In fuzzy set theory, fuzzy convex structures are important mathematical structures. In this paper, we focus on separation axioms in fuzzy convex spaces. Concretely, we introduce S0, S1 and S2 separation axioms in fuzzy convex spaces and establish their relationships. Furthermore, we investigate their hereditary and productive properties.
Convexity, which is a branch of geometry, analysis and linear algebra, plays an important role in mathematics. Convexity exists in numerous mathematical environment such as vector spaces, lattices, algebras, graphs and topological spaces. In 1993, M. van de Vel collected the theory of convexity systematically in his famous book [21].
With the development of fuzzy set theory, many mathematical structures have been generalized to fuzzy case, such as fuzzy topologies [1, 4] and fuzzy convergence structures [8, 9]. Following this trend, convex structures have also been endowed with fuzzy set theory. Up to now, there have been three typical kinds of fuzzy convex structures, including L-convex structures [7, 16], M-fuzzifying convex structures [19] and (L, M)-fuzzy convex structures [20]. Many researchers studied fuzzy convex structures from different aspects, such as fuzzy hull operators [10, 13], fuzzy interval operators [14, 34], categorical properties [12, 30], convergence structures [29], degree presentations [6, 31] and geometric properties [26, 27].
Separation is an essential property in the theory of convex spaces. Ellis [2] gave a common generalization of Kakutani Separation Property in real vector spaces and of the Stone Theorem in distributive lattices. Jamison [3] gave a restricted version of the Polytope Characterization in terms of screening with half-spaces to characterize separation properties of general convex spaces. In the fuzzy case, Liang and Li [5] first studied the separation property of M-fuzzifying convex spaces. In a degree approach, they equipped each M-fuzzifying convex space with some degree to be S0, S1 and S2 separated. By this motivation, we will consider separation axioms in fuzzy convex spaces, which is exactly [0,1]-convex spaces. Concretely, we will propose S0, S1 and S2 separation axioms in fuzzy convex spaces. Furthermore, we will study their mutual relationships as well as their hereditary and productive properties.
Preliminaries
Throughout this paper, let X be a nonempty set and 2X be the powerset of X. Each mapping A : X ⟶ [0, 1] is called a fuzzy set of X, the complement of A, denoted by A′, is defined by A′ (x) =1 - A (x) for each x ∈ X. The collection of all fuzzy sets of X is denoted by . The set of all fuzzy points xλ(i.e., a fuzzy set such that A (x) = λ ≠ 0 and A (y) =0 for y ≠ x) is denoted by . For each a ∈ [0, 1], denotes the constant mapping X ⟶ [0, 1] , x ⟼ a, which is called a constant fuzzy subset. is also a complete lattice by defining ⩽ on pointwisely. For each U ⊆ X, its characteristic function is defined as follows:
For S ⊆ [0, 1], write ⋁S for the least upper bound of S and ⋀S for the greatest lower bound of S. We say {Aj} j∈J is a directed subset of , if for each Aj1, Aj2 ∈ {Aj} j∈J, there exists Aj3 ∈ {Aj} j∈J such that Aj1, Aj2 ⩽ Aj3. Also, we adopt the convention that ⋁ ∅ =0 and ⋀ ∅ =1.
Given a mapping f : X ⟶ Y, define and by f→ (A) (y) = ⋁ f(x)=yA (x) for and y ∈ Y, and f← (B) = B ∘ f for .
Definition 2.1. ([21]) A subset of 2X is called a convex structure on X if it satisfies the following conditions:
;
if is nonempty, then ;
if is nonempty and totally ordered by inclusion, then .
For a convex structure on X, the pair is called a convex space and each is called a convex set.
Definition 2.2. ([21]) Let be a convex space. A subset H of X is called a biconvex set (half-space, hemispace) provided H is convex and H′ is convex.
is said to be S1 separated provided that all singletons in X are convex.
is said to be S2 separated provided that for all x, y ∈ X with x ≠ y, there is a biconvex set H of X with x ∈ H and y ∉ H.
Definition 2.4. ([7, 16]) A subset of is called a fuzzy convex structure on X if it satisfies the following conditions:
;
if is nonempty, then ;
if is directed, then .
For a fuzzy convex structure on X, the pair is called a fuzzy convex space and each is called a fuzzy convex set.
Definition 2.5. ([7, 16]) Let be a fuzzy convex space. For each , define
That is, co (A) is the least element of that contains A, called the fuzzy convex hull of A.
The following definitions and propositions are proposed for a more general lattice background. Here we only use the special case [0, 1] as the table of truth-values.
Definition 2.6. ([20]) Let be a fuzzy convex space and ∅ ≠ Y ⊆ X. Then is a fuzzy convex structure on Y. We call a subspace of .
Definition 2.7. ([12]) A fuzzy convex structure on X is called stratified if it satisfies:
∀a ∈ [0, 1], .
For a stratified fuzzy convex structure on X, the pair is called a stratified fuzzy convex space.
Definition 2.8. ([12]) Let and be fuzzy convex spaces. A mapping f: X ⟶ Y is called
fuzzy convexity preserving (fuzzy CP for short) if for each ;
fuzzy convex-to-convex (fuzzy CC for short) if for each ;
isomorphic if f is a fuzzy CP and fuzzy CC bijection.
Definition 2.9. ([15]) Let be a family of fuzzy convex spaces, , and let {pi : X ⟶ Xi} i∈I be the family of projection mappings. The fuzzy product convex structure on X is the one generated by the subbase
The resulting fuzzy convex space is called the product of the spaces , and is denoted by .
Proposition 2.10. ([15]) Let be a family of stratified fuzzy convex spaces, , and let {pi : X ⟶ Xi} i∈I be the family of projection mappings. Then is a fuzzy CC and fuzzy CP mapping for each i ∈ I.
S0 separation axiom in fuzzy convex spaces
In this section, we will introduce S0 separation axiom in fuzzy convex spaces and investigate its hereditary and product property.
Definition 3.1. A fuzzy convex space is S0 separated if it satisfies
for any with xλ ≠ yμ, there exists such that λ > A (x) and μ ⩽ A (y), or there exists such that λ ⩽ B (x) and μ > B (y).
In this sense, we call an S0 fuzzy convex space.
Remark 3.2. If [0, 1] is replaced by {0, 1}, then S0 can be reduced to the classical S0 separation axiom in classical convex spaces. That is,
for any x, y ∈ X with x ≠ y, there exists such that x ∉ U and y ∈ U or there exists such that y ∉ V and x ∈ V.
Hence, S0 fuzzy convex spaces can be considered as the generalizations of S0 convex spaces.
Given a classical convex space , we can induce a fuzzy convex space in the following way.
Proposition 3.3. ([28]) Let be a classical convex space and define as follows: . Then is a fuzzy convex space.
Next we have the following theorem.
Theorem 3.4.Let be a classical convex space. If is S0 separated, then so is .
Proof. Take any x, y ∈ X with x ≠ y. Then it follows that for each λ ∈ (0, 1], we have and xλ ≠ yλ. Since is S0 separated, there exists such that λ > χU (x) and λ ⩽ χU (y), or there exists such that λ ⩽ χV (x) and λ > χV (y). This implies that there exists such that x ∉ U and y ∈ U, or there exists such that y ∉ V and x ∈ V. Thus, is S0 separated.□
By Theorem 3.4, if is not S0 separated, then is not S0 separated. It means that there really exist some concrete fuzzy convex spaces which are not S0 separated. Then, we give some examples.
Example 3.5. Let X = {x, y}. We define as follows:
Then is not an S0 fuzzy convex space.
Proof. It is obvious that is a fuzzy convex structure on X. Take satisfying x0.5 ≠ y0.5. If there exists , such that 0.5 > A (x) and 0.5 ⩽ A (y), then it is a contradiction to A (x) = A (y). If there exists , such that 0.5 ⩽ B (x) and 0.5 > B (y), then it is a contradiction to B (x) = B (y). Hence is not an S0 fuzzy convex space.□
Example 3.6. Let X = {x, y}. We define as follows:
Then is an S0 fuzzy convex space.
Proof. It is obvious that is a fuzzy convex structure on X. Hence it suffices to verify that it satisfies S0 separation axiom. Take any with xλ ≠ yμ. We divide into two cases. Case(1) with λ ≠ μ. If λ > μ, then there exist λ1, μ1 such that λ > λ1 ⩾ μ1 > μ. Take satisfying A (x) = λ1, A (x) = μ1. Hence λ > A (x) and A (x) ⩾ μ. If μ > λ, then there exist λ2, μ2 such that μ > λ2 ⩾ μ2 > λ. Take satisfying B (x) = μ2, B (x) = λ2. Hence μ > B (x) and B (x) ⩾ λ. Case(2)x ≠ y. The proof is similar to Case(1). This proves that for any with xλ ≠ yμ, there exists such that λ > A (x) and μ ⩽ A (y) or there exists such that λ ⩽ B (x) and μ > B (y). Hence is an S0 fuzzy convex space.□
Next let us investigate the hereditary and productive property of S0 separation axiom.
Theorem 3.7.Let be a fuzzy convex space and be a subspace of . If is S0 separated, then so is .
Proof. Take any with xλ ≠ yμ. Define as follows: for any z ∈ X,
It follows that . Since is S0 separated, there exists such that λ > A (x*) and μ ⩽ A (y*), or there exists such that λ ⩽ B (x*) and μ > B (y*). Let C = A|Y and D = B|Y. This implies that there exists such that λ > C (x) and μ ⩽ C (y), or there exists such that λ ⩽ D (x) and μ > D (y). Thus is S0 separated.□
In [15], Pang and Xiu proposed the concept of product space of L-convex spaces, where L is a continuous lattice. In order to show the productive property of separation axioms when L = [0, 1], we need the following definitions and theorem.
Definition 3.8. ([15]) Let be a fuzzy convex space and . Then is called a base of provided that for each , there is a directed family such that .
Definition 3.9. ([15]) Let be a fuzzy convex space and . Then is called a subbase of provided that is a base of , where .
Definition 3.10. ([15]) Let be the product space of . Then
is a base of . For convenience, we give the following notation with respect to the product space.
Let {Xi} i∈I be a family of non-empty sets, I≠ ∅, , x = {xi} i∈I be a fixed point in X, s ∈ I. Denote
Definition 3.11. Let be the product space of . Then the subspace of is called a fuzzy plane that is parallel to the factor space through the point x.
Lemma 3.12.Let be the product space of , {pi ∣ X ⟶ Xi} i∈I be the family of projection mappings, x = {xi} i∈I be a fixed point in X. If ( is stratified for some s ∈ S, then is a fuzzy CC mapping.
Proof. By Definition 3.10, is a base of . Then it suffices to verify that
for and k ∈ K ⊆ I.
Take each ys ∈ Xs, we have
Next we divide into two cases.
Case(1)s ∉ K. Then it follows that
Case(2)s ∈ K. Then it follows that
This implies
Since is stratified, it follows that and . So we always have
as desired.□
Theorem 3.13. Let be the product space of , x = {xi} i∈I be a fixed point in X. If is stratified for some s ∈ S, then the fuzzy plane is isomorphic to . That is,
Proof. Since is the fuzzy plane that is parallel to the factor space through the point x, we define a mapping by
where zi = xi for each i ≠ s and zs = ys .
It is easy to see that all the coordinates are fixed except for the sth coordinate, thus f is a bijection. Next it suffices to show that f is fuzzy CP and fuzzy CC.
(1) Take each . Then there exists such that . For each ys ∈ Xs, it follows that And
By the definition of f, we have z = f (ys) for each such that zs = ys. This implies that
which means
By Lemma 3.12, we have
which means f is fuzzy CP.
(2) Take each . Then for each , it follows from the definition of f that
which means
Since is fuzzy CP, it follows that
This implies that
Hence, f is fuzzy CC.□
Theorem 3.14.Let be an isomorphic mapping. If is S0 separated, then so is .Proof. For any with xλ ≠ yμ, it is easy to check that and f← (xλ) ≠ f← (yμ). Since is S0 separated, there exists such that λ > A (f-1 (x)) and μ ⩽ A (f-1 (y)), or there exists such that λ ⩽ B (f-1 (x)) and μ > B (f-1 (y)). Furthermore, f is isomorphic, then it follows that there exists such that λ > f→ (A) (x) and μ ⩽ f→ (A) (y), or there exists such that λ ⩽ f→ (B) (x) and μ > f→ (B) (y). Thus, is an S0 fuzzy convex space.□
Theorem 3.15.Let be a family of fuzzy convex spaces and be the product space. If is S0 separated for each i ∈ I, then so is . Conversely, if is S0 separated, and is stratified for some i ∈ I, then is S0separated.
Proof. Take any such that xλ ≠ yλ. Then there exists i0 ∈ I such that pi0 (x) λ ≠ pi0 (y) μ. Since is S0 separated, there exists such that λ > Ai0 (pi0 (x)) and μ ⩽ Ai0 (pi0 (y)), or there exists such that λ ⩽ Bi0 (pi0 (x)) and μ > Bi0 (pi0 (y)). By the definition of product of fuzzy convex spaces, we can obtain that the projection mapping is a fuzzy CP mapping. Then implies . This means that there exists such that and or there exists such that and . Thus we obtain that is an S0 fuzzy convex space. Conversely, assume that is stratified for some i ∈ I, it follows from Theorem 3.13 that is isomorphic to , which is a subspace of . From Theorems 3.7 and 3.14, is S0 separated.□
S1 separation axiom in fuzzy convex spaces
In this section, we will present the definition of S1 separation axiom in fuzzy convex spaces and investigate its properties.
Definition 4.1. A fuzzy convex space is S1 separated if it satisfies
for any with xλ≰yμ, there exists such that λ > A (x) and μ ⩽ A (y).
In this sense, we call an S1 fuzzy convex space.
In the classical case, S1 convex space is described as follows: is S1 iff for each x ∈ X.
In the case of fuzzy convex spaces, S1 separation axiom can also be characterized in this way.
Proposition 4.2. A fuzzy convex space is S1 separated iff it satisfies
, .
Proof. (S1) ∗ ⇒ (S1): Obviously. (S1) ⇒ (S1) ∗: Take each such that yμ≰xλ. Then it follows from (S1) that there exists such that μ > A (y) and λ ⩽ A (x). Then we have yμ≰A and xλ ⩽ A. This implies
By the arbitrariness of yμ, we get
Then we have
This means
as desired.□
The next result will show the relationship between S0 and S1 separation axioms.
Theorem 4.3.Let be a fuzzy convex space. If is S1 separated, then is S0 separated.
Proof. It follows immediately from the definitions of S0 and S1 separation axioms.□
In Example (3.6), take x0.5≰y0.5, then there is no satisfying 0.5 > A (x) and 0.5 ⩽ A (y), but there exists such that 0.5 ⩽ B (x) and 0.5 > B (y). Hence we know that is an S0 fuzzy convex space but it is not an S1 fuzzy convex space.
Next, we will give an example of S1 fuzzy convex space.
Example 4.4. Let X = {x}. Take . Then is a stratified fuzzy convex space. For each , we have . Hence it is an S1 fuzzy convex space from Proposition 4.2. In the following, we will show the hereditary and productive property of S1 separation axiom.
Theorem 4.5.Let be a fuzzy convex space and be a subspace of . If is S1 separated, then so is .
Proof. Take each . Define as follows: for any z ∈ X,
Then it follows that . Since is S1 separated, we have from Proposition 4.2. It is easy to see that , which means . Thus we obtain for each . By Proposition 4.2, we know that is S1 separated.
□
Theorem 4.6.Let be an isomorphic mapping. If is S1 separated, then so is .
Proof. Take each . For f is a bijection, we have . Since is S1 separated, it follows that by Proposition 4.2. Furthermore, f is fuzzy CP, hence , as desired.□
Theorem 4.7. Let be a family of fuzzy convex spaces and be the product space. If is S1 separated for each i ∈ I, then so is . Conversely, if is S1 separated, and is stratified for some i ∈ I, then is S1 separated.
Proof. Take each . Then for each i ∈ I. Since is S1 separated, it follows from Proposition 4.2 that for each i ∈ I. By the definition of product of fuzzy convex spaces, we can obtain that the projection mapping is a fuzzy CP mapping. This means . Thus for each . By Proposition 4.2, we know that is S1 separated. Conversely, assume that is stratified for some i ∈ I, it follows from Theorem 3.13 that is isomorphic to , which is a subspace of . From Theorems 4.5 and 4.6, is S1 separated.□
S2 separation axiom in fuzzy convex spaces
In this section, we will propose S2 separation axiom in fuzzy convex spaces and investigate its relationships with S0 and S1 separation axioms as well as its hereditary and productive properties.
In order to give the definition of S2 fuzzy convex spaces, we first give the following concept.
Definition 5.1. Let be a fuzzy convex space and . A is called a fuzzy biconvex subset (fuzzy half-space, fuzzy hemispace) provided that and .
For convenience, the family of all fuzzy biconvex subsets in is denoted by .
Remark 5.2. If [0, 1] is replaced by {0, 1}, then A′ reduces to the complement of A. Then fuzzy biconvex subsets in fuzzy convex spaces will reduce to classical biconvex sets in classical convex spaces.
In [35], Yang and Shi proposed L-biconvex sets on some fuzzy algebraic substructures, where L is a complete residuated lattice. Here we list some results when L = [0, 1].
Let be a fuzzy meet semilattice convex space. Denote Then HS is the family of fuzzy biconvex subsets of .
Let be a fuzzy join semilattice convex space. Denote Then HS is the family of fuzzy biconvex subsets of .
Let X be a lattice. Denote Then HL is the family of fuzzy biconvex subsets of X.
Now let us present the definition of S2 fuzzy convex spaces.
Definition 5.4. A fuzzy convex space is S2 separated if it satisfies
for any with xλ≰yμ, there exists such that λ > A (x) and μ ⩽ A (y).
In this case, we call an S2 fuzzy convex space.
Remark 5.5. If [0, 1] is replaced by {0, 1}, then S2 separation axiom in Definition 5.3 will reduce to S2 separation axiom in classical convex spaces. That is,
for any x, y ∈ X with x ≠ y, there exists a biconvex set A such that x ∉ A and y ∈ A.
This shows that S2 fuzzy convex spaces are exactly the generalizations of S2 convex spaces.
The following result shows the relationship between S1 and S2 separation axiom.
Proposition 5.6.Let be a fuzzy convex space. If is S2 separated, then is S1 separated.
Proof. Since , that is, fuzzy biconvex subsets are fuzzy convex subsets. It follows immediately from the definitions of S1 and S2 separation axioms that S2 fuzzy convex spaces must be S1 fuzzy convex spaces.□
Example 5.7. It is easy to check that the fuzzy convex space in Example (4.4) is an S2 fuzzy convex space.
Next we show the hereditary of S2 separation axiom. We first give the following lemma.
Lemma 5.8.Let be a fuzzy convex space and let be a subspace of . If , then .
Proof. Since and , it follows that and . Furthermore, it is easy to check that (A|Y) ′ = A′|Y. This implies . Thus we have and , which means .□
Theorem 5.9.Let be a fuzzy convex space and be a subspace of . If is S2 separated, then so is .
Proof. Take any with xλ≰yμ. Define as follows: for any z ∈ X,
It follows that . By S2 separation axiom of , there exists such that λ > A (x*) and μ ⩽ A (y*). This implies λ > A|Y (x) and μ ⩽ A|Y (y). By Lemma 5.8, we have . This proves that for any with xλ≰yμ, there exists such that λ > A|Y (x) and μ ⩽ A|Y (y), as desired.□
In order to show the productive property of S2 separation axiom, the following lemma and theorem are necessary.
Lemma 5.10.Let f: be a fuzzy CP mapping. Then implies
Proof. Take each . Then and . Since f is a fuzzy CP mapping, it follows that and . This implies , as desired.□
Theorem 5.11.Let be an isomorphic mapping. If is S2 separated, then so is .
Proof. Take any with xλ≰yμ. Then and f← (xλ) ≰f← (yμ). By S2 separation axiom of , there exists such that λ > A (f-1 (x)) and μ ⩽ A (f-1 (y)). Since f is isomorphic, this implies that there exists such that λ > f→ (A) (x) and μ ⩽ f→ (A) (y). Hence, is an S2 fuzzy convex space.□
Theorem 5.12.Let be a family of fuzzy convex spaces and be the product space. If is S2 separated for each i ∈ I, then so is . Conversely, if is S2 separated, and is stratified for some i ∈ I, then is S2 separated.
Proof. Take any with xλ≰yμ. Then there exists i0 ∈ I such that pi0 (x) λ≰pi0 (y) μ. Since is S2 separated, there exists such that λ > Ai0 (pi0 (x)) and μ ⩽ Ai0 (pi0 (y)). Then and . Since pi0 is a fuzzy CP mapping, we have by Lemma 5.10. This shows that for any with xλ≰yμ, there exists such that and . Thus is S2 separated. Conversely, assume that is stratified for some i ∈ I, it follows from Theorem 3.13 that is isomorphic to , which is a subspace of . From Theorems 5.9 and 5.11, we have that is S2 separated.□
Conclusions
In this paper, we mainly introduced S0, S1 and S2 separation axioms in fuzzy convex spaces and discussed their hereditary and productive properties.
As we all know, in classical convex spaces, S3 and S4 separation axioms are also important properties and have close connections with other properties. S3 separation property can be characterized by Sand-glass Property under Join-hull Commutativity Property. The convexity of a Pasch-Peano space can be characterized by S4 separation property and Join-hull Commutativity Property. In [27], the Sand-glass Property and Pasch-Peano Property have already been generalized to the fuzzy case, therefore our further exploration is to study the high-level separation axioms in fuzzy convex spaces. That is to say, we will consider S3 and S4 separation axioms in fuzzy convex spaces and then investigate the relationships among the multiple properties mentioned above.
Footnotes
Acknowledgments
The author would like to express the sincere thanks to referees for their careful reading and constructive comments. The project is supported by the National Natural Science Foundation of China (12071033, 11871097).
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