.3 (P12) and Theorem 3.3, s (a → b) = s (a ∨ b → b) = s (a ∨ b) → s (b).

Conversely, suppose b ≤ a, then s (a → b) = s (a ∨ b) → s (b) = s (a) → s (b).

Theorem 3.5. Let E be prelinear and good. Then s : E ⟶ L with s (0) =0 is a type I state iff s (d_E (a, b)) = s (a ∨ b) → s (a ∧ b).

Proof. By Proposition 2.3 (P14), s (d_E (a, b)) = s (a ∨ b → a ∧ b) = s (a ∨ b) → s (a ∧ b).

Conversely, suppose b ≤ a, then s (a → b) = s ((a → b) ∧ (b → a)) = s (d_E (a, b)) = s (a ∨ b) → s (a ∧ b) = s (a) → s (b).

Theorem 3.6. Let s : E ⟶ L be a type I state. Then

(1) s (1) =1.

(2) s (¬ a) = ¬ s (a) , s (¬¬ a) = ¬¬ s (a).

(3) s ((a → b) → b) = s (a → b) → s (b).

(4) s (a → a²) = s (a) → s (a²).

(5) s (a → a ⊗ b) = s (a) → s (a ⊗ b).

If E is good, then

(6) s (((a → b) → b) → a) = s ((a → b) → b) → s (a). Specially, s (¬¬ a → a) = s (¬¬ a) → s (a).

If E, L is good, then

(7) s (a) ⊗ s (a → a ⊗ b) ≤ s (a ⊗ b).

If E is a ℓEQ-algebra, then

(8) s (a ∨ b → a) = s (b) → s (a ∧ b).

(9) s ((a → b) → b) = (s (a ∨ b) → s (b)) → s (b).

Proof. (1) s (1) = s (1 →1) = s (1) → s (1 ∧1) =1.

(2) s (¬ a) = s (a) → s (a ∧ 0) = s (a) → s (0) = ¬ s (a) , s (¬¬ a) = ¬ s (¬ a) = ¬¬ s (a).

By b ≤ a → b, a² ≤ a, a ⊗ b ≤ a and Theorem 3.3, we have (3),(4),(5).

(6) By Proposition 2.3 (P10) and Theorem 3.3, s (((a → b) → b) → a) = s ((a → b) → b) → s (a). Put b = 0, then s (¬¬ a → a) = s (¬¬ a) → s (a).

(7) By (5) and Proposition 2.3 (P11), s (a) ⊗ s (a → a ⊗ b) = s (a) ⊗ (s (a) → s (a ⊗ b)) ≤ s (a ⊗ b).

(8) By Proposition 2.3 (P12) and Theorem 3.3, s (a ∨ b) → s (a) = s (a ∨ b → a) = s (b → a) = s (b) → s (b ∧ a) = s (b) → s (a ∧ b).

(9) By Proposition 2.3 (P8), Theorem 3.3 and Theorem 3.4, s ((a → b) → b) = s (a → b) → s (b) = (s (a ∨ b) → s (b)) → s (b).

Proposition 3.7. Let s : E ⟶ L be an order-preserving type I state. Then, for all a, b ∈ E:

(1) s (a → b) ≤ s (a) → s (b).

(2) s (d_E (a, b)) ≤ d_E (s (a) , s (b)).

(3) s (a → b) ⊗ s (b → a) ≤ d_E (s (a) , s (b)).

Proof. (1) It is easy that s (a → b) = s (a) → s (a ∧ b) ≤ s (a) → s (b) by the order-preserving of s and Proposition 2.3 (P3).

(2) s (d_E (a, b)) = s ((a → b) ∧ (b → a)) ≤ s (a → b) ∧ s (b → a) ≤ (s (a) → s (b)) ∧ (s (b) → s (a)) = d_E (s (a) , s (b)).

(3) s (a → b) ⊗ s (b → a) ≤ (s (a) → s (b)) ⊗ (s (b) → s (a)) ≤ d_E (s (a), s (b)).

Remark 3.8. We recall that a standard MV-algebra is ([0, 1] , max, min, ⊗ _L, → _L, 0, 1), where a ⊗_L b = max {0, a + b - 1 }, a →_L b = min {1, 1 - a + b}([19]). When L is a standard MV-algebra, it is trivial that order-preserving type I states coincide with Bosbach states on E.

3.2 Type II states

Definition 3.9. A mapping s : E ⟶ L is called a generalized Bosbach state of type II (briefly, a type II state) if it satisfies the following conditions:

(1) s (0) =0;

(2) s (a) = s (a → b) → s (a ∧ b).

Example 3.10. Let E = {0, a, b, c, d, 1} be a chain. The operations “⊗” and “∼” are defined as:

⊗	a	b	c	d	1
0	0	0	0	0	0
a	0	0	0	0	a
b	0	0	0	a	b
c	0	0	a	a	c
d	0	a	a	a	d
1	a	b	c	d	1

∼	0	a	b	c	d	1
0	1	c	b	a	0	0
a	c	1	b	a	a	a
b	b	b	1	b	b	b
c	a	a	b	1	c	c
d	0	a	b	c	1	d
1	0	a	b	c	d	1

Then E is an EQ-algebra ([30]). Define s : E ⟶ E as s (0) = s (a) =0, s (b) = b, s (c) = s (d) = s (1) =1, It is routine to verify that s is a type II state.

Theorem 3.11. A mapping s : E ⟶ L with s (0) =0 is a type II state iff b ≤ a implies s (a) = s (a → b) → s (b).

Proof. Suppose b ≤ a, then s (a) = s (a → b) → s (a ∧ b) = s (a → b) → s (b).

Conversely, by Proposition 2.3 (P2) and a ∧ b ≤ a, s (a) = s (a → a ∧ b) → s (a ∧ b) = s (a → b) → s (a ∧ b).

Theorem 3.12. Let E be a ℓEQ-algebra. Then s : E ⟶ L with s (0) =0 is a type II state iff s (a ∨ b) = s (a → b) → s (b).

Proof. Suppose b ≤ a, then s (a) = s (a ∨ b) = s (a → b) → s (b). By Theorem 3.11, we know that s is a type II state.

Conversely, by b ≤ a ∨ b and Theorem 3.11, s (a ∨ b) = s (a ∨ b → b) → s (b) = s (a → b) → s (b).

Theorem 3.13. Let E be a ℓEQ-algebra, L a good EQ-algebra and s : E ⟶ L with s (0) =0. Then the following are equivalent:

(1) s (a → b) → s (b) = s (b → a) → s (a).

(2) b ≤ a implies s (a) = s (a → b) → s (b).

(3) s (a ∨ b) = s (a → b) → s (b).

Proof. (1)⇒(2)

If b ≤ a, then s (a) =1 → s (a) = s (b → a) → s (a) = s (a → b) → s (b).

(2)⇒(3)

See the proof of Theorem 3.12.

(3)⇒(1)

s (a → b) → s (b) = s (a ∨ b) = s (b ∨ a) = s (b → a) → s (a).

Proposition 3.14. Let s : E ⟶ L be a type II state. Then

(1) s (1) =1.

(2) s is order-preserving: b ≤ a implies s (b) ≤ s (a).

(3) s (a) = ¬ s (¬ a).

(4) s (a → b) = s ((a → b) → b) → s (b).

(5) If E is residuated, then s (a ⊗ b) = ¬ s (¬ (a ⊗ b)) = ¬ s (a → ¬ b).

Proof. (1) s (1) = s (1 →1) → s (1 ∧1) = s (1) → s (1) =1.

(2) Suppose b ≤ a, then s (b) ≤ s (a → b) → s (b) = s (a).

(3) By 0 ≤ a and Theorem 3.11, s (a) = s (a → 0) → s (0) = s (¬ a) →0 = ¬ s (¬ a).

(4) Analogous to (3), by b ≤ a → b, s (a → b) = s ((a → b) → b) → s (b).

(5) s (a ⊗ b) = ¬ s (¬ (a ⊗ b)) = ¬ s (a ⊗ b → 0) = ¬ s (a → (b → 0)) = ¬ s (a → ¬ b).

Theorem 3.15. Let E be a good ℓEQ-algebra. Then every type II state is an order-preserving type I state.

Proof. It is clear that s is order-preserving from Proposition 3.14 (2). By b ≤ (a → b) → b and Theorem 3.11, s ((a → b) → b) = s (((a → b) → b) → b) → s (b) = s (a → b) → s (b) = s (a ∨ b). Again b ≤ a → b, then s (a → b) = s ((a → b) → b) → s (b) = s (a ∨ b) → s (b). By Theorem 3.4, we know that s is a type I state.

Remark 3.16. The above result is useful. When the condition is satisfied, we will use the properties of type I states.

Theorem 3.17. Let E be a good ℓEQ-algebra, L a good EQ-algebra and s : E ⟶ L with s (0) =0, s (1) =1. Then s is a type II state iff s (a → b) = (s (b → a) → s (a)) → s (b).

Proof. Suppose that s is a type II state, it follows that s is a type I state from Theorem 3.15. Thus s (a → b) = s (a ∨ b) → s (b) = s (b ∨ a) → s (b) = (s (b → a) → s (a)) → s (b).

Conversely, replacing a with a ∧ b in the assumption, it follows that 1 = s (a ∧ b → b) = (s (b → a ∧ b) → s (a ∧ b)) → s (b) = (s (b → a) → s (a ∧ b)) → s (b). That is, s (b → a) → s (a ∧ b) ≤ s (b). Replacing b with a ∧ b, it follows that s (a → a ∧ b) = (s (a ∧ b → a) → s (a)) → s (a ∧ b). That is s (a → b) = s (a) → s (a ∧ b), Thus s (b → a) = s (b) → s (a ∧ b). Hence s (b → a) → s (a ∧ b) = (s (b) → s (a ∧ b)) → s (a ∧ b) ≥ s (b). This shows that s (b) = s (b → a) → s (a ∧ b).

Theorem 3.18. Let s : E ⟶ L be a type II state and E a good ℓEQ-algebra. Then

(1) b ≤ a implies s (a → b) = s (a) → s (b).

(2) s ((a → b) → b) = s ((b → a) → a) = s (a ∨ b).

(3) s (¬¬ a) = s (a) , s (¬ a) = ¬ s (a). If L is good, then ¬¬ s (a) = s (a).

(4) b ≤ a and s (a) = s (b) implies s (a → b) =1. Specially, s (¬¬ a → a) =1.

If L is separated, then

(5) b ≤ a and s (a) = s (b) implies s (a ∘ c) = s (b ∘ c) for ∘ ∈ {∧ , ∨}. Specially s (¬¬ a ∘ c) = s (a ∘ c).

(6) b ≤ a and s (a) = s (b) implies s (c → a) = s (c → b). Specially, s (c → ¬¬ a) = s (c → a). s (c → a ∨ b) = s (c → ((a → b) → b)).

(7) b ≤ a and s (a) = s (b) implies s (a → c) = s (b → c). Specially s (¬¬ a → c) = s (a → c).

(8) s (¬ a → ¬ b) = s (b → a).

Proof. (1) By Theorem 3.3 and Theorem 3.15.

(2) s (a ∨ b) = s (a → b) → s (b) = s ((a → b) → b). Similarly, s (b ∨ a) = s ((b → a) → a). Again s (a ∨ b) = s (b ∨ a). Thus s ((a → b) → b) = s ((b → a) → a) = s (a ∨ b).

(3) Put b = 0 in (2), we have s (¬¬ a) = s (a). And s (¬ a) = s (a → 0) = s ((a → 0) →0) → s (0) = s (¬¬ a) →0 = ¬ s (¬¬ a) = ¬ s (a). If L is good, then ¬¬ s (a) = ¬¬ ¬ s (¬ a) = ¬ s (¬ a) = s (a).

(4) By (1) and (3).

(5) Let ∘ ∈ {∧ , ∨}. It is clear that s (b ∘ c) ≤ s (a ∘ c) by the monotonicity of ∘ and the order-preservation of s. By Proposition 2.3 (P6),(P13), we have 1 = s (a) → s (b) = s (a → b) ≤ s (a ∘ c → b ∘ c) = s (a ∘ c) → s (b ∘ c). Thus s (a ∘ c) ≤ s (b ∘ c). Hence s (a ∘ c) = s (b ∘ c). It is easy that s (¬¬ a ∘ c) = s (a ∘ c) from (3) and a ≤ ¬¬ a.

(6) Suppose b ≤ a, then c → b ≤ c → a. By the order-preservation of s, we have s (c → b) ≤ s (c → a). It follows from Proposition 2,3 (P5) that 1 = s (a) → s (b) = s (a → b) ≤ s ((c → a) → (c → b)) = s (c → a) → (c → b). Hence s (c → a) ≤ s (c → b). Thus s (c → a) = s (c → b). By (2) and (3), we have s (c → ¬¬ a) = s (c → a). s (c → a ∨ b) = s (c → ((a → b) → b)).

(7) The proof is similar to that of (6) by using the inequalities a → b ≤ (b → c) → (a → c).

(8) By Proposition 2.3 (P9), ¬a → ¬ b = ¬ a → (b → 0) = b → ¬¬ a . Thus (8) follows from s (b → ¬¬ a) = s (b → a).

Theorem 3.19. Let E be a good ℓEQ-algebra, L an MV-algebra and s : E ⟶ L a mapping. If s is an order-preserving type I state, then s is a type II state.

Proof. By Proposition 2.3 (P8) and Theorem 3.3, s ((a → b) → b) = s (a → b) → s (b) = (s (a ∨ b) → s (b)) → s (b) =(s (b) → s (a ∨ b)) → s (a ∨ b) =1 → s (a ∨ b) = s (a ∨ b). Also s (a ∨ b) = s (b ∨ a). Thus s ((a → b) → b) = s ((b → a) → a). Again s (a → b) → s (b) = s ((a → b) → b) = s ((b → a) → a) = s (b → a) → s (a). Hence, by Theorem 3.13, s is a type II state.

Corollary 3.20. Let s : E ⟶ L be a mapping, E a good łEQ-algebra and L an MV-algebra. Then the following assertions are equivalent:

(1) s is an order-preserving type I state.

(2) s is a type II state.

Remark 3.21. When L is a standard MV-algebra, by Proposition 2.9 (3), we know that type II states coincide with Bosbach states on E.

3.3 Type III states

Definition 3.22. A mapping s : E ⟶ L is called a generalized Bosbach state of type III (briefly, a type III state) if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0, s (1) =1;

(2) s (b) → s (a) = s (a → b) → s (b → a).

Example 3.23. In Example 3.2, define s (0) =0, s (a) = s (b) = s (c) = s (d) = d, s (1) =1, it is routine to verify that s is a type III state.

Proposition 3.24. Let s : E ⟶ L be a type III state. Then

(1) s (a → b) ≤ s (a) → s (b).

(2) If L is sparated, a ≤ b implies s (a) ≤ s (b).

(3) If L is good, then b ≤ a implies s (a → b) = s (a) → s (b).

Proof. (1) s (a) → s (b) = s (b → a) → s (a → b) ≥ s (a → b).

(2) Suppose a ≤ b, by (1), 1 = s (1) = s (a → b) ≤ s (a) → s (b). Thus s (a) ≤ s (b).

(3) Suppose b ≤ a, then s (a) → s (b) = s (b → a) → s (a → b) =1 → s (a → b) = s (a → b).

Corollary 3.25. Let s : E ⟶ L be a type III state and L a good EQ-algebra. Then

(1) s (¬ a) = ¬ s (a).

(2) s ((a → b) → b) = s (a → b) → s (b).

(3) s (a → a²) = s (a) → s (a²).

(4) s (a → a ⊗ b) = s (a) → s (a ⊗ b).

(5) s (a → a ∧ b) = s (a) → s (a ∧ b).

Theorem 3.26. Let s : E ⟶ L be a type III state and L a good EQ-algebra. Then s is an order-preserving type I state.

Proof. It is clear from Proposition 3.24 (3) and Theorem 3.3.

Theorem 3.27. Let s : E ⟶ L be an order-preserving type I state, E a linearly ordered EQ-algebra and L a good EQ-algebra. Then s is a type III state.

Proof. Since E is linearly ordered, without loss of generality, for all a, b ∈ E, we assume that a ≤ b. Then s (b) → s (a) = s (b → a) =1 → s (b → a) = s (a → b) → s (b → a).

Theorem 3.28. Let s : E ⟶ L be a type II state, E a good ℓEQ-algebra and L a good EQ-algebra. Then s is a type III state.

Proof. Suppose that s is a type II state, then s (b) → s (a) = s (b) → (s (a → b) → s (a ∧ b)) = s (a → b) → (s (b) → s (a ∧ b)) = s (a → b) → s (b → a).

Theorem 3.29. Let s : E ⟶ L be a type III state such that s ((a → b) → b) = s (a ∨ b) and E be a ℓEQ-algebra. Then s is a type II state.

Proof. By Corollary 3.25 (2), s (a → b) → s (b) = s ((a → b) → b) = s (a ∨ b).

3.4 Type IV states

Definition 3.30. A mapping s : E ⟶ L is called a generalized Bosbach state of type IV (briefly, a type IV state) if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0, s (1) =1;

(2) s (a ∧ b) = s (a) ⊗ s (a → b).

Example 3.31. Let E = {0, a, b, c, d, 1} such that 0 < a, b < c < 1, 0 < b < d < 1, a, b and respective c, d incomparable. The operations “⊗” and “∼” are defined below:

⊗	a	b	c	d	1
0	0	0	0	0	0
a	a	0	a	0	a
b	0	0	0	b	b
c	a	0	a	b	c
d	0	b	b	d	d
1	a	b	c	d	1

∼	0	a	b	c	d	1
0	1	d	c	b	a	0
a	d	1	b	c	a	a
b	c	b	1	d	c	b
c	b	c	d	1	b	c
d	a	0	c	b	1	d
1	0	a	b	c	d	1

Then E is a good EQ-algebra ([27]). Define s : E ⟶ E as s (0) = s (a) =0, s (b) = s (c) = b, s (d) = d, s (1) =1. It is routine to verify that s is a type IV state.

Theorem 3.32. Let s : E ⟶ L be a mapping such that s (0) =0, s (1) =1. Then the following are equivalent, for all a, b ∈ E:

(1) s is a type IV state.

(2) s (a) ⊗ s (a → b) = s (b) ⊗ s (b → a).

(3) b ≤ a implies s (a) ⊗ s (a → b) = s (b).

Proof. (1)⇒(2)

It is clear by exchanging a and b.

(2)⇒(3)

Suppose b ≤ a, then s (a) ⊗ s (a → b) = s (b) ⊗ s (b → a) = s (b) ⊗1 = s (b).

(3)⇒(1)

Since a ∧ b ≤ a, then s (a) ⊗ s (a → a ∧ b) = s (a ∧ b). That is, s (a) ⊗ s (a → b) = s (a ∧ b).

Theorem 3.33. Let s : E ⟶ L be a mapping such that s (0) =0, s (1) =1 and E a prelinear good EQ-algebra. Then the following assertions are equivalent, for all a, b ∈ E:

(1) s (a ∨ b) ⊗ s (a → b) = s (b).

(2) s is a type IV state.

(3) s (a ∨ b) ⊗ s (d_E (a, b)) = s (a ∧ b).

Proof. (1)⇒(2)

Replacing b with a ∧ b, it follows that s (a ∧ b) = s (a ∨ (a ∧ b)) ⊗ s (a → a ∧ b) = s (a) ⊗ s (a → b).

(2)⇒(3)

s ((a ∨ b) ∧ (a ∧ b)) = s (a ∨ b) ⊗ s ((a ∨ b) → (a ∧ b)) = s (a ∨ b) ⊗ s (d_E (a, b)).

(3)⇒(1)

Replacing a with a ∨ b in the assumption, then s (b) = s ((a ∨ b) ∧ b) = s ((a ∨ b) ∨ b) ⊗ s ((a ∨ b → b) ∧ (b → a ∨ b)) = s (a ∨ b) ⊗ s (a → b).

Proposition 3.34. Let s : E ⟶ L be a type IV state. Then

(1) s is order-preserving.

(2) s (a) ⊗ s (¬ a) =0.

(3) If E is residuated, then s (a) ⊗ s (b) ≤ s (a ⊗ b), s (a) ⊗ s (a → b) = s (a ⊗ (a → b)).

Proof. (1) Suppose b ≤ a, then s (b) = s (a ∧ b) = s (a) ⊗ s (a → b) ≤ s (a).

(2) s (a) ⊗ s (a → 0) = s (a ∧ 0) =0.

(3) s (a) ⊗ s (b) ≤ s (a) ⊗ s (a → a ⊗ b) = s (a ∧ (a ⊗ b)) = s (a ⊗ b). And s (a) ⊗ s (a → b) ≤ s (a ⊗ (a → b)) ≤ s (a ∧ b) ≤ s (a) ⊗ s (a → b). Thus s (a) ⊗ s (a → b) = s (a ⊗ (a → b)).

Proposition 3.35. Let s : E ⟶ L be a type IV state and L satisfy a = b → a ⊗ b. Then s is a type II state.

Proof. s (a → b) → s (a ∧ b) = s (a → b) → (s (a) ⊗ s (a → b)) = s (a). This shows that s is a type II state.

Proposition 3.36. Let s : E ⟶ L be an order-preserving type I state and L satisfy a ∧ b = a ⊗ (a → b). Then s is a type IV state.

Proof. s (a) ⊗ s (a → b) = s (a) ⊗ s (a → a ∧ b) = s (a) ⊗ (s (a) → s (a ∧ b)) = s (a) ∧ s (a ∧ b) = s (a ∧ b).

3.5 Type V states

Definition 3.37. A mapping s : E ⟶ L is called a generalized Bosbach state of type V (briefly, a type V state) if it satisfies the following conditions:

(1) s (0) =0;

(2) s (a → b) = s (a) → (s (b) ⊗ s (b → a)).

Example 3.38. Let E = {0, a, b, 1} such that 0 < a < b < 1. The operations “⊗” and “∼” are defined as:

⊗	a	b	1
0	0	0	0
a	0	0	a
b	0	0	b
1	a	b	1

∼	0	a	b	c
0	1	0	0	0
a	0	1	1	1
b	0	1	1	1
1	0	1	1	1

Then E is an EQ-algebra ([33]). Define s : E ⟶ E as s (0) =0, s (a) = s (b) = a, s (1) =1. It is routine to verify that s is a type V state.

Proposition 3.39. Let s : E ⟶ L be a type V state. Then

(1) s (1) =1.

(2) s (a → b) ≤ s (a) → s (b).

(3) b ≤ a implies s (a → b) = s (a) → s (b).

(4) If L is separated, then s is order-preserving.

(5) If L is good, then s (a ∧ b) = s (a) ⊗ s (a → b).

Proof. (1) s (1) = s (0 → 1) = s (0) → (s (1) ⊗ s (1 → 0)) = 0 → (s (1) ⊗ s (1 → 0)) = 1.

(2) s (a → b) = s (a) → (s (b) ⊗ s (b → a)) ≤ s (a) → s (b).

(3) Suppose b ≤ a, then s (a → b) = s (a) → (s (b) ⊗ s (b → a)) = s (a) → (s (b) ⊗1) = s (a) → s (b).

(4) Suppose a ≤ b, then 1 = s (1) = s (a → b) = s (a) → (s (b) ⊗ s (b → a)), thus s (a) ≤ s (b) ⊗ s (b → a) ≤ s (b).

(5) Replacing a with a ∧ b, then 1 = s (a ∧ b → b) = s (a ∧ b) → (s (b) ⊗ s (b → a ∧ b)) = s (a ∧ b) → (s (b) ⊗ s (b → a)). Thus s (a ∧ b) ≤ s (b) ⊗ s (b → a). Then s (b ∧ a) ≤ s (a) ⊗ s (a → b). By (3), s (a → a ∧ b) = s (a) → s (a ∧ b). Thus s (a) ⊗ s (a → b) = s (a) ⊗ s (a → a ∧ b) = s (a) ⊗ (s (a) → s (a ∧ b)) ≤ s (a ∧ b). Hence s (a ∧ b) = s (a) ⊗ s (a → b).

Corollary 3.40. If s : E ⟶ L is a type V state, then s is a type I state.

Corollary 3.41. If s : E ⟶ L is a type V state and L is a good EQ-algebra, then s is a type IV state.

Theorem 3.42. Let s : E ⟶ L be a mapping such that s (0) =0 and L good. Then s is a type V state iff s is both a type I state and a type IV state.

Proof. Suppose s is both a type I state and a type IV state, then s (a → b) = s (a) → s (a ∧ b) = s (a) → (s (a) ⊗ s (a → b)) = s (a) → (s (b) ⊗ s (b → a)).

Conversely, it is clear from Proposition 3.39 (4), (5).

Theorem 3.43. Let s : E ⟶ L be a type V state, E a prelinear good EQ-algebra and L a residuated EQ-algebra. Then s is a type III state.

Proof. Suppose that s is a type V state, then by Theorem 3.42, we know that s is both a type I state and a type IV state. Thus s (a → b) → s (b → a) = s (a → b) → (s (a ∨ b) → s (a)) = (s (a → b) ⊗ s (a ∨ b)) → s (a) = s (b) → s (a). This shows that s is a type III state.

3.6 Type VI states

Definition 3.44. A mapping s : E ⟶ L is called a generalized Bosbach state of type VI (briefly, a type VI state) if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0, s (1) =1;

(2) s (b) = s (a → b) ⊗ (s (b → a) → s (a)).

Example 3.45. In Example 3.38, define s : E ⟶ E as s (0) =0, s (a) = s (b) = s (1) =1, it is easy to verify that s is a type VI state.

Proposition 3.46. Let s be a type VI state. Then, for all a, b ∈ E:

(1) a ≤ b implies s (a) ≤ s (b).

(2) s (b) = s (b → a) → s (b ∧ a).

(3) If E is a ℓEQ-algebra, then s (a ∨ b) = s (b → a) → s (a).

(4) If L is good, then s (a ∧ b) = s (a) ⊗ s (a → b).

Proof. (1) Suppose a ≤ b, then s (b) = s (a → b) ⊗ (s (b → a) → s (a)) = s (1) ⊗ (s (b → a) → s (a)) = s (b → a) → s (a) ≥ s (a).

(2) Replacing a with a ∧ b, then s (b) = s (a ∧ b → b) ⊗ s ((b → a ∧ b) → s (a ∧ b)) = 1⊗ (s (b → a) → s (a ∧ b)) = s (b → a) → s (b ∧ a).

(3) Replacing b with a ∨ b, then s (a ∨ b) = s (a → a ∨ b) ⊗ (s (a ∨ b → a) → s (a)) = 1⊗ (s (b → a) → s (a)) = s (b → a) → s (a).

(4) Replacing b with a ∧ b, then s (a ∧ b) = s (a → a∧ b) ⊗ (s (a∧ b → a) → s (a)) = s (a → b) ⊗ (s (1) → s (a)) = s (a) ⊗ s (a → b).

Corollary 3.47. Every type VI state is a type II state.

Corollary 3.48. Let s be a type VI state and L good. Then s is a type IV state.

Theorem 3.49. Let E be a prelinear ℓEQ-algebra and L good. Then s is a type VI state iff s is both a type II state and a type IV state.

Proof. Suppose s is a type VI state, by Proposition 3.46 (2),(4), we know that s is both a type II state and a type IV state.

Conversely, suppose s is both a type II state and a type IV state, then s (a → b) ⊗ (s (b → a) → s (a)) = s (a → b) ⊗ s (b ∨ a) = s (b).

Theorem 3.50. Let s : E ⟶ L be a mapping such that s (0) =0, s (1) =1, E a prelinear ℓEQ-algebra and L a good EQ-algebra. Then s is a type VI state iff s (b) = s (b → a) → (s (a) ⊗ s (a → b)).

Proof. Suppose s is a type VI state, by Theorem 3.49, s (b) = s (b → a) → s (b ∧ a) = s (b → a) → (s (a) ⊗ s (a → b)).

Conversely, replacing b with a ∧ b, then s (a ∧ b) = s (a ∧ b → a) → (s (a) ⊗ s (a → a ∧ b)) = 1→ (s (a) ⊗ s (a → b)) = s (a) ⊗ s (a → b). Thus s (b) = s (b → a) → s (b ∧ a). This show that s is both a type II state and a type IV state. By Theorem 3.49, we know that s is a type VI state.

Corollary 3.51. Let s : E ⟶ L be a mapping, E a prelinear good algebra and L a good EQ-algebra. If s is a type VI state, then s is a type V state.

Corollary 3.52. Let s : E ⟶ L be a mapping, E a prelinear good algebra and L an MV-algebra. Then s is a type V state iff it is a type VI state.

3.7 Type VII states

Definition 3.53. A mapping s : E ⟶ L is called a generalized Bosbach state of type VII (briefly, a type VII state) if it satisfies the following conditions, for all a, b ∈ E:

(1) s (0) =0, s (1) =1;

(2) s (b) → s (a → b) = s (a) → s (b → a).

Example 3.54. In Example 3.2, define s (0) =0, s (a) = s (b) = s (c) = a, s (d) = d, s (1) =1, it is routine to verify that s is a type VII state.

Proposition 3.55. Let s be a type VII state and L separated. Then a ≤ b implies s (a) ≤ s (b → a).

Proof. Suppose a ≤ b, then s (a) → s (b → a) = s (b) → s (a → b) = s (b) →1 = 1. Thus s (a) ≤ s (b → a).

Proposition 3.56. If s : E ⟶ L with s (0) =0, s (1) =1 is order-preserving, then s is a type VII state.

Proof. Suppose that s is order-preserving, then s (a) ≤ s (b → a) , s (b) ≤ s (a → b). Thus 1 = s (a) → s (b → a) = s (b) → s (a → b).

Corollary 3.57. Type II states, type IV states and type VI states are type VII states.

Corollary 3.58. If L is good, then type III states, type V states are type VII states.

Theorem 3.59. Let s : E ⟶ L be a type I state and L a good EQ-algebra. Then s is a type VII state.

Proof. Suppose s is a type I state, then s (b) → s (a → b) = s (b) → (s (a) → s (a ∧ b)) = s (a) → (s (b) → s (b ∧ a)) = s (a) → s (b → a). This shows that s is a type VII state.

4 Conclusions

In this paper, we propose the notions of generalized Bosbach states of type I and generalized Bosbach states of type II (resp. III, IV, V, VI, VII) on EQ-algebras and give some specific examples of these generalized Bosbach states. We obtain their equivalent characterizations and discuss the relationship between them.

In summary, we devote to the algebraic studies of generalized Bosbach states on EQ-algebras. We mainly concentrate on the academic departments. Inspried by [24 , 35], in the future work, we will consider combining EQ-algebras with relative knowledge of soft sets and use them in decision-making.

Footnotes

Acknowledgment

This work is supported by the National Natural Science Foundation of China, Grant No. 11701540, 61773019, 61379018.

References

Borzooei

R.A.

and Saffar

B.G.

, States on EQ-algebras, Journal of Intelligent & Fuzzy Systems 29 (2015), 209–221.

Botur

and Dvurečenskij

, State-morphism algebras– General approach, Fuzzy Sets & Systems 218 (2013), 90–102.

Ciungu

L.C.

, Bosbach and Riečan states on residuated lattices, Journal of Applied Functional Analysis 2 (2008), 175–188.

Ciungu

L.C.

, Dvurečenskij

and Hyčko

, State BL-algebras, Soft Computing 80 (2010), 619–634.

Ciungu

L.C.

, Georgescu

and Mureşsan

, Generalized Bosbach states: Part II, Archive for Mathematical Logic 52 (2013), 707–732.

Di Nola

and Dvurečenskij

, State-morphism MValgebras, Annals of Pure and Applied Logic 161 (2009), 161–173.

Di Nola

and Dvurečenskij

, Some classes of state morphism MV-algebras, Mathematica Slovaca 59 (2009), 517–534.

Dvurečenskij

, Every linear pseudo BL-algebras admits a state, Soft Computing 11 (2007), 495–501.

Dvurečcenskij

, States on MV-algebras and their applications, Journal of Logic & Computation 21 (2011), 407–427.

10.

Dvurečenskij

, States on pseudo MV-Algebras, Studia Logica an International Journal for Symbolic Logic 68 (2001), 301–327.

11.

Dvurecenskij

and Rachůnek

, Probabilistic averaging in bounded Rℓ-monoids, Semigroup Forum 72 (2006), 191–206.

12.

Dvurečenskij

and Rachůnek

, On Riečan and Bosbach states for bounded non-commutative Rℓ-monoids, Mathematica Slovaca 5 (2006),.

13.

Dvurečcenskij

, Rachůnek

and Šalounová

, States operators on generalization of fuzzy structures, Fuzzy Sets & Systems 187 (2012), 58–76.

14.

Dvurečcenskij

Kowalski

and Montagna

, State morphism MV-algebras, International Journal of Approximate Reasoning 52 (2011), 1215–1228.

15.

El-Zekey

, Representable good EQ-algebras, Soft Computing 14 (2010), 1011–1023.

16.

El-Zekey

, Novák

and Mesiar

, On good EQ-algebras, Fuzzy Sets & Systems 178 (2011), 1–23.

17.

Flaminio

and Montagna

, MV-algebras with internal states and probabilistic fuzzy logics, International Journal of Approximate Reasoning 50 (2009), 138–152.

18.

Georgescu

, Bosbach states on fuzzy structures, Soft Computing 8 (2004), 217–230.

19.

Georgescu

and Muresąn

, Generalized Bosbach states, Archive for Mathematical Logic 52 (2010), 335–376.

20.

, Xin

and Yang

, On state residuated lattices, Soft Computing 19 (2015), 2083–2094.

21.

Kondo

, States on bounded commutative residuated lattices, Mathematica Slovaca 64 (2014), 1093–1104.

22.

Liu

, On the existence of states on MTL-algebras, Information Sciences 220 (2013), 559–567.

23.

Liu

and Zhang

, States on R0-algebras, Soft Computing 12 (2008), 1099–1104.

24.

, Zhan

, Ali

M.I.

and Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49 (2018), 511–529.

25.

, Liu

and Zhan

, A survey of decision making methods based on certain hybrid soft set models, Artificial Intelligence Review 47 (2017), 507–530.

26.

Z.M.

and Fu

Z.W.

, Algebraic study to generalized Bosbach states on residuated lattices, Soft Computing 19 (2015), 2541–2550.

27.

Mohtashamnia

and Torkzadeh

, The lattice of prefilters of an EQ-algebra, Fuzzy Sets & Systems 311 (2017), 86–98.

28.

Mundici

, Averaging the truth-value in Łukasiewicz logic, Studia Logica 55 (1995), 113–127.

29.

Novák

, On fuzzy type theory, Fuzzy Sets & Systems 149 (2005), 235–273.

30.

Novák

and De

, Baets, EQ-algebras, Fuzzy Sets & Systems 160 (2009), 2956–2978.

31.

Riečan

, On the probalility on BL-algebras, Acta Math Nitra 4 (2000), 3–13.

32.

Turunen

and Mertanen

, States on semi-divisible residuated lattices, Soft Computing 12 (2008), 353–357.

33.

Wang

, Xin

X.L.

and Wang

J.T.

, EQ-algebras with internal states, Soft Computing 1 (2017), 1–17.

34.

Zhan

and Alcantud

J.C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review 4 (2018), 1–30.

35.

Zhan

and Alcantud

J.C.R.

, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review 4 (2017), 1–34.

36.

Zhou

and Zhao

, Generalized Bosbach and Riečan states based on relative negations in residuated lattices, Fuzzy Sets & Systems 187 (2012), 33–57.