Q -orders with no unit on Q

Abstract

For a usual commutative quantale Q (does not necessarily have a unit), we propose a definition of Q-ordered sets by introducing a kind of self-adaptive self-reflexivity. We study their completeness and the related Q-modules of complete lattices. The main result is that, the complete Q-ordered sets and the Q-modules of complete lattices are categorical isomorphic.

Keywords

Commutative quantale

1 Introduction

The history of fuzzy orders can be traced back to the late seventies of Zadeh’s paper [33] and even to the late fifties of Menger’s paper [22]. From then on, different kinds of fuzzy orders were introduced and studied by different authors [1 , 24] and have become increasingly important in applications such as fuzzy preference modeling [15], fuzzy topology [13], quantitative domains [21 , 29–32], fuzzy algebra [6 , 28] and fuzzy formal concept analysis [19 , 25].

At the very beginning stage of fuzzy orders and related structures, the truth value table is the unit interval [0,1] and axioms are defined by using the intrinsic operation on [0,1]. For example, Menger [22] used the usual multiplication between numbers to define the probability of indistinguishable relation (a kind of fuzzy equivalence relation), and Zadeh [33] used the min operation to define a kind of fuzzy orders. Then in 1978, Bezdek and Harris [3] used the Lukasiewiecz norm to define and study a kind of fuzzy partitions.

Since Chang introduced MV-algebra [7] and Pavelka put residuated lattices into the framework of fuzzy logic [23], researches have increasingly realized that fuzzy logic is an important tool to deal with fuzziness and uncertainty. At this regard, we can try to replace the usual operations on [0, 1] to some more general operations to represent the logical conjunction and implication, and also we can generalize [0, 1] to some more general logical algebras, for example, to residuated lattices and even to commutative unital quantales.

Logical algebra strongly relies on a continuous t-norm and the induced implication operation. In 1985, for F being a continuous t-norm on [0,1], Valverde introduced a concept of F-preorder as a kind of fuzzy orders [27]. For a special complete residuated lattice L, Höhle gave a concept of partial order on an L-underdeterminate set. In [4, 5], for T being a t-norm [0,1], Bodenhofer introduced and studied T-E-orders w.r.t. a T-equivalence relation E. Bělohlávek proposed L-orders, a kind of fuzzy orders, with a complete residuated lattice as the truth value table [1, 2]. In [9, 10], Demirci extended Bodenhofer’s T-E-orders to L-E-orders, from [0,1] to an integral, commutative, complete, quasi monoidal-lattice [8, 17] (an iccqm-lattice, for short) (L, * , ≤) and then studied vague lattices. An L-E-order unifies the notions of T-E-orders [4, 5], L-orders [1, 2] and partial orders on an L-underdeterminate set [16] within the same framework.

Besides other applications, fuzzy ordered structures are specific fundamental structures to study quantitative domain theory. In [12, 34], for L being a complete Heyting algebra, a kind of fuzzy order, namely the degree functions, was defined and studied by Fan and Zhang, which yields the fuzzy set approach to quantitative domain theory. In [29, 30] and consequently in [21], for Q being a commutative unital quantale, a kind of fuzzy ordered set, called Q-categories, is introduced to study quantitative domain structures, which leads to the enriched category approach to quantitative domain theory via fuzzy sets. Indeed, it is routine to show that the fuzzy order in the sense of Bělohlávek and that in the sense of Fan and Zhang are equivalent to each other (cf. Section 3 in [31]), both of which are special cases of Q-categories.

We know that a partial order on a set is a self-reflexive, transitive and antisymmetric binary relation. Accordingly, a fuzzy partial order on a set should be a kind of fuzzy binary relation with certain fuzzy versions of self-reflectivity, transitivity and antisymmetry. Roughly speaking, there are different kinds of fuzzy orders since there are different kinds of fuzzy versions of antisymmetry. While in this paper, let us now focus on the self-reflexivity instead of the antisymmetry.

Since the classical self-reflexivity of an ordered set (X, ≤) says that “x ≤ x for all x ∈ X”, the self-reflexivity of a fuzzy ordered set (X, e) should say that “e (x, x) is larger than or equal to the truth value” in the truth value table. For example, for case of a residuated lattice L, the axiom is “e (x, x) =1 (∀ x ∈ X)”; and for the case of a unital quantale (Q, k) (k is the unit), the axiom is “e (x, x) ≥ k (∀ x ∈ X)”. This basic axiom is crucial for further study on the properties and propositional reasoning of related structures. We can see that both of the top element 1 in a residuated lattice and the element k in a unital quantale are the unit of the related t-norm (the conjunction in logical sense), which is the truth value (the minimal true value) in the truth value table. We now have an interesting question, that is,

How can we define and deal with fuzzy orders for the truth value table without a unit (i.e., without an element corresponding to the value of truth)?

This question looks strange since it seems that we can do nothing in dealing with fuzzy structures without the fundamental base of fuzzy logic, while it is really important if we pay more attention to the algebraic features of fuzzy orders within a relatively lax logical framework.

In this paper we will try to give a model of fuzzy orders for a truth value table without a unit. In fact, let (Q, k) be a unital quantale, for a fuzzy ordered set, if we use e (x, y) to represent the degree of “x being less than or equal to y”, then we always write the self-reflexivity as k ≤ e (x, x). It is easily seen that λ ≥ k iff a ≤ a * λ holds for all a ∈ Q. That is to say, for a nonunital quantale-valued fuzzy ordered set (X, e), we may use the set {e (x, x) | x ∈ X} as an index set of nonfixed formal unit rather than the fixed unit k.

After the basic definitions and properties of nonunital quantale-valued Q-ordered sets, we will mainly prove that, for this kind of fuzzy ordered sets, there is a categorial isomorphism between complete Q-ordered sets and Q-modules over complete lattices. This result has been obtained by Stubbe in the setting of quantaloid enriched categories [26] (a quantaloid enriched category is an extension of a commutative unital quantale-valued fuzzy order).

2 Quantale-valued fuzzy ordered sets

In this section, for a usual commutative quantale Q, we introduce a concept of Q-ordered sets and then study their basic properties.

Definition 2.1. A commutative quantale is a pair (Q, *) satisfying that

(Q1) (Q, *) is a commutative semigroup;

(Q2) Q is a complete lattice;

(Q3) a * (⋁ _ib_i) = ⋁ _i (a * b_i) for all a, b_i ∈ Q (i ∈ I).

Condition (Q3) is equivalent to

(Q4) There is a binary operation → on Q satisfying that $a * b \leq c \Leftrightarrow a \leq b \to c (\forall a, b, c \in Q) .$

Proposition 2.2. Let Q be a commutative quantale. Then for all a, b, c ∈ Q and all {a_i| ı ∈ I} ⊆ Q, it holds that,

(1) a * (a → b) ≤ b;

(2) b ≤ a → (a * b);

(3) (a → b) * (b → r) ≤ a → b;

(4) (⋁ _ia_i) → b = ⋀ _i (a_i → b);

(5) b → (⋀ _ia_i) = ⋀ _i (b → a_i);

(6) (c → a) → (c → b) ≥ a → b;

(7) (a → c) → (b → c) ≥ b → a;

(8) a → (b → c) = (a * b) → c.

Example 2.3. (1) The unit interval equipped with a left-continuous t-norm is a commutative quantale.

(2) Every complete lattice is a commutative quantale by considering the join operation as the quantale product.

(3) Let Q = {0, a, b, 1} ordered by 0 < a, b < 1 and a ∥ b. The operations *, → on Q are listed in the following tabular:

*	a	b	1
0	0	0	0
a	a	1	1
b	1	b	1
1	1	1	1

→	0	a	b	1
0	1	1	1	1
a	0	a	0	1
b	0	0	b	1
1	0	0	0	1

Then Q is a commutative with no unit.

Proposition 2.4. Let Q be a commutative quantale and a ∈ Q. Then the following statements are equivalent:

(1) x ≤ a * x for any x ∈ X;

(2) a → x ≤ x for any x ∈ X.

Proof. (1) ⇒ (2): For any x ∈ X, a → x ≤ a * (a → x) ≤ x.

(2) ⇒ (1): For any x ∈ X, x ≤ a → (a * x) ≤ a * x.□

Definition 2.5. Let Q be a commutative quantale. A binary Q-relation e : X × X ⟶ Q is called a Q-order if

(QO1) a ≤ a * e (x, x) for all a ∈ Q, x ∈ X;

(QO2) e (x, y) * e (y, z) ≤ e (x, z) for all x, y, z ∈ X.

The pair (X, e) is called a Q-ordered set.

Example 2.6. In fuzzy order theory, it is important to make the truth value table itself to be a fuzzy ordered set. For a quantale Q. Define e_Q (a, b) = a → b. Then e_Q is a Q-order iff λ ≤ λ * (μ → μ) for all λ, μ ∈ Q. It is hard to make all quantale to be a Q-ordered set, but every unital quantale satisfies this condition, and especially the lattice in Example 2.3(3) is a standard nonunital quantale satisfying this condition.

Proposition 2.7. In a Q-ordered set (X, e), for any x ∈ X, it holds that $e (x, x) * e (x, y) = e (x, y) = e (y, y) * e (x, y) .$

Proof. Firstly, by (QO1), e (x, x) * e (x, y) ≥ e (x, y) ≤ e (y, y) * e (x, y). Secondly, by (QO2), e (x, x) * e (x, y) ≤ e (x, y) and e (y, y) * e (x, y) ≤ e (x, y).□

Proposition 2.8. In a Q-ordered set (X, e), for any x, y ∈ X, it holds that

(1) e (x, x) → e (x, y) = e (x, y);

(2) e (x, x) → e (y, x) = e (y, x).

Proof. We only need to prove (1). Firstly, e (x, y) * e (x, x) ≤ e (x, y) and then e (x, y) ≤ e (x, x) → e (x, y). Secondly, by Lemma 2.1, e (x, y) ≥ e (x, x) → e (x, y).□

Example 2.9. Propositions 2.7 and 2.8 indicate that a quantale may have no unit, while for a Q-ordered set (X, e), the elements e (x, x) and e (y, y) can act as the units of e (x, y).

Theorem 2.10. In a Q-ordered set (X, e), for any x, y ∈ X, it holds that

(1) e (x, y) = ⋀ _z∈Xe (z, x) → e (z, y);

(2) e (x, y) = ⋀ _z∈Xe (y, z) → e (x, z).

Proof. Here we only prove (1). Firstly, e (x, y) * e (z, x) ≤ e (z, y) for each z ∈ X, and then e (x, y) ≤ ⋀ _z∈Xe (z, x) → e (z, y). Secondly, ⋀_z∈Xe (z, x) → e (z, y) ≤ e (x, x) → e (x, y) ≤ e (x, y). Hence, e (x, y) = ⋀ _z∈Xe (z, x) → e (z, y).□

Let (X, e) be a Q-ordered set. For each A ∈ Q^X, define A^l, A^u ∈ Q^X by $A^{l} (x) = ⋀_{y \in X} A (y) \to e (x, y);$ $A^{u} (x) = ⋀_{y \in X} A (y) \to e (y, x) .$

Definition 2.11. Let (X, e) be a Q-ordered set. An element x is called a supremum of A ∈ Q^X, in symbols x = sup A, if

(J1) for any z ∈ X, A (z) ≤ e (z, x);

(J2) for any y ∈ X, A^u (y) ≤ e (x, y);

(J3) for any y ∈ X, A^u (y) = e (x, y).

Definition 2.12. Let (X, e) be a Q-ordered set. An element x is called an infimum of A ∈ Q^X, in symbols x = inf A, if

(M1) for any z ∈ X, A (z) ≤ e (x, z);

(M2) for any y ∈ X, A^l (y) ≤ e (x, y);

(M3) for any y ∈ X, A^l (y) = e (x, y).

Proposition 2.13. Let (X, e) be a Q-order set. Then

(1) (J1)+(J2) ⇔ (J3);

(2) (M1)+(M2) ⇔ (M3). Proof. (1) ⇒: We only need to verify A^u (y) ≥ e (x, y) for all x, y ∈ X. In fact, e (x, y) * A (z) ≤ e (x, y) * e (z, x) ≤ e (z, y). Then e (x, y) ≤ A (z) → e (z, y) and thus A^u (y) = ⋀ _z∈XA (z) → e (z, y) ≥ e (x, y).

⟸: We only need to show (J1). In fact, since e (x, x) = A^u (x) = ⋀ _z∈XA (z) → e (z, x), we have e (z, x) ≥ A (z) * e (x, x) ≥ A (z) (∀ z ∈ X).

(2) It can be proved similarly.□

A Q-ordered set is called antisymmetric if

(QA) for all x, y ∈ X, a ≤ (a * e (x, y)) ∧ (a * e (x, y)) (∀ a ∈ Q) implies x = y.

Proposition 2.14. For an antisymmetric Q-ordered set (X, e), then supremum and infimum of Q-subsets of X are unique if they exist, that is, for every A ∈ Q^X,

(1) if x = sup A = y, then x = y;

(2) if x = inf A = y, then x = y.

Proof. (1) We only need to show that a ≤ a * e (x, y) holds for every a ∈ Q. In fact, $\begin{matrix} a * e (x, y) & = & a * ⋀_{z \in X} A (z) \to e (z, y) \\ \geq & a * ⋀_{z \in X} e (z, y) \to e (z, y) \\ \geq & a * e (y, y) \geq a . \end{matrix}$

(2) It can be proved similarly.□

Proposition 2.15. Let (X, e) be a Q-ordered set and A ∈ Q^X. Then sup A = inf A^u, inf A = sup A^l, if they exist.

Proof. Let x = sup A. We need to show that for any y ∈ X. ⋀_z∈XA^u (z) → e (y, z) = e (y, x). Clearly, A^u (z) = e (x, z). Then ⋀_z∈XA^u (z) → e (y, z) = ⋀ _z∈Xe (x, z) → e (y, z) = e (y, x). Hence, x = inf A^u. Similarly, inf A = sup A^l.□

Definition 2.16. A Q-ordered set (X, e) is called complete if sup A exists for any A ∈ Q^X, or equivalently, inf A exists for any A ∈ Q^X.

3 Complete Q-ordered sets vs Q-modules of complete lattices

In [26], in the setting of quantaloid $Q$ , Stubbe proved that complete $Q$ -ordered sets and $Q$ -modules of complete lattices are categorical isomorphic. In this section, we will prove that this result holds for every Q-ordered sets.

Theorem 3.1. Let (X, e) be a Q-ordered set. Then the cut of Q-order, $| e | = {(x, y) \in X \times X | e (x, x) \leq e (x, y)},$ is a classical order on X, the related ordered set is denoted by |X|, i.e., |X| = (X, |e|).

Proof. If (x, y) ∈ |e|, then we write it as x ≤ _ey or y ≥ _ex. Firstly, (O1) is obvious. Secondly, for (O2), if x ≤ _ey and y ≤ _ez, then e (x, x) ≤ e (x, y) and e (y, y) ≤ e (y, z). Thus, $e (x, z) \geq e (x, y) * e (y, z) \geq e (x, x) * e (y, y) \geq e (x, x) .$ Hence, x ≤ _ez.□

Example 3.2. If the quantale Q has a unit k, then |e| = {(x, y) | k ≤ e (x, y)}. We only need to show that k ≤ e (x, y) iff e (x, x) ≤ e (x, y) for all x, y ∈ X. In fact, since it is clear k ≤ e (x, x), the leftward implication is obvious; for the rightward implication, if k ≤ e (x, y), then e (x, x) = e (x, x) * k ≤ e (x, x) * e (x, y) ≤ e (x, y). This result again indicates that if Q is a nonunital quantale, then {e (x, x) | x ∈ X} can be used as the indexed set for every Q-ordered set (X, e). We could call Condition (QO1) a dynamic or a self-adaptive version of self-reflexivity of Q-ordered sets.

Theorem 3.3. If e is a complete Q-order on X, then |e| is a complete order on X.

Proof. For each A ⊆ X, define $\tilde{A} (x) = e (x, x) (\forall x \in A)$ and $\tilde{A} = 0$ otherwise. Put $x = sup \tilde{A}$ . Let us prove that x = ⋁ A. For each y ∈ X, we have $\begin{matrix} e (x, y) & = & (\tilde{A})^{u} (y) \\ = & ⋀_{z \in X} \tilde{A} (z) \to e (z, y) \\ = & ⋀_{z \in A} e (z, z) \to e (z, y) \\ = & ⋀_{z \in A} e (z, y) . \end{matrix}$ For any z ∈ A, we have e (x, x) ≤ e (z, x) and then z ≤ _ex. That is to say, x is an upper bound of A in (X, |e|). Let y be an upper bound of A in (X, |e|). Then e (z, y) ≥ e (y, y) for any z ∈ A. Then e (x, y) = ⋀ _z∈Ae (z, y) ≥ e (y, y), and thus x ≤ _ey. Therefore, x = ⋁ A holds in (X, |e|). By the arbitrariness of A ⊆ X, |e| is a complete order on X.□

Theorem 3.4. In a complete Q-ordered set, for all {x_i| i ∈ I} ⊆ X, we have e (⋁ _ix_i, y) = ⋀ _ie (x_i, y) (∀ y ∈ X). Proof. Let A = {x_i| i ∈ I}. By the proof process of Theorem 3.3, e (⋁ _ix_i, y) = e (⋁ A, y) = ⋀ _z ∈Ae (z, y) = ⋀ _ie (x_i, y).□

Let (X, e) be a complete Q-ordered set. Define ⊗_e : Q × X ⟶ X by $a \otimes_{e} x = sup x_{a} (\forall a \in Q, \forall x \in X) .$

Theorem 3.5. Let (X, e) be a Q-ordered set. Then for all a ∈ Q and x, y ∈ X,

(1) e (a ⊗ _ex, y) = a → e (x, y);

(2) a ⊗ _ex ≤ _ey iff a ≤ e (x, y) .

Proof. (1) e (a ⊗ _ex, y) = e (sup x_a, y) = ⋀ _z∈Xx_a (z) → e (z, y) = a → e (x, y).

(2) a ⊗ _ex ≤ _ey iff e (y, y) ≤ e (a ⊗ _ex, y) = a → e (x, y) iff a ≤ e (y, y) → e (x, y) = e (x, y).□

A pair of order-preserving mapping f : V ⟶ W and g : W ⟶ V between two ordered sets is called a Galois adjunction if f (v) ≤ w ⇔ v ≤ g (w) (∀ v ∈ V, ∀ w ∈ W). In this case, f preserves existing joins and g preserves existing meets, that is to say, if A ⊆ V, B ⊆ W and ⋁A, ⋀ B existing, then f (⋁ A) = ⋁ f (A), g (⋀ B) = ⋀ g (B).

Corollary 3.6 Let (X, e) be a Q-ordered set. Then for all a ∈ Q and x ∈ X, $a \otimes_{e} x = ⋀ {y \in X | a \leq e (x, y)},$ $e (x, y) = ⋁ {a \in Q | a \otimes_{e} x \leq y} .$

Proof. By Theorem 3.5, we only need to prove that both (-) ⊗ _ex : L ⟶ |X| and e (x, -) : |X| ⟶ L are order-preserving, so that they form Galois adjunction and consequently, a ⊗ _ex = ⋀ {y ∈ X| a ≤ e (x, y)} and e (x, y) = ⋁ {a ∈ Q| a ⊗ _ex ≤ y}. In fact, for e (x, -), if y ≤ _ez, then e (y, y) ≤ e (y, z) and then e (x, y) = e (x, y) * e (y, y) ≤ e (x, y) * e (y, z) ≤ e (x, z). Hence, e (x, -) : |X| ⟶ L is order-preserving. For (-) ⊗ _ex, if a ≤ b in Q, then by b ⊗ _ex ≤ _eb ⊗ _ex, we have a ≤ b ≤ e (x, b ⊗ _ex) and then a ⊗ _ex ≤ b ⊗ _ex. Hence, (-) ⊗ _ex : L ⟶ |X| is order-preserving.□

Definition 3.7. Let X be a complete lattice. A Q-module on X is a Q-action ⊗ : Q × X ⟶ X satisfying that: for all {a_i| i ∈ I} ⊆ Q, {x_j| j ∈ J} ⊆ X, for all a, b ∈ Q and x ∈ X,

(QM1) (⋁ _ia_i) ⊗ x = ⋁ _i (a_i ⊗ x);

(QM2) a ⊗ (⋁ _jx_j) = ⋁ _j (a ⊗ x_j);

(QM3) a ⊗ (b ⊗ x) = (a * b) ⊗ x;

(QM4) a ≤ a * k_x, where k_x = ⋁ {λ ∈ Q| λ ⊗ x ≤ x}.

Example 3.8. If there is a unit k_Q ∈ Q, then (QM4) is equivalent to

(QM4^′) k_Q ⊗ x ≤ x (∀ x ∈ X).

Proof. (QM4)⇒(QM4^′): Let a = k_Q, then k_x = k_Q * k_x ≥ k_Q and then for any x ∈ X, we have $\begin{matrix} k_{Q} \otimes x & = & (⋁ {λ \in Q | λ \otimes x \leq x}) \otimes x \\ = & ⋁ {λ \otimes x | λ \in Q, λ \otimes x \leq x} \\ \leq & x . \end{matrix}$

(QM4^′)⇒(QM4): Clearly, k_x ≥ k_Q and then a * k_x ≥ a * k_Q = a.□

Theorem 3.9. Let (X, e) be a complete Q-ordered set. Then ⊗_e : Q × |X| ⟶ |X| is a Q-module.

Proof. (QM1) For any y ∈ X, we have (⋁ _ia_i) ⊗ _ex ≤ _ey iff ⋁_ia_i ≤ e (x, y) iff a_i ≤ e (x, y) (∀ i ∈ I) iff a_i ⊗ _ex ≤ _ey (∀ i ∈ I) iff ⋁_i (a_i ⊗ _ex) ≤ _ey. By the arbitrariness of y, we have (⋁ _ia_i) ⊗ _ex = ⋁ _i (a_i ⊗ _ex).

(QM2) For any y ∈ X, we have a ⊗ _e (⋁ _jx_j) ≤ _ey iff a ≤ e (⋁ _jx_j, y) = ⋀ _je (x_j, y) iff a ≤ e (x_j, y) (∀ j ∈ J) iff a ⊗ _ex_j ≤ _ey (∀ j ∈ J) iff ⋁_j (a ⊗ _ex_j) ≤ _ey. By the arbitrariness of y, we have a ⊗ _e (⋁ _jx_j) = ⋁ _j (a ⊗ _ex_j).

(QM3) $\begin{matrix} a \otimes_{e} (b \otimes_{e} x) & = & ⋀ {y \in X | a \leq e (b \otimes_{e} x, y)} \\ = & ⋀ {y \in X | a \leq b \to e (x, y)} \\ = & ⋀ {y \in X | a * b \leq e (x, y)} \\ = & (a * b) \otimes x . \end{matrix}$

(QM4) Let x ∈ X. Then $\begin{matrix} k_{x} & = & ⋁ {λ \in Q | λ \otimes_{e} x \leq_{e} x} \\ = & ⋁ {λ \in Q | λ \leq e (x, x)} \\ = & e (x, x) . \end{matrix}$ and then for any a ∈ Q, it holds that a ≤ a * e (x, x) = a * k_x.□

Let ⊗ : Q × X ⟶ X be a Q-module, define e_⊗ : X × X ⟶ Q by $e_{\otimes} (x, y) = ⋁ {a \in Q | a \otimes x \leq y} (\forall x, y \in X) .$

Theorem 3.10. Let ⊗ : Q × X ⟶ X be a Q-module. Then for all a ∈ Q and all x, y ∈ X,

(1) a ⊗ x ≤ y iff a ≤ e_⊗ (x, y);

(2) e_⊗ (a ⊗ x, y) = a → e_⊗ (x, y).

Proof. (1) The necessity is obvious. Conversely, if a ≤ e_⊗ (x, y), then $\begin{matrix} a \otimes x & \leq & e_{\otimes} (x, y) \otimes x \\ = & ⋁ {b \in Q | b \otimes x \leq y} \otimes x \\ = & ⋁ {b \otimes x | b \otimes x \leq y} \\ \leq & y . \end{matrix}$

(2) $\begin{matrix} e_{\otimes} (a \otimes x, y) & = & ⋁ {b \in Q | b \otimes (a \otimes x) \leq y} \\ = & ⋁ {b \in Q | b \leq e_{\otimes} (a \otimes x, y} \\ = & ⋁ {b \in Q | b \leq a \to e_{\otimes} (x, y)} \\ = & a \to e_{\otimes} (x, y) . \end{matrix}$ □

Theorem 3.11. Let ⊗ : Q × X ⟶ X be a Q-module. Then e_⊗ is a complete Q-order, where for any S ∈ Q^X, sup S = ⋁ _x∈XS (x) ⊗ x.

Proof. (QO1) Clearly, k_x = e_⊗ (x, x) and then by (QM4), a ≤ a * e_⊗ (x, x) for all x ∈ X and a ∈ Q.

(QO2) Let x, y, z ∈ X. Then $\begin{matrix} e_{\otimes} (x, y) * e_{\otimes} (y, z) \\ = & ⋁ {a \in Q | a \otimes x \leq y} * ⋁ {b \in Q | b \otimes y \leq z} \\ = & ⋁ {a * b | a, b \in Q | a \otimes x \leq y, b \otimes y \leq z} \\ \leq & ⋁ {a * b | a, b \in Q | b \otimes (a \otimes x) \leq z} \\ = & ⋁ {a * b | a, b \in Q | (a * b) \otimes x \leq z} \\ \leq & ⋁ {c \in Q | c \otimes x \leq z} \\ \leq & e_{\otimes} (x, z) . \end{matrix}$

The completeness of e_⊗ is because for any y ∈ X, we have $\begin{matrix} e_{\otimes} (⋁_{x \in X} S (x) \otimes x, y) & = & ⋀_{x \in X} e_{\otimes} (S (x) \otimes x, y) \\ = & ⋀_{x \in X} S (x) \to e_{\otimes} (x, y) \end{matrix}$ and then sup S = ⋁ _x∈XS (x) ⊗ x.□

Theorem 3.12. Let (X, e) be a complete Q-ordered set. Then e_{⊗_e} = e.

Proof. For all x, y ∈ X, we have $\begin{matrix} e_{\otimes_{e}} (x, y) & = & ⋁ {a \in Q | a \otimes_{e} x \leq_{e} y} \\ = & ⋁ {a \in Q | a \leq e (x, y)} \\ = & e (x, y) . \end{matrix}$ Hence, e_{⊗_e} = e.

Theorem 3.13. Let X be a complete lattice and ⊗ : Q × X ⟶ X be a Q-module. Then ⊗_{e
_⊗} =⊗.

Proof. Since a ⊗ _{e
_⊗}x = sup x_a, we need to prove that a ⊗ x = sup x_a in (X, e_⊗). In fact, for any y ∈ X, we have $e_{\otimes} (a \otimes x, y) = a \to e_{\otimes} (x, y) = ⋀_{z \in X} x_{a} (z) \to e_{\otimes} (z, y),$ as desired.□

By Theorems 3.9, 3.11, 3.12 and 3.13, we have already shown that there is a one-to-one correspondence between complete Q-orders and Q-modules of complete lattice structures on any set. In order to establish the isomorphism in the categorical sense, let us now consider the related morphisms.

A mapping f : (X, e₁) ⟶ (Y, e₂) between two Q-ordered sets is called Q-order-preserving if e₁ (x₁, x₂) ≤ e₂ (f (x₁) , f (x₂)) (∀ x₁, x₂ ∈ X). For the case of complete Q-ordered sets, f is called sup-preserving if f (sup A) = sup f^→ (A) (∀ A ∈ Q^X), where f^→ (A) ∈ Q^Y is defined by $f^{\to} (A) (y) = ⋁_{f (x) = y} A (x) (\forall y \in Y),$ called the Zadeh-type extension of f. It is easily shown that sup-preserving mapping are always Q-order-preserving. Denote by Q-COS the category of complete Q-ordered sets with sup-preserving mapping between them.

A mapping g : (X, ⊗ ₁) ⟶ (Y, ⊗ ₂) between two Q-modules of complete lattices is called a Q-module homomorphism if g (⋁ S) = ⋁ g (S) (∀ S ⊆ X) and g (a ⊗ ₁x) = a ⊗ ₂g (x) (∀ a ∈ Q, ∀ x ∈ X). Denote by Q-ModCL the category of Q-modules of complete lattices and Q-module homomorphisms between them.

Lemma 3.14. If f : (X, e₁) ⟶ (Y, e₂) is a Q-order preserving mapping between Q-ordered set, then for any x ∈ X and any y ∈ Y, it holds that e₁ (x, x) → e₂ (f (x) , y) = e₂ (f (x) , y).

Proof. Firstly, $\begin{matrix} e_{1} (x, x) \to e_{2} (f (x), y) \\ \geq & e_{2} (f (x), f (x)) \to e_{2} (f (x), y) = e_{2} (f (x), y) . \end{matrix}$ Secondly, if a ≤ e₁ (x, x) → e₂ (f (x) , y), then a ≤ a * e₁ (x, x) ≤ e₂ (f (x) , y) and thus e₁ (x, x) → e₂ (f (x) , y) ≤ e₂ (f (x) , y). Hence, $e_{1} (x, x) \to e_{2} (f (x), y) = e_{2} (f (x), y) . □$

Theorem 3.15. Let f : (X, ⊗ ₁) ⟶ (Y, ⊗ ₂) be a Q-module homomorphism. Then f : (X, e_⊗₁) ⟶ (Y, e_⊗₂) is sup-preserving.

Proof. $\begin{matrix} sup f (S) & = & ⋁_{y \in Y} f^{\to} (S) (y) \otimes_{2} y \\ = & ⋁_{y \in Y} (⋁_{f (x) = y} S (x)) \otimes_{2} y \\ = & ⋁_{y \in Y} ⋁_{f (x) = y} (S (x) \otimes_{2} y) \\ = & ⋁_{x \in X} (S (x) \otimes_{2} f (x)) \\ = & ⋁_{x \in X} f (S (x) \otimes_{1} x) \\ = & f (⋁_{x \in X} S (x) \otimes_{1} x) \\ = & f (sup S) . \end{matrix}$ □

Theorem 3.16. If g : (X, e₁) ⟶ (Y, e₂) is a sup-preserving mapping between two complete Q-ordered sets, then g : (X, ⊗ _{e
₁}) ⟶ (Y, ⊗ _{e
₂}) is a Q-module homomorphism.

Proof. Step 1. Let A ⊆ X, we need to show g (⋁ A) = ⋁ g (A). In fact, $g (⋁ A) = g (sup \tilde{A}) = sup g^{\to} (\tilde{A})$ and $⋁ g (A) = sup \tilde{g (A)}$ . For every y ∈ Y, $y = sup g^{\to} (\tilde{A})$ iff for any z ∈ Y, it holds that $\begin{matrix} e_{2} (y, z) & = & ⋀_{u \in Y} g^{\to} (\tilde{A}) (u) \to e_{2} (u, z) \\ = & ⋀_{u \in Y} ⋀_{g (x) = u} \tilde{A} (x) \to e_{2} (u, z) \\ = & ⋀_{x \in X} \tilde{A} (x) \to e_{2} (g (x), z) \\ = & ⋀_{x \in A} e_{1} (x, x) \to e_{2} (g (x), z) \\ = & ⋀_{x \in A} e_{2} (g (x), z) \\ = & ⋀_{x \in A} e_{2} (g (x), g (x)) \to e_{2} (g (x), z) \\ = & ⋀_{u \in g (A)} e_{2} (u, u) \to e_{2} (u, z) \\ = & ⋀_{u \in Y} \tilde{g (A)} (u) \to e_{2} (u, z) \end{matrix}$ iff $y = sup \tilde{g (A)}$ . Hence, $sup g^{\to} (\tilde{A}) = sup \tilde{g (A)}$ and then g (⋁ A) = ⋁ g (A).

Step 2. For any a ∈ Q and any x ∈ X, we have g (a ⊗ _{e
₁}x) = g (sup x_a) = sup g^→ (x_a) = sup(g (x)) _a = a ⊗ _{e
₂}g (x).

Hence, g : (X, ⊗ _{e
₁}) ⟶ (Y, ⊗ _{e
₂}) is a Q-module homomorphism.□

Theorem 3.17. For every commutative quantale Q, the categories Q-COS and Q-ModCL are isomorphic.

4 Conclusion

The title of this paper is Q-order without unit, where Q is a commutative quantale. The meaning is that the definition of Q-orders does not rely on the unit of Q even if Q has a unit. The self-reflectivity of Q-orders is a self-adaptive version of the usual self-reflexivity. This kind of Q-orders may help us to extend different kinds of fuzzy posets to more general cases, and will connect different mathematical structures in another viewpoint.

We would like to point out that a Q-ordered set is just a special case of quantaloid enriched categories explored in a series of papers by Stubbe (eg. [26])).

Define a category $Q_{idm}$ as follows: an object of $Q_{idm}$ is an idempotent element p of Q such that a ≤ p * a for all a ∈ Q; for $p, q \in Q_{idm}$ , a morphism from p to q is an element x of Q such that x * p = x = q * x; the composite of $x \in Q_{idm} (p, q)$ and $y \in Q_{idm} (q, r)$ is given by y * x; the identity on $p \in Q_{idm}$ is p itself; each hom-set $Q_{idm} (p, q)$ is equipped with the order inherited from Q. Then, $Q_{idm} (p, q)$ is a quantaloid. It is easily seen that a Q-ordered set in the sense of Definition 2.5 is exactly a category enriched over the quantaloid $Q_{idm}$ ; that is to say, a Q-ordered set is just a $Q_{idm}$ -category. Actually, this fact is the essence of Proposition 2.7 in the paper. The contribution and results of the paper are implicitness of those of the quantaloid $Q_{idm}$ in [26].

We also would like to point out that not every quantale can be supplied as a truth value table in this paper, as already has been pointed out by Remark 2.6. For examples, consider the quantale (Q, *) = ([0, 1/2] , ×). It is clear that $Q_{idm}$ is the empty category with no objects; consider the quantale (Q, *) = ([2, ∞] , *), where 2 * x = 2 for all x and x* ∞ = ∞ whenever x > 2. In this case, the quantaloid $Q_{idm}$ has only one object (namely, ∞) and two morphisms (namely, 2 and ∞). Put differently, $Q_{idm}$ is the Boolean algebra {0, 1}, viewed as a one-object quantaloid. So, a Q-ordered set in this case is nothing but a preordered set in the usual sense.

Footnotes

Acknowledgment

This paper is supported by National Natural Science Foundation of China (11871189, 1211101082), Natural Science Foundation of Hebei Province (A2020208008), Jiangsu Provincial Innovative and Entrepreneurial Talent Support Plan (JSSCRC202521) and the Startup Foundation for Introducing Talent of NUIST (2019r63).

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