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State maps on semihoops

Yu Long Fu
/ Xiao Long Xin
/ Jun Tao Wang
Published Online: 2018-09-18 | DOI: https://doi.org/10.1515/math-2018-0089

Abstract

In this paper, we introduce the notion of state maps from a semihoop H1 to another semihoop H2, which is a generalization of internal states (or state operators) on a semihoop H. Also we give a type of special state maps from a semihoop H1 to H1, which is called internal state maps (or IS-maps). Then we give some examples and basic properties of (internal) state maps on semihoops. Moreover, we discuss the relations between state maps and internal states on other algebras. Then we introduce several kinds of filters by state maps on semihoops, called SM-filters, state filters and dual state filters, respectively, and discuss the relations among them. Furthermore we introduce and study the notion of prime SM-filters on semihoops. Finally, using SM-filter, we characterize two kinds of state semihoops.

Keywords: Semihoop; State map; SM-filter; Prime SM-filter

MSC 2010: 06F35; 03G25; 08A72

1 Introduction

Residuated structures arise in many areas of mathematics, and are particularly common among algebras associated with logical systems. The essential ingredients are a partial order ≤, a binary operation of associative and commutative multiplication ⊙ that respects the partial order, and a binary (left-)residuation operation → characterized by xyz if and only if xyz. Semihoops  are very important and basic residuated structures in which the community of many-valued logicians got interested in the last years, as they are building blocks for several interesting structures being the algebraic semantics for relevant many-valued logics such as basic fuzzy logic (BL, for short). Apart from their logic interest, semihoops have interesting algebraic properties and include kinds of important classes of algebras: Hoops which were originally introduced by Bosbach [6, 7] under the name of complementary semigroups and Brouwerian semilattices-the models of the conjunction-implication fragment of the intuitionistic propositional calculus. A semihoop is called a hoop if x ⊙ (xy) = y ⊙ (yx) and a semihoop does not satisfy the divisibility condition xy = x ⊙ (xy). Therefore, semihoops are the most fundamental fuzzy structures. It will play an important role in studying fuzzy logics and the related algebraic structures.

In order to measure the average truth-value of propositions in Lukasiewicz logic, Mundici  presented an analogue of probability measure, called a state, as averaging process for formulas in Łukasiewicz logic. States on MV-algebras have been deeply investigated. Consequently, the notion of states has been extended to other logical algebras such as BL-algebras , MTL-algebras [20, 21], R0-algebras  and residuated lattices [12, 19, 23, 26].

Since MV-algebras with state are not universal algebras, they do not automatically induce an assertional logic. Flaminio and Montagna [15, 16] presented an algebraizable logic using a probabilistic approach, and its equivalent algebraic semantics is precisely the variety of state MV-algebras. We recall that a state MV-algebra is an MV-algebra whose language is extended by adding an operator(also called an internal state), whose properties are inspired by ones of states with the addition property. State MV-algebras generalize, for example, Hajek’s approach  to fuzzy logic with modality Pr (interpreted as probably) which has the following semantic interpretation: The probability of an event a is presented as the truth value of Pr(a). On the other hand, if s is a state, then s(a) is interpreted as the average appearance of the many valued event a. Consequently, the notion of internal states has also been extended to other algebraic structures. For example, the concept of a state BL-algebra was introduced by Ciungu et al., as an extension of the concept of a state MV-algebra. Subsequently, the concept of internal states was extended by Dvurečenskij et al. to R-monoids (not necessarily commutative). More generally, the state residuated lattices were introduced by He and Xin .

We observed that the states and internal state on MV-algebras, BL-algebras, BCK-algebras and residuated lattices are maps from an algebra X to [0, 1] and X to X, respectively. From the viewpoint of universal algebras, it is meaningful to study a state map from an algebra X to another algebra Y. In particular, if Y = [0, 1], a state can be seen as a state map from X to [0, 1], and if X = Y, a state operator can also be seen as a state map from XX. Based on this idea, we can conclude that a state map is not only a generalization of internal states but also preserves the usual properties of states. Therefore, it is meaningful to introduce state map to the more general fuzzy structures semihoops and providing an algebraic foundation for reasoning about probabilities of fuzzy events in a new way. This is the motivation for us to investigate state maps on semihoop.

This paper is structured in five sections. In order to make the paper as self-contained as possible, we recapitulate in Section 2 the definition of semihoops, and review their basic properties that will be used in the remainder of the paper. In Section 3, we introduce the notion of state maps (or simply, S-maps), which is a generalization of states on semihoops. Also, we give a characterization of two kinds of semihooops. In Section 4, we discuss the relations between state maps on semihoops and internal states on other algebras, respectively. In Section 5, we introduce several kinds of filters by state maps on semihoops, called SM-filters, state filters and dual state filters, respectively, and discuss the relations among them. Using SM-filter, we characterize two kinds of state semihoops.

2 Preliminaries

In this section, we summarize some definitions and results about semihoop, which will be used in the following sections of the paper.

Definition 2.1

(). An algebra (H, ⊙, →, ∧, 1) of type (2, 2, 2, 0) is called a semihoop if it satisfies the following conditions:

1. (H, ∧, 1) is a ∧-semilattice with upper bound 1,

2. (H, ⊙, 1) is a commutative monoid,

3. (xy) → z = x → (yz), for all x, y, zH.

In what follows, by H we denote the universe of a semihoop (H, ⊙, →, ∧, 1). For any xH and a natural number n, we define x0 = 1 and xn = xn−1x for n ≥ 1.

On a semihoop (H, ⊙, →, ∧, 1) we define xy iff xy = 1. It is easy to check that ≤ is a partial order relation on H and for all xH, x ≤ 1. A semihoop H is bounded if there exists an element 0 ∈ H such that 0 ≤ x for all xH. In a bounded semihoop (H, ⊙, →, ∧, 0, 1), we define the negation ∗ : x = x → 0 for all xL. If x∗∗ = x, for all xH, then the bounded semihoop H is said to have the Double Negation Property, or (DNP) for short. We define a relation ⊥ on H by xy iff y∗∗x. If xx = x, that is, x2 = x for all xH, then the semihoop H is said to be idemopent. A semihoop H is called a hoop if x ⊙ (xy) = y ⊙ (yx) for all x, yH. Also, in every hoop H, xy = x ⊙ (xy) for all x, yH, see .

Proposition 2.2

([14, 30]). In any semihoop (H, ⊙, →, ∧, 1), the following properties hold: for all x, y, zH,

1. xyz iff xyz,

2. xyxy, xyx,

3. 1 → x = x, x → 1 = 1,

4. x ⊙ (xy) ≤ y,

5. If xy, then yzxz, zxzy and xzyz,

6. x ≤ (xy) → y,

7. ((xy) → y) → y = xy,

8. x → (yz) = y → (xz),

9. xy ≤ (zx) → (zy), xy ≤ (yz) → (xz),

10. x → (xy) = xy,

11. xy = x ⊙ (xxy).

Proposition 2.3

([4, 30]). In a bounded semihoop (H, ⊙, →, ∧, 0, 1), the following properties hold: for all x, y, zH,

1. 1 = 0, 0 = 1,

2. xx∗∗, where x∗∗ = (x),

3. xx = 0, x∗∗∗ = x,

4. xy implies yx,

5. xyyx,

6. (xy∗∗)∗∗ = xy∗∗,

7. x∗∗y∗∗ ≤ (xy)∗∗,

8. (x∗∗y) = (xy).

Proposition 2.4

(). Let (H, ⊙, →, ∧, 1) be a semihoop and for all x, yH, we define xy = ((xy) → y) ∧ ((yx) → x). Then the following conditions are equivalent:

1. is an associative operation on H,

2. xy implies xzyz for all x, y, zH,

3. x ⊔ (yz) ≤ (xy) ∧ (xz) for all x, y, zH,

4. is the join operation on H.

Definition 2.5

(). A semihoop is called a ⊔-semihoop if it satisfies one of the equivalent conditions of Proposition 2.4.

Proposition 2.6

(). In a ⊔-semihoop, the following properties hold: for all x, y, zH,

1. x ⊙ (yz) = (xy) ⊔ (xz),

2. x ⊔ (yz) ≥ (xy) ⊙ (xz),

3. xyn ≥ (xy)n and xmyn ≥ (xy)mn for any natural numbers m, n.

Proof

The proofs are easy, and we hence omit the details. □

Definition 2.7

([1, 2]). Let (H, →, ⊙, 1) be a hoop. H is called:

1. a basic hoop if (xy) → z ≤ ((yx) → z) → z for any x, y, zH.

2. a Wajsberg hoop (xy) → y = (yx) → x for any x, yH.

3. a Gödel hoop if xx = x for any xH.

Proposition 2.8

(). Let (H, ⊙, →, 1) be a bounded hoop. Then

1. bounded basic hoops are definitionally equivalent to BL-algebras.

2. bounded Wajsberg hoops are definitionally equivalent to MV-algebras.

Let (H, ⊙, →, ∧, 1) be a semihoop. A nonempty set F of H is called a filter of H if it satisfies: (1) x, yF implies xyF; (2) xF, yH and xy imply yF. A filter F of H is called a proper filter if FH. A proper filter F of H is called a maximal filter if it is not contained in any proper filter of H. A nonempty set F of H is a filter of H if and only if 1 ∈ F and if x, xyF, then yF. A proper filter F of a semihoop H is called a prime filter of H, if for any filters F1, F2 of H such that F1F2F, then F1F or F2F. For more details about filters in semihoops, see .

Definition 2.9

([4, 28]). Let (H, →, ⊙, ∧, 1) be a semihoop. H is called:

1. a simple semihoop if it has exactly two filters: {1} and H.

2. a local semihoop if it has only one maximal filter.

3 State maps on semihoops

In this section, we introduce the notion of state maps on a semihoop and investigate some related properties of state maps.

Definition 3.1

Let (X, ⊙1, →1, ∧1, 11) and (Y, ⊙2, →2, ∧2, 12) be two semihoops. A map σ : XY is called a state map from X to Y, which is denoted simply by S-map, if it is satisfies the following conditions:

• (SM1)

x1 y implies σ (x) ≤2 σ (y);

• (SM2)

σ(x1 y) = σ((x1 y) →1 y) →2 σ(y);

• (SM3)

σ(x1 y) = σ(x) ⊙2 σ(x1 (x1 y));

• (SM4)

σ(x) ⊙2 σ(y) ∈ σ(X);

• (SM5)

σ (x) ∧2 σ (y) ∈ σ(X);

• (SM6)

σ (x) →2 σ (y) ∈ σ(X).

for all x, yX.

The pair (X, Y, σ) is said to be a S-map semihoop. Moreover, if X = Y and σ2 = σ, then σ is called an internal state map on X, simply IS-map on X, in this case, (H,σ) is said to be an IS-map semihoop.

Now, we present some examples for S-maps on semihoops.

Example 3.2

Let H1 and H2 be two semihoops. Then the map 1H1, defined by 1H1(x) = 12 for all xH1, is a S-map from H1 to H2.

Example 3.3

Let H be a semihoop. One can check that idH is a S-map on H.

Example 3.4

Let H1 = {01, a1, b1, c1, 11} and H2 = {02, a2, b2, c2, 12}, where 01a1b1, c1 ≤ 11 and 02a2b2c2 ≤ 12. Define operationsi andi for i = 1, 2 as follows:

$→101a1b1c111011111111111a10111111111b101c111c111c101b1b111111101a1b1c111⊙101a1b1c111010101010101a101a1a1a1a1b101a1b1a111c101a1a1c1111101a1b1c111→202a2b2c212021212121212a20212121212b202c2121212c202b2b212121202a2b2c212⊙202a2b2c212020202020202a202a2a2a2a2b202a2a2a2b2c202a2a2c2c21202a2b2c212$

Then (H1, →1, ⊙1, ∧1, 11) and (H2, →2, ⊙2, ∧2, 12) are semihoops. Now, we define a map σ : H1H2 as follows:

$σ(x)=02,x=01a2,x=a1,b112,x=c1,11.$

One can check that σ is a S-map from H1 to H2.

Example 3.5

Let H = [0, 1] be the real interval. If for x, yH, we define xy = max{0, x + y−1} and xy = min{1, 1 − x + y}, then (H, ⊙, →, 0, 1) becomes a hoop, and hence it is a semihoop. Now we define σ : H1H as follows:

$σ(x)=0,x=01;12,x=a1,b1;1,x=c1,11$

where H1 is given in Example 3.4. One can easily check that σ is a S-map from H1 to H.

Next, we present some properties of S-maps on semihoops.

Proposition 3.6

Let Hi, i = 1, 2 be semihoops and σ be a S-map from H1 to H2. Then we have: for any x, yH1,

1. σ(11) = 12;

2. σ(x1 y) ≥ σ(x) ⊙2 σ(y);

3. σ(x1 y) ≤2 σ(x) →2 σ(y) and if x1y, then σ(x1 y) = σ(x) →2 σ(y);

4. σ(H1) is a subalgebra of H2.

Proof

1. Applying (SM2), we have σ(11) = σ(011 01) = σ((011 01) →1 01) →2 σ(01) = σ(111 01) →2 σ(01) = σ(01) →2 σ(01) = 12.

2. From x1 yx1 y, we get y1 x1 (x1 y) by Proposition 2.2(1). By (SM1), we have σ(y) ≤2 σ(x1( x1 y)). Applying (SM3), we get σ(x1 y) = σ(x) ⊙2 σ(x1 (x1 y)) ≥2 σ(x) ⊙2 σ(y).

3. By (SM2), we deduce σ(x1 y) = σ((x1 y) →1 y) →2 σ(y) ≤2 σ(x) →2 σ(y) by (5) and (6) of Proposition 2.2. If x1 y, then σ(x) ≤2 σ(y). This means σ(x) →2 σ(y) = 1. Moreover, σ(x1 y) = σ((x1 y) →1 y) →2 σ(y) = σ(111 y) →2 σ(y) = σ(y) →2σ(y) = 12. Thus σ(x1 y) = σ(x) →2 σ(y).

4. It follows from (SM4), (SM5), (SM6) and (1). □

Definition 3.7

Let H1 and H2 be two bounded semihoops. A S-map σ from H1 to H2 is called a regular if it satisfies σ(01) = 02.

Note that the S-map σ given in Example 3.2 is not regular and the S-map σ given in Example 3.4 is regular.

In the following we give some characterizations for a S-map becoming regular.

Theorem 3.8

Let Hi, i = 1, 2 be two bounded Wajsberg semihoops and σ be a S-map from H1 to H2. Then the following are equivalent:

1. σ is regular,

2. σ(x1) = (σ(x))2 for any x, yH1,

3. x1 y implies σ(x) ⊥2 σ(y) for any x, yH1.

Proof

(1) ⇒ (2) By (1) and (SM2), we get σ(x1) = σ(x1 01) = σ((x1 01) →1 01) →2 σ(01) = σ(x) →2 02 = (σ(x))2.

(2) ⇒ (3) Suppose that x1 y. Then y111 x1, it follows that σ(y11) ≤1 σ(x1). By (2) we have (σ(y))222 (σ(x)). Hence we have σ(x) ⊥2 σ(y).

(3) ⇒ (1) Since 0111 = 11, we get 111 01. By (3) we have σ(11) ⊥2σ(01), and so σ(01)222σ(11)2. From Proposition 3.6(1), σ(01)222σ(11)2 = 122 = 02, and hence σ(01)22 = 02. It follows that σ(01)222 = 12. By Proposition 2.2(7), σ(01)222 = σ(01)2 = 12, that is, σ(01) →2 02 = 12. This shows that σ(01) ≤2 02, so σ(01) = 02. □

Proposition 3.9

Let H be a semihoop and σ be an IS-map on H. Then we have: for any x, yH,

1. σ(1) = 1;

2. σ(xy) ≥ σ(x) ⊙ σ(y);

3. σ(xy) ≤ σ(x) → σ(y) and if xy, then σ(xy) = σ(x) → σ(y);

4. σ(σ(x) ⊙ σ(y)) = σ(x) ⊙ σ(y);

5. σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y);

6. σ(σ(x) → σ(y)) = σ(x) → σ(y);

7. σ(H) = Fix(σ), where Fix(σ) = {xHσ(x) = x};

8. σ(H) is a subalgebra of H;

9. Ker(σ) is a filter of H, where Ker(σ) = {xHσ(x) = 1}.

Proof

1. It follows from Proposition 3.6(1).

2. It follows from Proposition 3.6(2).

3. It follows from Proposition 3.6(3).

4. From (SM4), we have σ(x) ⊙ σ(y) = σ(z) for some zH. Hence σ(σ(x) ⊙ σ(y)) = σ2(z) = σ(z) = σ(x) ⊙ σ(y) by the definition of the IS-maps.

5. It is similar to (4).

6. It is similar to (1).

7. Let ∈ σ(H). Then x = σ(z) for some zH. Hence σ(x) = σ2(z) = σ(z) = x. So xFix(σ). Conversely assume xFix(σ). Then x = σ(x) ∈ σ(H). This shows that (7) is true.

8. It follows from (1), (2), (3) and (4).

9. It is straightforward. □

Next, we consider properties of IS-map to characterize two kinds of semihoops. The following results and the next one are proved in , where (SM2) replace by (SM2’)σ(xy) = σ(x) → σ(xy). We can show the same results without the identity (SM2’).

Theorem 3.10

Let H be a semihoop. Then the following are equivalent:

1. H is a hoop;

2. every IS-map σ on H satisfies σ(x) ⊙ σ(xy) = σ(y) ⊙ σ(yx) for all x, yH.

Proof

The proof is similar to that of He et al .(Theorem 4.7 ). □

Theorem 3.11

Let H be a semihoop. Then the following are equivalent:

1. H is idemopent;

2. every IS-map σ on H satisfies σ(xy) = σ(y) ⊙ σ(y) = σ(x) ⊙ σ(xy) for all x, yH.

Proof

The proof is similar to that of He et al .(Theorem 4.8 ). □

Here, we give relations between IS-map and Riečan states on semihoops.

Definition 3.12

(). Let H be a bounded semihoop. A Riečan state on H is a founction s : H ⟶ [0, 1] such that the following conditions hold: for all x, yH,

1. s(1) = 1,

2. if xy, then s(x + y) = s(x) + s(y).

Let H be a semihoop, σ be an IS-map on H and s be a Riečan state on H. Then s is called σ-compatible if σ(x) = σ(y) ⇒ s(x) = s(y) for all x, yH.

We denote by RS[H] and RSσ[H] the set of all Riečan states and σ-compatible Riečan states on H, respectively.

Theorem 3.13

Let H be a semihoop and σ be an IS-map on H. Then there is a one-to-one correspondence between σ-compatible Riečan states on H and Riečan states on σ(H).

Proof

1. Suppose that s is a Riečan state on σ(H). Define a mapping φ : RS[σ(H)] → RSσ[H] as follows: φ(s)(x) := s(σ(x)) for all xH. We will prove that φ(s) is a Riečan state on H. Clearly, φ(s)(1) = s(σ(1)) = s(1) = 1. Next, we will show that φ(s)(x + y) = φ(s)(x)+φ(s)(y) when xy. In order to do this, we prove that σ(x + y) = σ(x) + σ(y) for xy. Now, suppose that xy. From Theorem 3.8(3), we have σ(x) ⊥ σ(y). Then σ(x) + σ(y) = (σ(x)) → (σ(y))∗∗. Moreover, σ(x + y) = σ(x y∗∗) = σ(x) → σ(xy∗∗). Since xy, then y∗∗x. It follows that σ(x + y) = σ(x) → σ(y∗∗) = (σ(x)) → (σ(y))∗∗ = σ(x) + σ(y). Now, we prove that φ(s)(x + y) = φ(s)(x)+φ(s)(y) when xy. Since σ(x + y) = σ(x) + σ(y) for xy, we have that φ(s)(x + y) = s(σ(x + y)) = s(σ(x) + σ(y)) = s(σ(x)) + s(σ(y)) = φ(s)(x)+φ(s)(y). Therefore, φ(s) is a Riečan state on H. Moreover, let σ(x) = σ(y) for all x, yH, then φ(s)(x) = s(σ(x)) = s(σ(y)) = φ(s)(y). Thus, φ(s) is a σ-compatible state on H. Therefore, the mapping φ is well defined.

2. Assume that s is a σ-compatible Riečan state on H. The mapping ψ : RSσ[H] → RS[σ(H)] is defined by ψ(s)(σ(x)) := s(x) for all xH. Let σ(x) = σ(y), then s(x) = s(y) for all x, yH. Now, we show that ψ(s) is a Riečan state on σ(H). Let σ(x) ⊥ σ(y). Then σ(σ(x) + σ(y)) = σ((σ(x)) → (σ(y))∗∗) = σ(σ(x) → σ(y∗∗)) = σ(x)σ(y)∗∗ = σ(x) + σ(y). Based on this, we have that ψ(s)(σ(x) + σ(y)) = ψ(s)(σ(σ(x) + σ(y))) = s(σ(x) + σ(y)) = s(σ(x)) + s(σ(y)) = ψ(s)(σ(σ(x))) + ψ(s)(σ(σ(y))) = ψ(s)(σ(x)) + ψ(s)(σ(y)). Moreover, ψ(s)(σ(1)) = s(1) = 1. That means that ψ(s) is a Riečan state on σ(H). Therefore, ψ is a mapping of RSσ[H] into RS[σ(H)].

3. Let s1, s2 be σ-compatible states on H and ψ(s1) = ψ(s2). Then we have ψ(s1)(σ(x)) = ψ(s2)(σ(x)), which implies s1(x) = s2(x) for all xH. Thus, s1 = s2. Now, suppose that s is a Riečan state on σ(H), then we have that (ψ(φ(s))(σ(x)) = φ(s)(x) = s(σ(x)). Therefore, ψ is a bijective mapping from RSσ[H] onto RS[σ(H)] and ψ−1 = φ. □

4 Relations between state maps on semihoops and states on other algebras

Definition 4.1

(). A Bosbach state on a bounded pseudo-hoop (A, ⊙, →, ⇝, 0, 1) is a function s : A → [0, 1] such that the following conditions hold: for any x, yA:

• (B1)

s(x) + s(xy) = s(y) + s(yx);

• (B2)

s(x) + s(xy) = s(y) + s(yx);

• (B3)

s(0) = 0 and s(1) = 1.

Proposition 4.2

(). Let A be a bounded pseudo-hoop and s be a Bosbach state on A. Then for all x, yA the following properties hold:

1. yx implies s(y) ≤ s(x) and s(xy) = s(xy) = 1 − s(x) + s(y);

2. s(x) = s(x) = 1 − s(x), where x = x → 0 and x = x ⇝ 0.

Definition 4.3

A state-morphism map on a bounded hoop A is a function s : A → [0, 1] such that:

• (SM1)

m(0) = 0;

• (SM2)

m(xy) = min{1, 1 − m(x) + m(y)}.

Proposition 4.4

Every state-morphism map on a bounded hoop A is a Bosbach state on A.

Proof

Let m be a state-morphism map on A. Then m(1) = m(0) = m(0 → 0) = min{1, 1 − m(0) + m(0) = 1}. (B3) holds. Consider m(x) + m(xy) = m(x)+min{1, 1 − m(x) + m(y) = min{1 + m(x), 1 + m(y)} = m(y) + m(yx). Hence (B1) is true. Since A is a hoop, (B2) is true, too. Combining the above arguments we get that m is a Bosbach state on A. □

Proposition 4.5

Every state-morphism on a bounded hoop A is a state map from A to the hoop H = ([0, 1], ⊙, →, 0, 1) given in Example 3.5.

Proof

Assume m is a state-morphism map on a bounded hoop A. By Propositions 4.2 and 4.4, (SM1) holds.

Now we check (SM2). Let x, yA. By definition of 4.3, we have

$m((x→y)→y)→m(y)=min{1,1−m((x→y)→y)+m(y))}=min{1,1−(1−m(x→y)+m(y))+m(y))}=m(x→y).$

For (SM3), we have

$m(x)⊙m(x→x⊙y)=0∨(m(x)+m(x→x⊙y)−1)=0∨(m(x)+(1−m(x)+m(x⊙y))−1)=0∨m(x⊙y)=m(x⊙y).$

For (SM4), we have m(x) ⊙ m(y) = max{0, m(x) + m(y) − 1} = 1 − m(y) + m(x) = min{1, 1 − m(y) + m(x)} = m(yx) and hence m(x) ⊙ m(y) ∈ m(A). This shows that (SM4) holds.

Note that m(x) → m(y) = min{1, 1 − m(x) + m(y)} = m(xy). It follows that m(x) → m(y) ∈ m(A), that is (SM6).

For (SM5), we have m(x) ∧ m(y) = m(x) ⊙ (m(x) → m(y)). From (SM4) and (SM6), we get that (SM5) holds. □

Definition 4.6

(). A state semihoop is a pair (H,σ) where H is a bounded semihoop and σ : HH is a mapping, called state operator, such that for any x, yH the following conditions are satisfied:

1. σ(0) = 0;

2. xy implies σ(x) ≤ σ(y);

3. σ(xy) = σ(x) → σ(xy);

4. σ(xy) = σ(x) ⊙ σ(xxy);

5. σ(σ(x) ⊙ σ(y)) = σ(x) ⊙ σ(y);

6. σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y).

Theorem 4.7

Let H be a bounded semihoop and σ : HH be a mapping on H preserving → . Then the following conditions are equivalent:

1. (H,σ) is an IS-map semihoop;

2. (H,σ) is a state semihoop.

Proof

(1) ⇒ (2) If H is a bounded semihoop and σ : HH is a mapping on H preserving →. Then σ(xy) = σ(xxy) = σ(x) → σ(xy). From proposition 3.9 and definition 4.6, we can obtain that (H,σ) a state semihoop.

(2) ⇒ (1) Let (H,σ) be a state semihoop and σ preserving →. We only need to prove that (SM2) holds. Since ((xy) → y) → y = xy, so we have σ((xy) → y) → σ(y) = σ(((xy) → y) → y) = σ(xy). Thus σ is an IS-map on H and hence (H,σ) is an IS-map semihoop. □

Inspired by Ciungu’s state BL-algebras , He and Xin enlarged the language of residuated lattice by introducing a new operator, an internal state on residuated lattice in .

Definition 4.8

(). A state residuated lattice is a pair (A, σ) where A is a residuated lattice and σ: AA is a mapping, called state operator, such that for any x, yA the following conditions are satisfied:

1. σ(0) = 0;

2. xy = 1 implies σ(x) → σ(y) = 1;

3. σ(xy) = σ(x) → σ(xy);

4. σ(xy) = σ(x) ⊙ σ(xxy);

5. σ(σ(x) ⊙ σ(y)) = σ(x) ⊙ σ(y);

6. σ(σ(x) → σ(y)) = σ(x) → σ(y);

7. σ(σ(x) ∨ σ(y)) = σ(x) ∨ σ(y);

8. σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y).

Let (H; ⊙, →, 0, 1) be a bounded ⊔-semihoop. For any x, yH, we set xy = ((xy) → y) ∧ ((yx) → x). Then (H, ∧, ⊔, →, ⊙, 0, 1) is a residuated lattice. (see [2, 3])

Theorem 4.9

Let H be a bounded ⊔-semihoop and σ : HH be a mapping on H preserving → . Then the following conditions are equivalent:

1. σ is an IS-map on H;

2. (H,σ) is a state residuated lattice.

Proof

(1) ⇒ (2) If H is a bounded ⊔-semihoop and σ : HH is a mapping on H preserving →. Then σ(xy) = σ(xxy) = σ(x) → σ(xy). Moreover, by Proposition 3.9(5),(6), we have σ(σ(x) ⊔ σ(y)) = σ(((σ(x) → σ(y)) → σ(y)) ∧ ((σ(y) → σ(x)) → σ(x))) = σ(x) ⊔ σ(y). Therefore, (H,σ) is a state residuated lattice.

(2) ⇒ (1) Let (H,σ) be a state residuated lattice and σ preserving →. We only need to prove that (SM2) holds. Since ((xy) → y) → y = xy, so we have σ((xy) → y) → σ(y) = σ(((xy) → y) → y) = σ(xy). Thus σ is an IS-map on H. □

A state operator σ on a BL-algebra L was introduced in Ciungu et al. (2011) as a mapping σ : LL satisfying conditions (1) and (3)–(6) in Definition 4.8. We know that BL-algebras are special cases of residuated lattices satisfying the conditions of divisibility and prelinearity. Consequently, a BL-algebra satisfies the property: xy = ((xy) → y) ∧ ((yx) → x) for any x, yL. Therefore, in the case of BL-algebras, condition (4) implies the validity of (2) and conditions (5) and (6) imply the validity of (7) and (8). Hence the notion of a state residuated lattice essentially generalizes that of a state BL-algebra. Moreover, it has been proved (Ciungu et al. 2011) that a mapping σ : L → L is a state operator on an MV-algebra L (Flaminio and Montagna 2007, 2009) if and only if it is a state operator on L taken as a BL-algebra. From this point of view, the notion of a state residuated lattice also generalizes that of a state MV-algebra. Based on this, we have the following results .

Corollary 4.10

Let H be a bounded basic hoop and σ : HH be a mapping on H preserving → . Then the following conditions are equivalent:

1. σ is an IS-map on H;

2. (H,σ) is a state BL-algebra.

Proof

It follows from Proposition 2.8(1) and Theorem 4.9. □

Corollary 4.11

Let H be a bounded Wajsberg hoop and σ : HH be a mapping on H. Then the following conditions are equivalent:

1. σ is an IS-map on H preserving;

2. (H,σ) is a state MV-algebra.

Proof

It follows from Proposition 2.8(2) and Theorem 4.9. □

As we know, every hoop H is a BCK-meet semilattice in which a partial order over H can be defined as usual.

Definition 4.12

(). A state BCK-meet semilattice is a pair (A, σ) where A is a BCK-meet semilattices and σ : AA is a mapping, called state operator, such that for any x, yA the following conditions are satisfied:

1. xy = 1 implies σ(x) → σ(y) = 1;

2. σ(xy) = σ(xy) → y) → σ(y);

3. σ(σ(x) → σ(y)) = σ(x) → σ(y);

4. σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y).

Proposition 4.13

Let H be a hoop and σ : HH be an IS-map on H. Then the {→, ∧} subreduct of (H,σ) is a state BCK-meet semilattice.

Proof

It follows from Definition 3.1 and Definition 4.12. □

Since the class of equality algebra and the class of BCK-∧-semilattice with meet are categorically equivalent, then we have the following result.

Definition 4.14

(). A state equality algebra is a pair (A, σ) where A is an equality algebra and σ : AA is a mapping, called state operator, such that for any x, yA the following conditions are satisfied:

1. xy implies σ(x) ≤ σ(y);

2. σ(x∼ xy) = σ(x∼ xy)∼ y)∼σ(y);

3. σ(σ(x) → σ(y)) = σ(x) → σ(y);

4. σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y).

Proposition 4.15

Let H be a hoop and σ : HH be an IS-map on H. Then the {∼, ∧} subreduction of (H,σ) is a state equality algebra, where xy = x → (xy).

Proof

It follows from Definition 4.14. □

5 State map filters in semihoops

In this section, we introduce state map filters of semihoops.

Definition 5.1

Let H1 and H2 be semihoops, σ : H1H2 be a S-map from H1 to H2, F be a filter of H1. If σ−1(σ(F)) ⊆ F, we call F to be a SM-filter of (H1, H2, σ).

Example 5.2

Consider the Example 3.4, one can easily check that the SM-filter of (H1, H2, σ) are {a1, b1, c1, 1}, {11} and H1.

Example 5.3

Let H1 and H2 be semihoops and σ be a S-map from H1 to H2. Then Ker(σ) = {xH1σ(x) = 12} is a SM-filter of (H1, H2, σ).

Proof

Let K = Ker(σ) and x, yK. Then σ(x) = 12 and σ(y) = 12. By Proposition 3.6(2) we have σ(x1 y) ≥2 σ(x) ⊙2σ(y) = 12212 = 12. This means x1 yK. Let xK and xy. Then 12 = σ(x) ≤ σ(y) and hence σ(y) = 12. This shows that yK. It follows that K is a filter of H1. Moreover let xσ−1σ(K). Then σ(x) ∈ σ(K) = {12} and hence σ(x) = 12. Therefore xK. This shows that σ−1σ(K) ⊆ K, or K is a SM-filter of (H1, σ). □

Definition 5.4

Let H be a semihoop and σ be an IS-map on H.

1. A filter F of H is called state filter of (H,σ) if xF implies σ(x) ∈ F for all xH ,

2. A filter F of H is called dual state filter of (H, σ) if σ(x) ∈ F implies xF for all xH,

3. A filter F of H is called strong state filter of (H, σ) if it is both a state filter and a dual state filter of (H, σ).

Proposition 5.5

Let H be a semihoop and σ be an IS-map on H. Then each SM-filter of H is a state filter on H.

Proof

Let xF. Then σ(x) ∈ σ (F). Therefore, σ(σ(x)) ∈ σ(F), that is σ(x) ∈ σ−1(σ(F)) ⊆ F. So σ(x) ∈ F. □

However, the converse of Proposition 5.5 is not true in general.

Example 5.6

Let H = {0, a, b, 1} with 0 ≤ a, b ≤ 1. Consider the operationandas follows:

$⊙0ab100000a0a0ab00bb10ab1→0ab101111ab1b1baa1110ab1$

Then H is semihoop. Now, we define σ follows: σ0 = a, σ a = a, σ b = 1, σ 1 = 1. One can easily check that σ is an IS-map on H. It is clear that {a, 1} is a state filter of (H, σ), but it is not a SM-filter of (H, σ).

Proposition 5.7

Let H be a semihoop, σ be an IS-map on H and FH. Then the following are equivalent:

1. F is a SM-filter of H,

2. F is a strong state filter on H.

Proof

(1) ⇒ (2) Let F be a SM-filter of (H, σ). By Proposition 5.5 we only need to prove that σ(x) ∈ F implies xF. Let σ(x) ∈ F. Then σ(x) = σ(σ(x)) ∈ σ(F). Hence there is tF such that σ(x) = σ(t). It follows from (1) that xσ−1(σ(t)) ⊆ σ−1(σ(F)) ⊆ F. That is xF.

(2)⇒ (1) Assume that F is a strong state filter on H. For xσ−1(σ(F)), we have σ(x) ∈ σ(F). Since F is strong filter of H, we get xF and hence σ−1(σ(F)) ⊆ F. □

Let H1 and H2 be two semihoops and σ be a S-map from H1 to H2. For any nonempty set X of H1, we denote by 〈Xσ the SM-filter of (H1, σ) generated by X, that is, 〈Xσ is the smallest SM-filter of (H1, σ) containing X.

Let H be be a semihoop and σ be an IS-map on H. For any nonempty set X of H, we denote by 〈XS (〈XDS) the state filter (the dual state filter) of (H, σ) generated by X, that is, 〈XS (〈XDS) is the smallest state filter (the dual state filter) of (H, σ) containing X.

Denote (X)DS = {xHσ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xmσ(xm), xiX}. In the following we discuss the structures of 〈XS, 〈XDS and 〈Xσ.

Theorem 5.8

Let H be a semihoop, σ be an IS-map on H and XH. Then

1. XS = {xHxx1σ(x1) ⊙ ⋯ ⊙ xnσ(xn), xiX, mN},

2. (X)DS is a dual state filter of (H, σ) containing X, and henceXDS ⊆ (X)DS,

3. Xσ = 〈XS ∪ (X)DS.

Proof

1. The proof is similar to that of He et al .(Theorem 4.13).

2. Let x, y ∈ (X)DS. Then σ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for some xiX, nN and σ(y) ≥ y1σ(y1) ⊙ ⋯ ⊙ ymσ(ym) for some yjX, mN. Hence σ(xy) ≥ σ(x) ⊙ σ(y) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) ⊙ y1σ(y1) ⊙ ⋯ ⊙ ymσ(ym). So xy ∈ (X)DS. Assume xy and x ∈ (X)DS. Then σ(y) ≥ σ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for some xiX. It follows that y ∈ (X)DS. This shows that (X)DS is a filter of H. Moreover, let σ(x) ∈ (X)DS. Then σ(σ(x)) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for some xiX and hence σ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for some xiX. This shows that x ∈ (X)DS and hence (X)DS is a dual state filter of (H, σ). Clearly X ⊆ (X)DS.

3. Denote B = 〈XS ∪ (X)DS. Let x, yB. If x, y ∈ 〈XS, then xy ∈ 〈XSB by (1). If x, y ∈ (X)DS, then xy ∈ (X)DSB by (2). Let x ∈ 〈XS and y ∈ (X)DS. Then σ(x) ∈ 〈XS since 〈XS is a state filter of (H, σ) by (1). Hence σ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for some xiX and σ(y) ≥ y1σ(y1) ⊙ ⋯ ⊙ ymσ(ym) for some yjX and hence σ(xy) ≥ σ(x) ⊙ σ(y) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) ⊙ y1σ(y1) ⊙ ⋯ ⊙ ymσ(ym) for some xi,yjX. It follows that xy ∈ (X)DSB. Combining the above arguments we get that B is closed on ⊙. It is easy to check that if xB and xy then yB. Clearly XB. Now we prove that B is a state filter. Let xB. If x ∈ 〈XS, then σ(x) ∈ 〈XS since 〈XS is a state filter. If x ∈ (X)DS, then σ(x) ∈ 〈XSB. So B is a state filter. Moreover we prove that B is a dual state filter. Let σ(x) ∈ B. If σ(x) ∈ 〈XS, then x ∈ (X)DSB. Let σ(x) ∈ (X)DS. Then x ∈ (X)DS since (X)DS is a dual state filter by (2). This shows that B a dual state filter. By Proposition 5.7, B a SM-filter. Let F be a SM-filter of (H, σ) containing X and xB. If x ∈ 〈XS, then xx1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for xiX. Since XF and F is a SM-filter of (H, σ), we have x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) ∈ F. So xF. If x ∈ (X)DS, then σ(x) ∈ 〈XS by (1). If σ(x) ≥ x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) for xiX. Since F is a SM-filter of (H, σ) containing X, then x1σ(x1) ⊙ ⋯ ⊙ xnσ(xn) ∈ F and hence σ(x) ∈ F. Note that F is also a dual state filter, we have xF. Combining the above arguments we get BF. It follows that B = 〈Xσ. □

Proposition 5.9

Let H be a semihoop, σ be an IS-map and F be state filters of (H, σ) and aF. Then

1. aσ = {xHx ≥ (aσ(a))n, n ≥ 1} ∪ {xHσ(x) ≥ (aσ(a))n, n ≥ 1},

2. F, {a}〉σ = {xHxf ⊙ (aσ(a))n, fF, n ≥ 1} ∪ {xHσ(x) ≥ f ⊙ (aσ(a))n, fF, n ≥ 1},

3. if ab, thenbσ ⊆ 〈aσ,

4. aaσ = 〈aσ,

5. σ(a)〉σ = 〈aσ,

6. aσ(a)〉σ = 〈aσ,

7. if H is a ⊔-semihoop, thenaσ ∩ 〈bσ = 〈(aσ(a)) ⊔ (bσ(b))〉σ.

Proof

The proofs of (1)–(4) are obvious.

(5) Let x ∈ 〈σ(a)〉σ. Then x ≥ (σ(a) ⊙ σ2(a))n = (σ(a))2n ≥ (aσ(a))2n or σ(x) ≥ (aσ(a))2n and hence x ∈ 〈aσ. Conversely, let x ∈ 〈aσ. Then x ≥ (aσ(a))n or σ(x) ≥ (aσ(a))n. Hence σ(x) ≥ (σ(a) ⊙ σ2(a))n. It follows that σ(x) ∈ 〈σ(a)〉σ. Since 〈σ(a)〉σ is a dual state filter we have x ∈ 〈σ(a)〉σ.

(6) Since aσ(a) ≤ a we have 〈aσ ⊆ 〈aσ(a)〉σ by (3). Conversely, by use of (3), (4) and (5) we have 〈aσ(a)〉σ = 〈σ(aσ(a))〉σ ⊆ 〈σ(a) ⊙ σ2(a)〉σ = 〈σ(a) ⊙ σ(a)〉σ = 〈 σ(a)〉σ = 〈aσ.

(7) Suppose that H is a ⊔-semihoop. From aσ(a) ≤ (aσ(a)) ⊔ (bσ(b)), we have that 〈(aσ(a)) ⊔ (bσ(b))〉σ ⊆ 〈aσ(a)〉σ = 〈aσ. Similarly, we can prove 〈(aσ(a)) ⊔ (bσ(b))〉σ ⊆ 〈bσ. Thus, 〈(aσ(a)) ⊔ (bσ(b))〉σ ⊆ 〈aσ ∩ 〈bσ. Conversely, let x ∈ 〈aσ ∩ 〈bσ. Then there exist n, m, s, t ≥ 1, such that x ≥ (aσ(a))n or σ(x) ≥ (aσ(a))n, and x ≥ (bσ(b))n or σ(x) ≥ (bσ(b))n. To complete the proof, we divide four cases as following:

1. Let x ≥ (aσ(a))n and x ≥ (bσ(b))n. Then x ≥ (aσ(a))n ⊔ (bσ(b))n ≥ ((aσ(a)) ⊔ (bσ(b)))ns ≥ (((aσ(a)) ⊔ (bσ(b))) ⊙ σ((aσ(a)) ⊔ (bσ(b))))ns by Proposition 2.6(3). We deduce that x ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ.

2. Let σ(x) ≥ (aσ(a))n and σ(x) ≥ (bσ(b))n. Similarly to (a) we can get σ(x) ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ. Since 〈(aσ(a)) ⊔ (bσ(b))〉σ is a dual state filter we have x ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ.

3. Let x ≥ (aσ(a))n and σ(x) ≥ (bσ(b))n. Then σ(x) ≥ (σ(a) ⊙ σ(a))n ≥ (aσ(a))2n and σ(x) ≥ (bσ(b))n. Thus σ(x) ≥ (aσ(a))2n ⊔ (bσ(b))n ≥ ((aσ(a)) ⊔ (bσ(b)))2nt ≥ (((aσ(a)) ⊔ (bσ(b))) ⊙ σ((aσ(a)) ⊔ (bσ(b))))2nt. It follows that σ(x) ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ. Since 〈(aσ(a)) ⊔ (bσ(b))〉σ is a dual state filter we have x ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ.

4. Let σ(x) ≥ (aσ(a))n and x ≥ (bσ(b))n. Similarly to the case (c) we can get x ∈ 〈(aσ(a)) ⊔ (bσ(b))〉σ.

Combining the above arguments we can prove 〈aσ ∩ 〈bσ ⊆ 〈(aσ(a)) ⊔ (bσ(b))〉σ. Therefore 〈aσ ∩ 〈bσ = 〈(aσ(a)) ⊔ (bσ(b))〉σ. □

Definition 5.10

Let H1 and H2 be two semihoops and σ be a S-map from H1 to H2. A proper SM-filter F of (H1, H2, σ) is called a prime SM-filter of (H1, H2, σ), if for all SM-filters F1, F2 of (H1, σ) such that F1F2F, then F1F or F2F.

Let H1 and H2 be two semihoops and σ be a S-map from H1 to H2. We denote by PSMF[H] the set of all prime SM-filters of (H1, σ).

Example 5.11

Consider the Example 3.4, one can check that F = {a1, b1, c1, 11} is a prime SM-filter of (H1, σ).

Theorem 5.12

Let H be a ⊔-semihoop, σ be an IS-map and F be a proper SM-filter of (H, σ). Then the following are equivalent:

1. F is a prime SM-filter of (H, σ),

2. if ((xσ(x)) ⊔ (yσ(y))) ⊙ σ((xσ(x)) ⊔ (yσ(y))) ∈ F for some x, yH, then xF or yF.

Proof

(1) ⇒ (2) Let ((xσ(x)) ⊔ (yσ(y))) ⊙ σ((xσ(x)) ⊔ (yσ(y))) ∈ F for some x, yH. Then 〈xσ ∩ 〈yσ = 〈(xσ(x)) ⊔ (yσ(y))〉σF. Since F is a prime SM-filter of (H, σ), then 〈xσF or 〈yσF. Therefore, xF or yF.

(2) ⇒ (1) Suppose that F1, F2 ∈ SMF[L] such that F1F2F and F1F and F2F. Then there exist xF1 and yF2 such that x, yF. Since F1, F2 are SM-filter of (H, σ), then xσ(x) ∈ F1 and yσ(y) ∈ F2. From xσ(x), yσ(y) ≤ (xσ(x)) ⊔ (yσ(y)), we obtain (xσ(x)) ⊔ (yσ(y)) ∈ F1F2F and hence ((xσ(x)) ⊔ (yσ(y))) ⊙ σ((xσ(x)) ⊔ (yσ(y))) ∈ F1F2F. By (2), we get that xF or yF, which is a contradiction. Therefore, F is a prime SM-filter of (H, σ). □

Definition 5.13

Let H1 and H2 be two semihoops and σ be a S-map from H1 to H2. A proper SM-filter of (H1, H2, σ) is called a maximal SM-filter if it not strictly contained in any proper SM-filter of (H1, H2, σ).

Example 5.14

Let H1 and H2 be two semihoops and σ be a S-map from H1 to H2 in Example 3.4. One can easily check that F = {a1, b1, c1, 11} is a maximal SM-filter of (H1, H2, σ).

Proposition 5.15

Let H be a bounded ⊔-semihoop, σ be an IS-map and F be a proper SM-filter of (H, σ). Then the following are equivalent:

1. F is a maximal SM-filter of (H, σ),

2. for any aF, there is an integer n ≥ 1 such that (σ(a)n)F.

Proof

(1) ⇒ (2) Suppose that F is a maximal SM-filter of (H, σ), and let aF. Then 〈F,aσ = H, which implies 0 ∈ 〈F, aσ. Then there is fF and an integer n ≥ 1 such that 0 = f ⊙ (aσ(a))n. So we have 0 = σ(0) ≥ σ(f) ⊙ σ(a)2n. Therefore, σ(f) ≥ (σ(a)2n). Thus, (σ(a)2n)F.

(2) ⇒ (1) Let a satisfy the condition. Since (σ(a)n) ⊙ (aσ(a))n ≤ (σ(a)n) ⊙ (σ(a))n = 0 and (σ(a)n)F, we obtain 0 ∈ 〈F,aσ, that is, 〈F, aσ = H. Therefore, F is a maximal SM-filter of (H, σ). □

Proposition 5.16

Let H1 and H2 be two bounded semihoops and σ be a S-map from H1 to H2.

1. If F2 is filter of σ(H1), then σ−1(F2) is a SM-filter of (H1, H2, σ).

2. If σ is an IS-map on H and F is a maximal filter of σ H, then σ−1(F) is a maximal SM-filter of H.

Proof

1. Suppose that F2 is a filter of σ(H1). If x, yσ−1(F2), then σ(x),σ(y) ∈ F2. It follows that σ(x) ⊙ σ(y) ∈ F2. Since σ(xy) ≥ σ(x) ⊙ σ(y) and σ(xy) ∈ σ(H1), we have σ(xy) ∈ F2, that is, xyσ−1(F2). Let x, yH1 such that xσ−1(F2) and xy. Then σ(x) ≤ σ(y). Since σ(x) ∈ F and σ(y) ∈ σ(H1), we can obtain that σ(y) ∈ F2, that is, yσ−1(F2). Thus, σ−1(F2) is a filter of H1. Note that σ(σ−1(x) = x for any xH1. Hence σ−1(σ(σ−1(F)) = σ−1(F). Thus σ−1(F2) is a SM-filter of H1.

2. Now, suppose that F is a maximal filter of σ(H). Let aσ−1(F), thus σ(a) ∉ F. By the maximality of F, there is an integer n ≥ 1 such that (σ(a)n)Fσ(H). Since σ ((σ(a)n)) = (σ(a)n)F, we have (σ(a)n)σ−1(F). Therefore, σ−1(F) is a maximal SM-filter of H. □

Proposition 5.17

Let H be a bounded semihoop and σ be an IS-map on H preserving ⊙.

1. If F is a SM-filter of (H, σ), then σ(F) is a SM-filter of (σ(H),σ).

2. If F is a maximal SM-filter of (H, σ), then σ(F) is a maximal SM-filter of (σ(H),σ).

Proof

1. Let σ(x),σ(y) ∈ σ(F), then x, yσ−1σ(F) ⊆ F. Since F is a filter, thus xyF and hence σ(x) ⊙ σ(y) = σ(xy) ∈ σ(F). Let σ(x), σ(y) ∈ σ(H) such that σ(x) ∈ σ(F) and σ(x) ≤ σ(y). Since σ(x) ∈ σ(F) we have xσ−1σ(F) ⊆ F. So xF. By Proposition 5.7 we have σ(x) ∈ F. Since σ(x) ≤ σ(y) we get σ(y) ∈ F. Using Proposition 5.7 again we obtain yF, and so σ(y) ∈ σ(F). Thus, σ(F) is a filter of σ(H). Now let xσ(F). Then x = σ(t) for some tF and hence σ(x) = σ2(t) = σ(t) = xσ(F). It follows that σ(F) is a state filter of (H, σ). Let xσ(H) and σ(x) ∈ σ(F). Then x = σ(t) for some tH. Hence x = σ(t) = σ2(t) = σ(σ(t)) = σ(x) ∈ σ(F). This means that σ(F) is a dual state filter of (σ(H),σ). Therefore σ(H) is a strong state filter of (σ(H),σ). By Proposition 5.7 we have that σ(F) is a SM-filter of (σ(H),σ).

2. Now, let F be maximal and σ(a) ∉ σ(F). Then aF, and there is an integer n ≥ 1 such that (σ(a)n)F and hence σ ((σ(a)n)) = (σ(a)n)σ(F). Since σ(σ(a)n) ≥ (σσ(a))n = (σ(a))n, we have (σ(a)n)σ ((σ(a)n)). Hence (σ(a)n)σ(F). Therefore, σ(F) is a maximal SM-filter of (σ(H),σ). □

Corollary 5.18

Let H be a bounded semihoop and σ be an IS-map on H.

1. If F is a (maximal) filter of σ(H), then σ−1(F) is a strong state (maximal)filter of (H, σ).

2. If σ is preservingand F is a strong state (maximal) filter of (H, σ), then σ(F) is a strong state (maximal) filter of (σ(H),σ).

Proof

1. It follows from Proposition 5.7 and 5.16.

2. It follows from Proposition 5.7 and 5.17. □

Now, we introduce two kinds of semihoops and give some characterizations of them.

Definition 5.19

Let H be a semihoop and σ : HH be an IS-map on H. If (H, σ) has exactly one maximal SM-filter, we call (H, σ) to be state local.

Theorem 5.20

Let H be a semihoop and σ be an IS-map on H. Then the following are equivalent:

1. (H, σ) is state local;

2. σ(H) is local.

Proof

(1) ⇒ (2) Let F be the only maximal SM-filter of (H, σ). We prove that σ(F) is the only maximal filter of σ(H). First, σ(F) is a proper filter of σ(H). In fact, if σ(F) = σ(H), then 0 ∈ σ(F), which implies 0 ∈ F, a contradiction. Now, let G be a filter of σ(H), Gσ(H) and let xG. It follows from Corollary 5.18(1) that σ−1(G) is a SM-filter of (H, σ). Thus σ−1(G) is a proper SM-filter of (H, σ). Moreover, if σ−1(G) = H, then 0 ∈ σ−1(G), so 0 ∈ G, a contradiction. It follows that σ−1(G) ⊆ F. if x = σ(x) ∈ G, then xσ−1(G), it follows that xF. But x = σ(x), so xσ(H). Thus Gσ(F). Hence σ(G) is the only maximal filter of σ(H). Therefore, σ(H) is local.

(2) ⇒ (1) Suppose that G is the only maximal filter of σ(H). By Corollary 5.18(1), we have that σ−1(G) is a maximal SM-filter of (H, σ). We will prove that σ−1(G) is the only maximal SM-filter of (H, σ). Let G be a SM-filter of (H, σ), FL. Then σ(F) is a proper filter of σ(H), so σ(F) ⊆ G. Let xF then σ(x) ∈ σ(F) ⊆ G. Thus, xσ−1(G). It follows that Fσ−1(G). Therefore, (H, σ) is state local. □

Definition 5.21

Let H a be semihoop and σ : HH be an IS-map on H. If (H, σ) has two SM-filters {1} and H, we call (H, σ) to be simple.

Theorem 5.22

Let H a be semihoop and σ : HH be an IS-map on H such that σ preserving ⊙ . Then the following are equivalent:

1. (H, σ) is simple;

2. σ(H) is simple and Ker(σ) = {1}.

Proof

(1) ⇒ (2) Let F be a filter of σ(H) and F ≠ {1}. It follows from Corollary 5.18(1) that σ−1F is a SM-fiter of (H, σ). Since (H, σ) is state simple, we have that σ−1(F) = {1} or σ−1(F) = H. Notice that Fσ−1F (if xF, then σ x = x, that is, xσ−1F, we obtain that σ−1F ≠ {1}. Thus, σ−1F = H. Then 0 ∈ σ−1F, that is, 0 = σ 0 ∈ F. So we obtain that F = σ H. Therefore, σ H is simple.

By Example 5.3 we have Ker(σ) is a SM-filter of (H, σ) and Ker(σ) ≠ H. It follows that Ker(σ) = {1}.

(2) ⇒ (1) Let F be a SM-filter of (H, σ) and F ≠ {1}. By Corollary 5.18(2), we obtain that σ F is a filter of σ H. Since σ H is simple, we obtain that σ F = {1} or σ F = σ x. Since Ker(σ) = {1}, we have F ≠ {1}. Thus, σ F = σ x. Then 0 ∈ σ F, that is, 0 ∈ F. It follows that F = H. Therefore (H, σ) is state simple. □

6 Conclusion

We observed that the states and state operators on MV-algebras, BL-algebras and BCK-algebras, are maps from an algebra X to [0, 1] and X to X, respectively. From the viewpoint of universal algebras, it is meaningful to study a state map from an algebra X to anther algebra Y. Indeed, if Y = [0, 1], a state can be seen as a state map from X to [0, 1], and if X = Y, a state operator can also be seen as a state map from XX. Based on this idea, we introduce a notion of state maps on semihoops by extending the codomain of a state (or internal state) to a more general algebraic structure, that is, from a semihoop H1 to an arbitrary semihoop H2. We give a type of special state map from a semihoop H to H, called internal state map (or IS-map), which is a generalization of internal states (or state operators) on some types of semihoops. We try to give a unified model of states and internal states on some important logic algebras. By the arguments in the paper we can see that state maps on an semihoops are generalization of internal states on BL-algebras, MV-algebras, equality algebras and BCK-algebras. In the next work, it is worthy to portray some types of logic algebras and corresponding logics by use of state maps.

Acknowledgement

This research is partially supported by a grant of National Natural Science Foundation of China (11571281, 61602359), China Postdoctoral Science Foundation (2015M582618), China 111 Project (B16037).

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Accepted: 2018-06-22

Published Online: 2018-09-18

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1061–1076, ISSN (Online) 2391-5455,

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