On strongly convex L -fuzzy subsets of an ordered semigroup

Abstract

This paper addresses the strongly convex lattice-valued fuzzy (L-fuzzy) subsets of an ordered semigroup. It is shown that the set of all strongly convex L-fuzzy subsets of an ordered semigroup S forms a quantale and that S can be embedded into the quantale. The properties of L-fuzzy ideals of an ordered semigroup, a special class of strongly convex L-fuzzy subsets, are discussed. Specially, two approaches are developed to construct an L-fuzzy (left) ideal by an arbitrary L-fuzzy subset of an ordered semigroup.

Keywords

Ordered semigroup quantale embedding mapping

1 Introduction

It is well known that the concept of ordered semigroups is a combination of algebraic structures and partially ordered structures. Since ordered semigroups have a close relation with theoretical computer science, especially with formal language, coding theory and theory of automata, they have been extensively investigated by many researchers (see e.g. [8, 11 , 30]). In the past ten years, ordered semigroup theory in the framework of fuzzy set has also been considered by many researchers. The pioneering work on this topic can be traced to Kehayopulu et al. [16], in which they defined and investigated fuzzy analogies for several notations. Taking advantage of fuzzy subsets, Kehayopulu et al. [17] proved that each ordered groupoid can be embedded into an ordered groupoid having the greatest element (a poe-groupoid). They also characterized regular ordered semigroups in terms of fuzzy bi-ideals (See [18]) and in terms of fuzzy right, left ideals and fuzzy quasi-ideals (See [19]), respectively. This research direction was continued afterwards by Davvaz, Jun, Khan, Shabir, Tang, Xie, Yin and others. The reader can refer to [5– 7 , 35] for details.

Recently, the research on strongly convex fuzzy subsets of ordered semigroups has received considerable attention. In [31], Xie and Tang introduced the notion of strongly convex fuzzy subsets of an ordered semigroup and analysed some of their basic properties. Then, by using a special case of strongly convex fuzzy subsets that are called ordered fuzzy points, Xie and Tang gave a characterization of prime fuzzy ideals of an ordered semigroup. In order to generalize the work of Kehayopulu et al. [16, 17] and Xie et al. [31], Han and Zhao [9] introduced the concept of a Q-fuzzy subset of an ordered semigroup, in which the structure of truth values is a quantale. Han and Zhao proved that every ordered semigroup S can be embedded into a quantale that consists of all Q-fuzzy subsets of S.

In this paper, we deal with L-fuzzy subsets of an ordered semigroup, which involves an interesting topic on fuzzy algebras called lattice-valued algebras (for some recent examples in this direction, see [1, 25]). In our study the structure of truth values is a frame. The reason for this is twofold. One is that the notion of a frame-valued fuzzy set generalizes that of a classical fuzzy set, and usually can lead to more general and interesting results. The other is that every frame fulfills an additional axiom, i.e., the infinitely distributive law, which will be useful in our discussion. The purpose of this paper is to give a well-informed discussion on strongly convex L-fuzzy subsets of an ordered semigroup. We will first characterize strongly convex L-fuzzy subsets of an ordered semigroup in terms of L-fuzzy points and ordered L-fuzzy points, respectively. In particular, we prove that every ordered semigroup S can be embedded into a quantale which is composed of all strongly convex L-fuzzy subsets of S. Then, we will introduce the concept of L-fuzzy ideals of an ordered semigroup and develop two approaches to construct an L-fuzzy (left) ideal of an ordered semigroup by an arbitrary L-fuzzy subset.

2 Preliminaries

In this section, we review and develop some notions and results that are necessary in the sequel.

By an ordered semigroup we mean a partially ordered set (S, ≤) with an associative multiplication operation, denoted by ‘·’ or more simply by juxtaposition, which is compatible with the ordering, i.e., for any x, y, z ∈ S, x ≤ y implies xz ≤ yz and zx ≤ zy. An ordered semigroup S is called commutative if xy = yx for all x, y ∈ S. An element e ∈ S is called identity if ex = xe = x for all x ∈ S.

For two subsets A, B of S, we denote $\begin{matrix} (A] & = & {t \in S : t \leq h, for some h \in A} \\ AB & = & {ab : a \in A, b \in B} \end{matrix}$

Let (S, ≤ , ·) be an ordered semigroup and I ⊆ S. I is called a left (resp. right) ideal of S iff it satisfies the following conditions:

SI ⊆ I (resp. IS ⊆ I).

for any a, b ∈ S, a ≤ b and b ∈ I imply a ∈ I.

If I is both a right and a left ideal of S, then I is called an ideal of S.

Let us recall that a quantale is a triple (Q, ≤ , ∗) such that (Q, ≤) is a complete lattice, (Q, ∗) is a semigroup and for any x ∈ Q and {y_i} _i∈I ⊆ Q, x ∗ (⋁ _iy_i) = ⋁ _i (x ∗ y_i) and (⋁ _iy_i) ∗ x = ⋁ _i (y_i ∗ x).

It is evident that each quantale is an ordered semigroup. However, the converse is not true in general.

We say that an ordered semigroup (S, ≤ , ·) can be embedded into a quantale (Q, ≤ , ∗) if there exists a mapping φ : S → Q (called an embedding) satisfying the following properties: For any x, y ∈ S,

x ≤ y if and only if φ (x) ≤ φ (y);

φ (x · y) = φ (x) ∗ φ (y).

A frame is a complete lattice (L, ∨ , ∧ , 1, 0) satisfying the infinite distributive law, i.e., a ∧ (⋁ A) = ⋁ (a ∧ A) for every a ∈ L and A ⊆ L. It is well known that L is a frame if and only if it is a complete Heyting algebra. Thus, for each a, b ∈ L, the relative pseudo complement of a with respect to b exists, which is the element a → b = ⋁ {c ∈ L : a ∧ c ≤ b}.

In the following we formulate some important properties of frames. For more details relevant to frames and Heyting algebras we refer the reader to [3].

Proposition 2.1. [10] For any frame, the following properties hold:

α ∧ β ≤ γ ⇔ α ≤ β → γ.

1 → α = α.

(α ∧ β) → γ = α → (β → γ).

α → (⋀ _iβ_i) = ⋀ _i (α → β_i).

(⋁ _iα_i) → β = ⋀ _i (α_i → β).

In this paper, if there is no further statement, S and L always denote an ordered semigroup and a frame, respectively.

An L-fuzzy subset of S is an arbitrary mapping f : S → L. The set of all L-fuzzy subsets of S is denoted by L^S. We define a partial order “⊆” on L^S as follows: f ⊆ g if and only if f (x) ≤ g (x) for all x ∈ S. Then (L^S, ⊆) is clearly a frame, where (⋃ _if_i) (x) = ⋁ _if_i (x), (⋂ _if_i) (x) = ⋀ _if_i (x) and (f → g) (x) = f (x) → g (x). For f ∈ L^S and α ∈ L, an L-fuzzy subset αf of S is given as follows: (αf) (x) = α ∧ f (x), ∀x ∈ S.

Let A be a nonempty subset of S. We denote by f_A the characteristic function of A, which is a special L-fuzzy subset of S defined by $f_{A} (x) = {\begin{matrix} 1, x \in A, \\ 0, otherwise . \end{matrix}$

For any x ∈ S and α ∈ L, an L-fuzzy point x_α of S is defined by $x_{α} (y) = {\begin{matrix} α, x = y, \\ 0, otherwise . \end{matrix}$

We say that an L-fuzzy point x_α belongs to an L-fuzzy subset f, denoted by x_α ∈ f, if f (x) ≥ α.

For any x ∈ S and α ∈ L, an ordered L-fuzzy point $x_{α}^{↓}$ of S is defined by $x_{α}^{↓} (y) = {\begin{array}{l} α, if y \leq x, \\ 0, otherwise . \end{array}$

Clearly, for any x ∈ S and α ∈ L, $x_{α} \in x_{α}^{↓}$ .

For a given x ∈ S, define S_x = {(y, z) ∈ S × S : x ≤ yz}. A binary operation ∘ and a unary operation (·] on L^S are defined, respectively, as follows:

(1) For any f, g ∈ L^S and x ∈ S, $(f ° g) (x) = {\begin{array}{l} \underset{(y, z) \in S_{x}}{⋁} f (y) \land g (z), if S_{x} \neq \emptyset, \\ 0, otherwise . \end{array}$

(2) For any f ∈ L^S and x ∈ S, $(f] (x) = ⋁_{x \leq y} f (y) .$

In what follows, we develop some basic properties of the operations ∘ and (·].

Proposition 2.2. Let S be an ordered semigroup. Then

For any f, g, h ∈ L^S, (f ∘ g) ∘ h = f ∘ (g ∘ h).

For any f, g, h ∈ L^S, f ⊆ g implies f ∘ h ⊆ g ∘ h, h ∘ f ⊆ h ∘ g and (f] ⊆ (g].

For any f ∈ L^S, f ⊆ (f].

For any f, g ∈ L^S, (f ∘ g] = f ∘ g.

For any A, B ⊆ S, (f_A] ∘ (f_B] = f_A ∘ (f_B] = (f_A] ∘ f_B = (f_AB].

Proof. We only prove (4). Let f and g be two L-fuzzy subsets of S. For any x ∈ S, if S_x =∅, then (f ∘ g) (x) =0 = (f ∘ g] (x). If S_x≠ ∅, then we have $\begin{matrix} (f \circ g] (x) = & ⋁_{x \leq y} (f \circ g) (y) \\ = & ⋁_{x \leq y} ⋁_{(a, b) \in S_{y}} f (b) \land g (w) \\ \leq & ⋁_{(a, b) \in S_{x}} f (a) \land g (b) \\ = & (f \circ g) (x) . \end{matrix}$

This implies that (f ∘ g] ⊆ f ∘ g. On the other hand, item (3) implies that f ∘ g ⊆ (f ∘ g].

Therefore, we have (f ∘ g] = f ∘ g.□

An L-fuzzy subset of S is called strongly convex if (f] = f (See [31]). The set of all strongly convex L-fuzzy subsets of S is denoted by Sconv (S), that is, Sconv (S) = {f ∈ L^S : (f] = f}.

Example 2.3. Let L be a frame given as in Fig. 1 and S = {a, b, c} an ordered semigroup defined by the following multiplication “·” and order “≤”:

·	a	b	c
a	a	a	a
b	a	b	c
c	a	c	b

$\leq : = {(a, a), (b, b), (c, c), (a, b), (a, c)} .$

Then the L-fuzzy subset $f_{1} = \frac{1}{a} + \frac{α}{b} + \frac{α}{c}$ is a strongly convex L-fuzzy set, while $f_{2} = \frac{α}{a} + \frac{α}{b} + \frac{β}{c}$ is not, since $(f_{2}] = \frac{γ}{a} + \frac{α}{b} + \frac{β}{c} \neq f_{2}$ .

It is easy to verify that every ordered L-fuzzy point of S is strongly convex, but an L-fuzzy point of S is generally not strongly convex.

The following results provide some characterizations of a strongly convex L-fuzzy subset of S.

Proposition 2.4. [31] Let f be an L-fuzzy subset of S. Then f is strongly convex if and only if x ≤ y ⇒ f (x) ≥ f (y) for all x, y ∈ S.

Proposition 2.5. [31] Let f be a strongly convex L-fuzzy subset of S. Then $f = ⋃_{x_{α}^{↓} \in f} x_{α}^{↓}$ .

Proposition 2.6. [9] Let f be a strongly convex L-fuzzy subset of S. Then $f = ⋃_{x \in S} x_{f (x)}^{↓}$ .

3 Main results

In this section, we focus on investigating the properties of strongly convex L-fuzzy subsets of an ordered semigroup.

3.1 Strongly convex L-fuzzy subsets

Proposition 3.1. Let f be an L-fuzzy subset of an ordered semigroup S. Then f is strongly convex if and only if for any x, y ∈ S and α ∈ L ∖ {0}, x ≤ y and y_α ∈ f imply x_α ∈ f.

Proof. Straightforward.□

Proposition 3.2. Let f be an L-fuzzy subset of an ordered semigroup S. Then f is strongly convex if and only if for any x ∈ S and α ∈ L ∖ {0}, x_α ∈ f implies $x_{α}^{↓} \subseteq f$ .

Proof. Let f be strongly convex and x_α ∈ f, i.e., f (x) ≥ α. For any y ∈ S, if y ≰ x, then $f (y) \geq 0 = x_{α}^{↓} (y)$ . If y ≤ x, by Proposition 2.4, we have $f (y) \geq f (x) \geq α = x_{α}^{↓} (y)$ . So $x_{α}^{↓} \subseteq f$ .

Conversely, for any x, y ∈ S with y ≤ x, we set α = f (x). Then x_α ∈ f. By assumption, $x_{α}^{↓} \subseteq f$ . Thus $f (y) \geq x_{α}^{↓} (y) = α = f (x)$ . Therefore, f is a strongly convex L-fuzzy subset of S.□

Proposition 3.3. Let f be an L-fuzzy subset of an ordered semigroup S. Then f is strongly convex if and only if for any x, y ∈ S and α ∈ L ∖ {0}, x_α ∈ f implies y_α ∈ (f_{x}] → f.

Proof. Let f be a strongly convex L-fuzzy subset of S. For x, y ∈ S and α ∈ L ∖ {0}, if x_α ∈ f, then f (x) ≥ α. Consider the following cases.

Case 1: y ≰ x. In this case, we have (f_{x}] (y) =0, which implies α ∧ (f_{x}] (y) =0 ≤ f (y).

Case 2: y ≤ x. In this case, we have (f_{x}] (y) =1 and f (y) ≥ f (x). This implies α ∧ (f_{x}] (y) = α ≤ f (x) ≤ f (y).

Thus, in both cases, we have α ∧ (f_{x}] (y) ≤ f (y). This is equivalent to (f_{x}] (y) → f (y) = ((f_{x}] → f) (y) ≥ α, i.e., y_α ∈ (f_{x}] → f, as required.

Conversely, let x, y ∈ S be such that y ≤ x. Then (f_{x}] (y) =1. Now, if we put α = f (x), then x_α ∈ f. Thus, by the assumption, we have y_α ∈ (f_{x}] → f, which yields that f (y) =1 → f (y) = (f_{x}] (y) → f (y) ≥ α = f (x). So, by Proposition 2.4, f is a strongly convex L-fuzzy subset.□

Next, we intend to construct strongly convex L-fuzzy subsets from arbitrary L-fuzzy subsets of S. Specially, for a given L-fuzzy subset f of S, we try to develop the greatest strongly convex L-fuzzy subset of S contained in f.

Proposition 3.4. Let f be an L-fuzzy subset of S. Then sc (f) is the greatest strongly convex L-fuzzy subset of S contained in f, where $sc (f) (x) = ⋁ {α : α \in L, x_{α}^{↓} \subseteq f}$ for all x ∈ S.

Proof. We split the proof into the following three parts.

(i) sc (f) is a strongly convex L-fuzzy subset of S. The proof is as follows. For x, y ∈ S with y ≤ x. Let $A_{x} = {α : α \in L, x_{α}^{↓} \subseteq f}$ and $A_{y} = {α : α \in L, y_{α}^{↓} \subseteq f} .$

For any α ∈ L, it is easy to deduce $y_{α}^{↓} \subseteq x_{α}^{↓}$ , which implies A_x ⊆ A_y. Consequently, sc (f) (y) = ⋁ A_y ≥ ⋁ A_y = sc (f) (x). So, by Proposition 2.4, sc (f) is a strongly convex L-fuzzy subset.

(ii) sc (f) ⊆ f. The verification is as follows. For any α ∈ A_x, we have $x_{α} \in x_{α}^{↓} \subseteq f$ . It follows that x_α ∈ f, i.e., f (x) ≥ α. This implies f (x) ≥ ⋁ A_x = sc (f) (x), and hence, sc (f) ⊆ f.

(iii) g ⊆ sc (f) for any strongly convex L-fuzzy subset g that is contained in f. The proof is as follows. For any element x ∈ S, it is easily seen that g (x) = ⋁ {α ∈ L : g (x) ≥ α} = ⋁ {α ∈ L : x_α ∈ g}. Meanwhile, for any x_α ∈ g, it comes from Proposition 3.2 that $x_{α}^{↓} \subseteq g$ . Thus, we have $\begin{matrix} g (x) = & ⋁ {α \in L : x_{α} \in g} \\ \leq & ⋁ {α \in L : x_{α}^{↓} \subseteq g} \\ \leq & ⋁ {α \in L : x_{α}^{↓} \subseteq f} = sc (f) (x) . \end{matrix}$

This implies that g ⊆ sc (f).□

Definition 3.5. Let S be an ordered semigroup and f ∈ L^S. We define two L-fuzzy subsets as follows.

f^l ∈ L^S is defined by $\forall x \in S, f^{l} (x) = ⋀_{x \notin (y]} f (y) \to 0 .$

f^u ∈ L^S is defined by $\forall x \in S, f^{u} (x) = ⋀_{y \notin (x]} f (y) \to 0 .$

Remark 3.6. If f is a crisp subset of S, that is f ⊆ S, then we have $f^{l} = {x \in S : x \leq y forall y \in f}$ and $f^{u} = {x \in S : x \geq y forall y \in f} .$

So f^l (resp. f^u) can be interpreted as the set of all lower (resp. upper) bounds of f.

Proposition 3.7. Let S be an ordered semigroup and f ∈ L^S. Then f^l is a strongly convex L-fuzzy subset of S.

Proof. Straightforward, by Proposition 2.4 and Definition 3.5.□

In [9], Han and Zhao obtained the following important results.

Theorem 3.8. [9] Let S be an ordered semigroup. Then (L^S, ⊆ , ∘) is a quantale.

Theorem 3.9. [9] Each ordered semigroup S can be embedded into a quantale (L^S, ⊆ , ∘).

From Theorems 3.8 and 3.9, one may observe that each ordered semigroup S can be embedded into a quantale, where the quantale is composed of all L-fuzzy subsets of S. Then, a natural question is raised: is there any quantale which is smaller than L^S such that S can be embedded into it? In what follows, we will prove that the answer to this question is yes. Before doing this, let us first develop an auxiliary lemma.

Lemma 3.10. Let {f_i : i ∈ I} be a nonempty family of strongly convex L-fuzzy subsets of S. Then so are ⋃_i∈If_i and ⋂_i∈If_i.

Proof. Straightforward.□

Theorem 3.11. Let S be an ordered semigroup. Then (Sconv (S) , ⊆ , ∘) is a quantale.

Proof. Lemma 3.10 implies that (Sconv (S) , ⊆) is a complete lattice, and the items (1) and (4) of Proposition 2.2 indicate that (Sconv (S) , ∘) is a semigroup. Now, to show that (Sconv (S) , ⊆ , ∘) is a quantale, it suffices to prove that f ∘ (⋃ _ig_i) = ⋃ _if ∘ g_i and (⋃ _ig_i) ∘ f = ⋃ _ig_i ∘ f for any f ∈ Sconv (S) and {g_i} _i∈I ⊆ Sconv (S). Indeed, for any x ∈ S, we have $\begin{matrix} (f \circ (⋃_{i} g_{i})) (x) & = & ⋁_{(y, z) \in S_{x}} f (y) \land (⋃_{i} g_{i}) (z) \\ = & ⋁_{(y, z) \in S_{x}} f (y) \land (⋁_{i} g_{i} (z)) \\ = & ⋁_{(y, z) \in S_{x}} ⋁_{i} (f (y) \land g_{i} (z)) \\ = & ⋁_{i} ⋁_{(y, z) \in S_{x}} (f (y) \land g_{i} (z)) \\ = & ⋁_{i} (f \circ g_{i}) (x) \\ = & (⋃_{i} f \circ g_{i}) (x) . \end{matrix}$

This implies f ∘ (⋃ _ig_i) = ⋃ _if ∘ g_i. Similarly, we can prove that (⋃ _ig_i) ∘ f = ⋃ _ig_i ∘ f.

Therefore, (Sconv (S) , ⊆ , ∘) is a quantale.□

Theorem 3.12. Each ordered semigroup S can be embedded into a quantale (Sconv (S) , ⊆ , ∘).

Proof. For any α ∈ L, define a mapping from S to Sconv (S) as follows: $π_{α} : S \to Sconv (S), x \mapsto x_{α}^{↓} .$

Now, analogous to the proof in [9] (cf. Theorem 2.18), it is easy to verify that π_α is an embedding mapping from S into Sconv (S).□

3.2 L-fuzzy ideals

In the sequel, we turn our attention to L-fuzzy ideals of an ordered semigroup, which is a special class of strongly convex L-fuzzy subsets as shown in the following definition.

Definition 3.13. Let S be an ordered semigroup and f ∈ L^S . Then a strongly convex L-fuzzy subset f of S is called an L-fuzzy left (resp. right) ideal of S if f_S ∘ f ⊆ f (resp. f ∘ f_S ⊆ f).

An L-fuzzy subset f of S is called an L-fuzzy ideal of S if it is both an L-fuzzy left and an L-fuzzy right ideal of S. We use the symbol Id (S) (resp. LId (S), RId (S)) to denote the set of all L-fuzzy ideals (resp. L-fuzzy left ideals, L-fuzzy right ideals) of S.

It is not difficult to check that a strongly convex L-fuzzy subset f of S is an L-fuzzy left (resp. right) ideal of S if and only if for any x, y ∈ S, f (xy) ≥ f (y) (resp. f (xy) ≥ f (x)).

Moreover, an L-fuzzy left (resp. right) ideal can be characterized by using ordered L-fuzzy points.

Proposition 3.14. A strongly convex L-fuzzy subset f of S is an L-fuzzy left (resp. right) ideal of S if and only if for any x, y ∈ S and α ∈ L ∖ {0}, $x_{α}^{↓} \subseteq f$ implies $(yx)_{α}^{↓} \subseteq f$ (resp. $(xy)_{α}^{↓} \subseteq f$ ).

Proof. Straightforward.□

In [31], Xie and Tang introduced the fuzzy ideal generated by an ordered fuzzy point of an ordered semigroup. Let $x_{α}^{↓}$ be an ordered L-fuzzy point of an ordered semigroup S. Then the fuzzy ideal generated by $x_{α}^{↓}$ , denoted by $〈 x_{α}^{↓} 〉$ , is $〈 x_{α}^{↓} 〉 (y) = {\begin{array}{l} α, if y \in (x \cup x S \cup S x \cup S x S] \\ 0, otherwise . \end{array}$

In what follows, we develop a method to generate an L-fuzzy ideal by an arbitrary L-fuzzy subset of an ordered semigroup.

Definition 3.15. Let f be an L-fuzzy subset of S. Then the least L-fuzzy ideal (resp. L-fuzzy left ideal, L-fuzzy right ideal) of S containing f is called the L-fuzzy ideal (resp. L-fuzzy left ideal, L-fuzzy right ideal) generated by f and denoted by 〈f〉 (resp. 〈f〉_L, 〈f〉_R).

Clearly, for any f ∈ L^S, we have $\begin{matrix} 〈 f 〉 & = & ⋂ {g \in Id (S) : f \subseteq g}, \\ {〈 f 〉}_{L} & = & ⋂ {g \in LId (S) : f \subseteq g} \end{matrix}$ and ${〈 f 〉}_{R} = \cap {g \in RId (S) : f \subseteq g} .$

In the sequel, we use the notation A ⊆ _FinB to mean that A is a finite subset of B.

Theorem 3.16. For any f ∈ L^S, $〈 f 〉 = ⋃_{A \subseteq_{Fin} S} (⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}] .$

Proof. Let f be an arbitrary L-fuzzy subset of S. Then we obtain:

(i) 〈f〉 is an L-fuzzy ideal of S. The proof is as follows. For any A ⊆ _FinS, it is easy to verify that $(⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}]$ is strongly convex. Then it follows from Lemma 3.10 that 〈f〉 is also a strongly convex L-fuzzy subset. Now, since f_S ∘ (f_{A∪SA∪AS∪SAS}] = (f_{SA∪SSA∪SAS∪SSAS)}] ⊆ (f_{A∪SA∪AS∪SAS}] for all A ⊆ _FinS, we have $\begin{matrix} f_{S} \circ 〈 f 〉 \\ = & f_{S} \circ (⋃_{A \subseteq_{Fin} S} (⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}]) \\ = & ⋃_{A \subseteq_{Fin} S} (⋀_{x \in A} f (x)) f_{S} \circ (f_{A \cup SA \cup AS \cup SAS}] \\ \subseteq & ⋃_{A \subseteq_{Fin} S} (⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}] = 〈 f 〉 . \end{matrix}$

Hence, 〈f〉 is an L-fuzzy left ideal of S. Similarly, we can proved that 〈f〉 is also an L-fuzzy right ideal of S.

(ii) f ⊆ 〈f〉. The verification is as follows. For any y ∈ S, it is clear that {y} ⊆ _FinS and f (y) = f (y) ∧1 = (⋀ _x∈{y}f (x)) ∧ (f_{{y}∪S{y}∪{y}S∪S{y}S}] (y).

Thus $\begin{matrix} f (y) \leq & ⋁_{A \subseteq_{Fin} S} (⋀_{x \in A} f (x)) \land (f_{A \cup SA \cup AS \cup SAS}] (y) \\ = & 〈 f 〉 (y) . \end{matrix}$

This implies that f ⊆ 〈f〉.

(iii) 〈f〉 ⊆ g for any L-fuzzy ideal g of S such that f ⊆ g. The proof is as follows. For any A ⊆ _FinS and any y ∈ S, we let α = ⋀ _x∈Af (x) . Then $\begin{matrix} ((⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}]) (y) \\ = & (α (f_{A \cup SA \cup AS \cup SAS}]) (y) \\ = & (α (f_{A}] \cup α (f_{SA}] \cup α (f_{AS}] \cup α (f_{SAS}]) (y) . \end{matrix}$

On the other hand, we have $\begin{matrix} α (f_{SA}] (y) = & ⋀_{x \in A} f (x) \land (⋁_{y \leq z} f_{SA} (z)) \\ \leq & ⋀_{x \in A} g (x) \land (⋁_{y \leq z} f_{SA} (z)) \\ = & ⋁_{y \leq z} ⋀_{x \in A} (g (x) \land f_{SA} (z)) \\ = & ⋁_{y \leq ax', a \in S, x' \in A} ⋀_{x \in A} g (x) \\ \leq & ⋁_{y \leq ax', a \in S, x' \in A} g (x') \\ \leq & ⋁_{y \leq ax', a \in S, x' \in A} g (ax') \\ \leq & ⋁_{y \leq ax', a \in S, x' \in A} g (y) = g (y), \end{matrix}$ and, in a similar way, we can prove α (f_A] (y) ≤ g (y), α (f_AS] (y) ≤ g (y) and α (f_SAS] (y) ≤ g (y).

Summing up the above arguments, we have $((⋀_{x \in A} f (x)) (f_{A \cup SA \cup AS \cup SAS}]) (y) \leq g (y) .$

Since A is an arbitrary finite subset of S, we get $〈 f 〉 (y) \leq g (y) .$

Therefore, 〈f〉 ⊆ g.□

Remark 3.17. (1) If f is an ordered L-fuzzy point, i.e., $f = x_{α}^{↓}$ , then 〈f〉 is equivalent to that given in Xie and Tang [31].

(2) In an analogous way, we have ${〈 f 〉}_{L} = \underset{A \subseteq_{Fin} S}{\cup} (\underset{x \in A}{\land} f (x)) (f_{A \cup S A}]$ and ${〈 f 〉}_{R} = \underset{A \subseteq_{Fin} S}{\cup} (\underset{x \in A}{\land} f (x)) (f_{A \cup A S}] .$

It is natural for us to study the greatest L-fuzzy ideals contained in a given L-fuzzy subset. For this, we obtain the following result.

Proposition 3.18. Let f be an L-fuzzy subset f of S. Then the L-fuzzy subset $l (f) (x) = ⋁ {α \in L : x_{α}^{↓} \subseteq f, (ax)_{α}^{↓} \subseteq f (\forall a \in S)}$ is the greatest L-fuzzy left ideal of S contained in f.

Proof. For x, y ∈ S, define $\begin{matrix} B_{x} & = & {α \in L : x_{α}^{↓} \subseteq f, (ax)_{α}^{↓} \subseteq f (\forall a \in S)}, \\ B_{y} & = & {α \in L : y_{α}^{↓} \subseteq f, (by)_{α}^{↓} \subseteq f (\forall b \in S)} \end{matrix}$ and $B_{xy} = {α \in L : (xy)_{α}^{↓} \subseteq f, (cxy)_{α}^{↓} \subseteq f (\forall c \in S)} .$

Then, we split the proof into three parts as follows.

(i) l (f) is an L-fuzzy left ideal of S. In fact, let x, y, a ∈ S be such that x ≤ y. Then ax ≤ ay. Thus, for every α ∈ B_y, we have $x_{α}^{↓} \subseteq y_{α}^{↓} \subseteq f, and (ax)_{α}^{↓} \subseteq (ay)_{α}^{↓} \subseteq f .$

This implies α ∈ B_x. Thus, B_y ⊆ B_x. It follows that l (f) (x) = ⋁ B_x ≥ ⋁ B_y = l (f) (y). By Proposition 2.4, l (f) is a strongly convex L-fuzzy subset.

In a similar way, we can prove B_y ⊆ B_xy, and hence l (f) (xy) = ⋁ B_xy ≥ ⋁ B_y = l (f) (y). Therefore, l (f) is an L-fuzzy left ideal of S.

(ii) l (f) ⊆ f. Indeed, for any x ∈ S and α ∈ B_x, we have $x_{α} \in x_{α}^{↓} \subseteq f$ , which derives that f (x) ≥ α. Thus f (x) ≥ ⋁ B_x = l (f) (x), i.e., l (f) ⊆ f.

(iii) g ⊆ l (f) for any L-fuzzy left ideal g that is contained in f. For any L-fuzzy point x_α ∈ g, we have g (x) ≥ α. Since g is an L-fuzzy left ideal, we get g (ax) ≥ g (x) ≥ α for any a ∈ S. Indeed, we have proved that x_α ∈ g implies (ax) _α ∈ g for any a ∈ S, and further, by Proposition 3.2, we have $(ax)_{α}^{↓} \subseteq g$ for any a ∈ S. Thus, for any x ∈ S, $\begin{matrix} g (x) = & ⋁ {α \in L : x_{α} \in g} \\ = & ⋁ {α \in L : x_{α} \in g, (ax)_{α} \in g (\forall a \in S)} \\ \leq & ⋁ {α \in L : x_{α}^{↓} \subseteq g, (ax)_{α}^{↓} \subseteq g (\forall a \in S)} \\ \leq & ⋁ {α \in L : x_{α}^{↓} \subseteq f, (ax)_{α}^{↓} \subseteq f (\forall a \in S)} \\ = & l (f) (x) . \end{matrix}$

This implies g ⊆ l (f), as required.□

As a direct consequence of Theorems 3.16 and Proposition 3.18, we conclude the following results.

Corollary 3.19. Let S be an ordered semigroup and f ∈ L^S. Then

f is an L-fuzzy ideal of S ⇔ 〈f〉 = f.

f is an L-fuzzy left ideal of S ⇔ l (f) = f.

Recall that an operator k on a partially ordered set (P, ≤) is called a projection if for any x, y ∈ P, k (k (x)) = k (x) and x ≤ y implies k (x) ≤ k (y). A projection is said to be a closure operator (resp. kernel operator) if x ≤ k (x) (resp. x ≥ k (x)) for all x ∈ P.

Corollary 3.20. Let S be an ordered semigroup. Then

$〈 \cdot 〉, {〈 \cdot 〉}_{L}$ and ${〈 \cdot 〉}_{R}$ are closure operators on the partially ordered set (L^S, ⊆).

l (·) is a kernel operator on (L^S, ⊆).

Lemma 3.21. Let {f_i : i ∈ I} be a nonempty family of L-fuzzy ideals of S. Then so are ⋃_i∈If_i and ⋂_i∈If_i.

Proof. It is straightforward.□

Lemma 3.22. Let f and g be two L-fuzzy ideals of S. Then so is f ∘ g.

Proof. It is straightforward.□

Theorem 3.23. Let S be an ordered semigroup. Then (Id (S) , ⊆ , ∘) is a quantale.

Proof. It is similar to that of Theorem 3.11.□

Note that both (LId (S) , ⊆ , ∘) and (RId (S) , ⊆ , ∘) can be similarly proved to be quantales.

Theorem 3.24. Let S be an ordered semigroup with the identity e, and Id_e (S) the set of all L-fuzzy ideals of S such that f (e) =1 for any f ∈ Id_e (S). Then (Id_e (S) , ∩ , ∘) is a lattice ordered by ⊆.

Proof. For any f, g ∈ Id_e (S), it follows from Lemma 3.21 and 3.22 that f ∩ g ∈ Id (S) and f ∘ g ∈ Id (S). Since (f ∩ g) (e) =1 and (f ∘ g) (e) =1, we have f ∩ g ∈ Id_e (S) and f ∘ g ∈ Id_e (S).

Clearly, f ∩ g is the greatest lower bound of f and g. Next we show that f ∘ g is the least upper bound of f and g. In fact, for any x ∈ S, we have $\begin{matrix} (f \circ g) (x) = & ⋁_{(y, z) \in S_{x}} (f \land g) (y) \\ \geq & f (x) \land g (e) = f (x) \land 1 = f (x) . \end{matrix}$

This implies that f ⊆ f ∘ g. Similarly, we can prove g ⊆ f ∘ g. Now, let h ∈ Id_e (S) such that f ⊆ h and g ⊆ h. Then, by Proposition 2.2(2), f ∘ g ⊆ h ∘ h ⊆ h ∘ f_S ⊆ h. This proves that f ∘ g is the least upper bound of f and g.

Therefore, (Id_e (S) , ∩ , ∘) is a lattice.□

In the sequel, we consider the homomorphism properties of strongly convex L-fuzzy subsets (L-fuzzy ideals) of an ordered semigroup.

Let (S, ≤ , ·) and (T, ≤ , ∗) be two ordered semigroups. A mapping φ : S → T is called a homomorphism if it satisfies the following conditions:

x ≤ y implies φ (x) ≤ φ (y);

φ (x · y) = φ (x) ∗ φ (y).

Let φ be any mapping from S to T. Then two mappings $φ^{\to} : L^{S} \to L^{T}$ and $φ^{\leftarrow} : L^{T} \to L^{S}$ are given respectively as follows:

For any f ∈ L^S and y ∈ T, $φ^{\to} (f) (y) = \underset{y \leq φ (x)}{\lor} f (x)$

For any g ∈ L^T and x ∈ S, $φ^{\leftarrow} (g) (x) = g (φ (x)) .$

Theorem 3.25. Let (S, ≤ , ·) and (T, ≤ , ∗) be two ordered semigroups and φ : S → T a homomorphism.

For any f ∈ L^S, $φ^{\to} (f)$ is a strongly convex L-fuzzy subset of T.

If g is an L-fuzzy ideal of T, then $φ^{\leftarrow} (g)$ is an L-fuzzy ideal of S.

Proof. (1) Straightforward.

(2) Let g be an L-fuzzy ideal of T and x, y ∈ S with x ≤ y. Then φ (x) ≤ φ (y), which deduces that $φ^{\leftarrow} (g) (x) = g (φ (x)) \geq g (φ (y)) = φ^{\leftarrow} (g) (y)$ . In addition, for any x, y ∈ S, we have $φ^{\leftarrow} (g) (x y) = g (φ (x y)) \geq g (φ (x) * φ (y)) \geq g (φ (y)) = φ^{\leftarrow} (g) (y)$ . So $φ^{\leftarrow} (g)$ is an L-fuzzy left ideal of S.

In a similar way, we can prove that $φ^{\leftarrow} (g)$ is also an L-fuzzy right ideal of S.□

3.3 Some remarks

Since the unit interval, i.e., [0, 1], is a special frame, whenever a statement is true for frame-valued fuzzy sets, it is clearly true for classical fuzzy sets. However, as a generalization of the unit interval, a frame drops some additional operations (for example, addition and multiplication) and wrecks the totally ordered structure of [0, 1], which makes some results in frame-valued algebras different from those in classical fuzzy algebras.

Let us consider the notion of fuzzy subsemigroups of an ordered semigroup, which is introduced by Kehayopulu et al. [18] and Shabir et al. [27]. A fuzzy subset f of S is called a fuzzy subsemigroup if it satisfies: $(A) f (xy) \geq min {f (x), f (y)}, \forall x, y \in S .$

It is easily seen that the condition (A) is equivalent to: $(B) f (xy) \geq f (x) or f (xy) \geq f (y), \forall x, y \in S .$

So we conclude that f is a fuzzy subsemigroup of S if and only if (B) holds. However, this is not true for frame-valued subsemigroups of an ordered semigroup, as shown in the following example.

Example 3.26. Let L be a frame given as in Fig. 2 and S = {a, b, c, d} an ordered semigroup defined by the following multiplication “·” and the order “≤”:

·	a	b	c	d
a	a	a	a	a
b	a	a	d	a
c	a	d	a	a
d	a	a	a	a

$\leq : = {(a, a), (b, b), (c, c), (d, d), (a, d)} .$

We define an L-fuzzy set $f = \frac{1}{a} + \frac{α}{b} + \frac{γ}{c} + \frac{β}{d}$ . Then it is clear that f is an L-fuzzy subsemigroup of S. But the condition (B) does not hold, since f (bc) = f (d) = β ≱ α = f (b) and f (bc) = f (d) = β ≱ γ = f (c).

Moreover, let x_α be a fuzzy point and f a fuzzy set. We say that x_α is quasi-coincident with f, denoted as x_αqf, if f (x) + α > 1. Jun et al. [14] generalized the concept of x_αqf as x_αq_kf if f (x) + α + k > 1, where k ∈ [0, 1). Using these concepts, many results and notions in fuzzy ordered semigroups have been extended to their (∈ , ∈ ∨ q)-fuzzy or (∈ , ∈ ∨ q_k)-fuzzy counterparts, for example, in [5 , 20, 28]. Unfortunately, without the aid of addition in a frame, these ideas can not be used to generalize the frame-valued fuzzy subsets of an ordered semigroup.

4 Conclusions and future research

As we know, fuzzy subsets play an important role in the research of ordered semigroups. In the present paper, we discussed some new properties of strongly convex L-fuzzy subsets of an ordered semigroup, where L is a frame. In particular, we give further details on some results in [9] (see Theorems 3.11 and 3.12) and generalize some results in [31] (see Theorem 3.16). Since a frame is a generalization of the unit interval, we also showed that some results related to fuzzy algebra do not hold for frame-valued fuzzy algebra.

Recently, the concept of fuzzy partial orders, which is proposed by B $\overset{˘}{e}$ lohl $\overset{´}{a}$ vek [2] and others as a generalization of classical partial orders, has been successfully applied to domain theory (see, e.g. [33, 36]) and group theory (see, e.g. [4, 25]). These works constructed certain links between fuzzy orderings and other structures, and effectively extended various results in classical ordered structures. In future work, we would focus on investigating the fuzzy subsets (strongly convex fuzzy subsets, fuzzy ideals and fuzzy filters) of fuzzy ordered semigroups which are based on fuzzy partial orders and exploit their applications in some theoretical or applied fields.

Footnotes

Acknowledgments

The authors are highly grateful to editors and anonymous referees for their valuable comments and suggestions for improving the paper.

This work is supported by the National Natural Science Foundation of China, Grant Nos. 11371130, 61273018 and by the Research Fund for the Doctoral Program of Higher Education of China, Grant Nos. 20120161110017.

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