Split Hausdorff internal topologies on posets

Shuzhen Luo; Xiaoquan Xu

doi:10.1515/math-2019-0136

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Split Hausdorff internal topologies on posets

Shuzhen Luo and Xiaoquan Xu

Published/Copyright: December 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation ρ : X ⇀ Y satisfies a certain condition, ρ is weak generalized finitely regular if and only if (φ_ρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φ_ρ(X, Y), ⊆) is split T₂; (2) the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T₂.

Keywords: split T2; T2 property; weak generalized finitely regular; weak quasi-hypercontinuous poset

MSC 2010: 06B35; 54H10; 06A11

1 Introduction

In domain theory, the interval topology and the Lawson topology are two important "two-sided" topologies on posets. A basic problem (see [1, 2, 3, 4, 5]) is: When do the interval topology and the Lawson topology have T₂ properties? In [5] (see also [3, 4]), Gierz and Lawson have discussed this problem for the Lawson topology, and proved that a complete lattice is a quasicontinuous lattice if and only if the Lawson topology is T₂. However, T₂ properties for the interval topology on posets have attracted a considerable deal of attention (see [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]). Especially, Erné [1] obtained several equivalent characterizations about T₂ properties of the interval topology on posets. For a complete lattice L, Gierz and Lawson [5] proved that the interval topology on L is T₂ if and only if L is a generalized bicontinuous lattice.

The regularity of binary relations was first characterized by Zareckiǐ [18]. In [18] he proved the following remarkable result: a binary relation ρ on a set X is regular if and only if the complete lattice (Φ_ρ(X), ⊆) is completely distributive, where Φ_ρ(X) = {ρ(A) : A ⊆ X}, ρ(A) = {y ∈ X : ∃a ∈ A with (a, y) ∈ ρ}. Further criteria for regularity were given by Markowsky [19] and Schein [20] (see also [21] and [22]). Motivated by the fundamental works relative Zareckiǐ on regular relations, Xu and Liu [23] introduced the concepts of finitely regular relations and generalized finitely regular relations, respectively. It is proved that a relation ρ is generalized finitely regular if and only if the interval topology on (Φ_ρ(X), ⊆) is T₂. Especially, in complete lattices, this condition turns out to be equivalent both to the T₂ interval topology and to the quasi-hypercontinuous lattices.

In this paper, we mainly concentrate on the T₂ interval topology of posets by using the regularity of binary relations. Therefore, we introduce the concepts of the split T₂ interval topology on posets and weak generalized finitely regular relations. Meanwhile, in order to characterize split T₂ interval topology of posets by a order structure, like the equivalence of the T₂ interval topology and quasi-hypercontinuous lattices in [23], we give the notion of a weak quasi-hypercontinuous poset. It is proved that when a binary relation ρ : X ⇀ Y satisfies property M, ρ is weak generalized finitely regular if and only if (φ_ρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φ_ρ(X, Y), ⊆) is split T₂, where φ_ρ(X, Y) = {ρ(x): x ∈ X}. For a poset P, the relation ≰ on P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T₂, which generalizes the corresponding works in [12, 16, 17].

2 Preliminaries

In this section, we recall some basic concepts needed in this paper; other non-explicitly stated elementary notions please refer to [4, 23, 24].

Let P be a poset. For all x ∈ P, A ⊆ P, let ↑ x = {y ∈ P : x ≤ y} and ↑ A = ⋃_a∈A ↑ a; ↓ x and ↓ A are defined dually. A^↑ and A^↓ denote the sets of all upper and lower bounds of A, respectively. Let A^δ = (A^↑)^↓ and δ(P) = {A^δ : A ⊆ P}. To avoid ambiguities, we also denote A^↑, A^↓ and A^δ on P by AP↑,AP↓ and APδ , respectively. (δ(P), ⊆) is called the normal completion, or the Dedekind-MacNeille completion of P (see [25]). The topology generated by the collection of sets P ∖ ↓ x (as a subbase) is called the upper topology and denoted by υ(P); the lower topology ω(P) on P is defined dually. The topology θ(P) = υ(P) ∨ ω(P) is called the interval topology on P. For any set X, let X^(<ω) = {F ⊆ X : F is nonempty and finite}.

For two sets X and Y, we call ρ : X ⇀ Y a binary relation if ρ ⊆ X × Y. When X = Y, ρ is usually called a binary relation on X.

Definition 2.1

Let ρ : X ⇀ Y, τ : Y ⇀ Z be two binary relations. Define

τ ∘ ρ = {(x, z) : ∃y ∈ Y, (x, y) ∈ ρ, (y, z) ∈ τ}. The relation τ ∘ ρ : X ⇀ Z is called the composition of ρ and τ.
ρ^–1 = {(y, x) ∈ Y × X : (x, y) ∈ ρ}.
ρ^c = X × Y ∖ ρ.
ρ(A) = {y ∈ Y : ∃x ∈ A with (x, y) ∈ ρ}, we call it the image of A under a binary relation ρ. Instead of ρ({x}), we write ρ(x) for short.
Φ_ρ(X, Y) = {ρ(A) : A ⊆ X}.
φ_ρ(X, Y) = {ρ(x) : x ∈ X}.
φ_y = {ρ(u) ∈ φ_ρ(X, Y) : y ∉ ρ(u)}.

Clearly, φ_ρ(X, Y) ⊆ Φ_ρ(X, Y), and (Φ_ρ(X, Y), ⊆) is a complete lattice in which the join operation ∨ is the set union operator ∪. But in general (φ_ρ(X, Y), ⊆) is not a complete lattice. For example, let X = {x₁, x₂, x₃} and Y = {y₁, y₂, y₃}. Define a relation ρ = {(x₁, y₁), (x₁, y₂), (x₂, y₂), (x₂, y₃), (x₃, y₁), (x₃, y₃)}. Then ρ(x₁) = {y₁, y₂}, ρ(x₂) = {y₂, y₃} and ρ(x₃) = {y₁, y₃}. It is easy to see that there is no lease upper bound of ρ(x₁), ρ(x₂) in (φ_ρ(X, Y), ⊆).

Definition 2.2

[12] Let P be a poset and x ∈ P, A ⊆ P.

Define a relation ≺ on P by A ≺_P x iff x ∈ int_υ(P) ↑ A. Without causing confusion, we write A ≺ x for short.
P is called quasi-hypercontinuous if for all x ∈ P, ↑ x = ⋂ {↑ F : F is finite and F ≺ x} and {F ∈ P^(<ω) : F ≺ x} is directed.

A complete lattice which is quasi-hypercontinuous as a poset is called a quasi-hypercontinuous lattice (see [12]). In [12], it has been proved that L is a quasi-hypercontinuous lattice if for all x ∈ L, and U ∈ υ(L) with x ∈ U, there exists F ∈ L^(<ω) such that x ∈ int_υ(L) ↑ F ⊆ ↑ F ⊆ U.

Definition 2.3

[12] A binary relation ρ : X ⇀ Y is called generalized finitely regular, ∀(x, y) ∈ ρ, ∃{u₁, u₂, …, u_n} ∈ X^(<ω) and {v₁, v₂, …, v_m} ∈ Y^(<ω) such that

(u_i, y) ∈ ρ, (x, v_j) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m), and
∀ {s₁, s₂, …, s_m} ∈ X^(<ω), {t₁, t₂, …, t_n} ∈ Y^(<ω), if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s_j, v_j) ∈ ρ (j = 1, 2, …, m), then ∃ (l, k) ∈ {1, 2, …, m} × {1, 2, …, n} such that (s_l, t_k) ∈ ρ.

Theorem 2.4

[12] Let ρ : X ⇀ Y be a binary relation. Then the following conditions are equialent:

ρ is generalized finitely regular;
(Φ_ρ(X, Y), ⊆) is a quasi-hypercontinuous lattice.

Definition 2.5

[12] Let τ and δ be two topologies on a poset P. α = τ ∨ δ is called split T₂ or split Hausdorff about τ and δ, if for any x, y with x ≰ y, there exists (U, V) ∈ τ × δ such that x ∈ U, y ∈ V with U ∩ V = ∅. We call it split T₂ internal topology on a poset P, if the internal topology θ(P) is split T₂ about υ(P) and ω(P).

In [12, 24], it is pointed that split T₂ is strictly stronger than T₂ property.

3 Weak generalized finitely regular relations

In this section, we consider the split T₂ interval topology of posets by using the regularity of binary relations, and obtain that the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T₂.

Definition 3.1

A poset P is called weak quasi-hypercontinuous, if ↑ x = ⋂ {↑ F : F ∈ P^(<ω), F ≺ x} for all x ∈ P.

In contrast to quasi-hypercontinuous posets, a weak quasi-hypercontinuous poset need not be the case that the set {F ∈ P^(<ω) : F ≺ x} is directed. Clearly, P is a quasi-hypercontinuous poset ⇒ P is weak quasi-hypercontinuous, and If P is a sup-semilattice, then they are equivalent.

Definition 3.2

A binary relation ρ : X ⇀ Y is called weak generalized finitely regular, w-generalized finitely regular for short, if for any (x, y) ∈ ρ, there are {u₁, u₂, …, u_n} ∈ X^(<ω) and {v₁, v₂, …, v_m} ∈ Y^(<ω) such that

(u_i, y) ∈ ρ, (x, v_j) ∈ ρ(i = 1, 2, …, n; j = 1, 2, …, m), and
∀s ∈ X, {t₁, t₂, …, t_n} ⊆ Y, if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s, v_j) ∈ ρ (j = 1, 2, …, m), then there is a k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ.

Obviously, if ρ is generalized finitely regular, then ρ is w-generalized finitely regular.

Proposition 3.3

For a binary relation ρ : X ⇀ Y, the following conditions are equivalent:

ρ is w-generalized finitely regular;
∀(x, y) ∈ ρ, ∃(U, V) ∈ X^(<ω) × Y^(<ω) such that
1. U ⊆ ρ^–1(y), V ⊆ ρ(x);
2. ∀(s, T) ∈ X × Y^(<ω), if U ⊆ ρ^–1(T) and V ⊆ ρ(s), then T ∩ ρ(s) ≠ ∅.

Proof

(1) ⇒ (2) For any (x, y) ∈ ρ, since ρ is w-generalized finitely regular, ∃{u₁, u₂, …, u_n} ∈ X^(<ω) and {v₁, v₂, …, v_m} ∈ Y^(<ω) such that

(u_i, y) ∈ ρ, (x, v_j) ∈ ρ(i = 1, 2, …, n; j = 1, 2, …, m), and
∀s ∈ X, {t₁, t₂, …, t_n} ⊆ Y, if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s, v_j) ∈ ρ (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ.

Let U = {u₁, u₂, …, u_n}, V = {v₁, v₂, …, v_m}. Then (U, V) ∈ X^(<ω) × Y^(<ω). By the condition (a), we have that U ⊆ ρ^–1(y), V ⊆ ρ(x), i.e., the condition (i) in (2) is satisfied. Now we check the condition (ii) in (2). ∀(s, T) ∈ X × Y^(<ω), if U ⊆ ρ^–1(T) and V ⊆ ρ(s), then ∀ i ∈ {1, 2, …, n}, ∃t_i ∈ T such that (u_i, t_i) ∈ ρ, and ∀ j ∈ {1, 2, …, m}, (s, v_j) ∈ ρ, by the condition (b), ∃k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ. Thus T ∩ ρ(s) ≠ ∅.

(2) ⇒ (1) Let (x, y) ∈ ρ. By (2), ∃ (U, V) ∈ X^(<ω) × Y^(<ω) such that

U ⊆ ρ^–1(y), V ⊆ ρ(x), and
∀(s, T) ∈ X × Y^(<ω), if U ⊆ ρ^–1(T) and V ⊆ ρ(s), then T ∩ ρ(s) ≠ ∅.

Let U = {u₁, u₂, …, u_n}, V = {v₁, v₂, …, v_m}. Then by condition (i), we have that (u_i, y) ∈ ρ, (x, v_j) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m). For any s ∈ X, {t₁, t₂, …, t_n} ⊆ Y, if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s, v_j) ∈ ρ (j = 1, 2, …, m), let T = {t₁, t₂, …, t_n}. Then U ⊆ ρ^–1(T) and V ⊆ ρ(s). By the condition (ii), T ∩ ρ(s) ≠ ∅, i.e., ∃k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ. Thus (1) holds.□

Definition 3.4

Let ρ : X ⇀ Y be a binary relation. We call ρ satisfies property M if for any y ∈ Y, φ_y = ∅ or φ_y has the greatest element, where φ_y = {ρ(u) ∈ φ_ρ(X, Y) : y ∉ρ(u)}.

Example 3.5

Let E be a binary relation on a set X with reflexive and transitive. Then the relation E^c = X² ∖ E satisfies property M.

In fact, for any y ∈ X, since E is reflexive, y ∉ E^c(y). Thus φ_y ≠ ∅. Let u ∈ X with y ∉ E^c(u), i.e., (u, y) ∈ E. Suppose that E^c(u) ⊈ E^c(y), then there is a t ∈ E^c(u) such that t ∉ E^c(y), i.e., (u, t) ∉ E and (y, t) ∈ E, we have (u, t) ∈ E since E is transitive, which contradicts (u, t) ∉ E. Thus E^c(y) is the greatest element of φ_y. Hence, the relation E^c satisfies property M.
Let X be a set and Y = {y}. Define a function f : X → Y by f(x) = y for any x ∈ X. Then f satisfies property M, since φ_y = ∅ for any y ∈ Y.
Let X, Y be two sets and g : X → Y a injective function. If |X| > 2, then g is not satisfy property M, since for any x₁, x₂ ∈ X, g(x₁) ⊈ g(x₂).

For any poset P, the relation ≤ on P is reflexive and transitive, by Example 3.5(1), we have the following corollary.

Corollary 3.6

For any poset P, the relation ≰ on P satisfies property M.

Lemma 3.7

Let ρ : X ⇀ Y be a binary relation. If ρ satisfies property M, then δ((φ_ρ(X, Y), ⊆)) is order isomorphism to (Φ_ρ(X, Y), ⊆).

Proof

For any A ⊆ X, define η : δ((φ_ρ(X, Y), ⊆)) → (Φ_ρ(X, Y), ⊆) by η( {ρ(x):x∈A}φρ(X,Y)δ = ρ(A) and ψ : (Φ_ρ(X, Y), ⊆) → δ((φ_ρ(X, Y), ⊆)) by ψ(ρ(A)) = {ρ(x):x∈A}φρ(X,Y)δ .

1^∘ η is order preserving. Let {ρ(x):x∈A}φρ(X,Y)δ ⊆ {ρ(y):y∈B}φρ(X,Y)δ . Then {ρ(y):y∈B}φρ(X,Y)↑ ⊆ {ρ(x):x∈A}φρ(X,Y)↑ . Now we have to show that ρ(A) ⊆ ρ(B). For any w ∈ ρ(A), there is a x_w ∈ A such that w ∈ ρ(x_w). Since ρ satisfies property M, let N_w be the greatest element of φ_w (if φ_w = ∅, let N_w = ∅). Then ρ(x_w) ⊈ N_w. Thus N_w ∉ {ρ(x):x∈A}φρ(X,Y)↑ . Note that {ρ(y):y∈B}φρ(X,Y)↑ ⊆ {ρ(x):x∈A}φρ(X,Y)↑ , we have N_w ∉ {ρ(y):x∈B}φρ(X,Y)↑ , it follows from that there is a b ∈ B such that ρ(b) ⊈ N_w. By the definition of N_w, w ∈ ρ(b) ⊆ ρ(B). Hence ρ(A) ⊆ ρ(B).

2^∘ ψ is order preserving. Let ρ(A) ⊆ ρ(B). We only have to show {ρ(y):y∈B}φρ(X,Y)↑ ⊆ {ρ(x):x∈A}φρ(X,Y)↑ . Suppose not, there is a ρ(w) ∈ {ρ(y):y∈B}φρ(X,Y)↑ such that ρ(w) ∉ {ρ(x):x∈A}φρ(X,Y)↑ . Thus for any y ∈ B, ρ(y) ⊆ ρ(w) and ρ(x₀) ⊈ ρ(w) for some x₀ ∈ A, it follows that there is a z₀ ∈ ρ(x₀) with z₀ ∉ ρ(w). Since ρ(x₀) ⊆ ρ(A) ⊆ ρ(B), there exists y₀ ∈ B such that z₀ ∈ ρ(y₀). Note that ρ(y) ⊆ ρ(w) for any y ∈ B. Thus z₀ ∈ ρ(w), which contradicts z₀ ∉ ρ(w). Thus {ρ(y):y∈B}φρ(X,Y)↑ ⊆ {ρ(x):x∈A}φρ(X,Y)↑ . Therefore, ψ(ρ(A)) ⊆ ψ(ρ(B)).

Obviously, η ∘ ψ = id_{(Φ_ρ(X,Y),⊆)} and ψ ∘ η = id_{δ((φ_ρ(X,Y),⊆))}. All there show that δ((φ_ρ(X, Y), ⊆)) ≅ (Φ_ρ(X, Y), ⊆).□

From the Lemma 3.7, we can see that if ρ satisfies property M, then (Φ_ρ(X, Y), ⊆) is the normal completion of (φ_ρ(X, Y), ⊆).

Definition 3.8

[24] A poset P is called S-poset, if for any F, G ∈ P^(<ω) ∖ {∅}, F ⊆ G^↓, there exists u ∈ P such that F ⊆ ↓ u ⊆ G^↓.

Lemma 3.9

[24] Let P be a sup-semilattice (inf-semilattice). Then P is an S-poset.

Lemma 3.10

Let ρ : X ⇀ Y be a relation with property M. 𝓕 ∈ φ_ρ(X, Y)^(<ω) and ρ(x) ∈ φ_ρ(X, Y). Consider the following conditions.

𝓕 ≺_{Φ_ρ(X,Y)} ρ(x);
𝓕 ≺_{φ_ρ(X,Y)}ρ(x).

Then (1) ⇒ (2). If φ_ρ(X, Y) is an S-poset, then they are equivalent.

Proof

(1) ⇒ (2) Let 𝓕 ≺_{Φ_ρ(X,Y)}ρ(x). Then there exist ρ(A₁), ρ(A₂), …, ρ(A_m) ⊆ Φ_ρ(X, Y) such that ρ(x) ∈ Φ_ρ(X, Y) ∖ ↓_{Φ_ρ(X,Y)} {ρ(A₁), ρ(A₂), …, ρ(A_m)} ⊆ ↑_{Φ_ρ(X,Y)} 𝓕. Thus for any j ∈ {1, 2, …, m}, ρ(x) ⊈ ρ(A_j), and thus there is a z_j ∈ ρ(x) with z_j ∉ ρ(A_j). Obviously, φ_{z_j} ≠ ∅. Let N_j be the greatest of φ_{z_j}. Then ρ(x) ⊈ N_j for any j ∈ {1, 2, …, m}, and thus ρ(x) ∈ φ_ρ(X) ∖ ↓_{φ_ρ(X)}{N₁, N₂, …, N_j}. Now we show that φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)}{N₁, N₂, …, N_j} ⊆ ↑_{φ_ρ(X,Y)} 𝓕. Let ρ(w) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)}{N₁, N₂, …, N_j}. Then for any j ∈ {1, 2, …, m}, ρ(w) ⊈ N_j. By the definition of N_j, we have z_j ∈ ρ(w), and thus ρ(w) ⊈ρ(A_j) (since z_j ∉ρ(A_j)). Hence ρ(w) ∈ Φ_ρ(X, Y) ∖ ↓_{Φ_ρ(X,Y)} {ρ(A₁), ρ(A₂), …, ρ(A_m)}, it follows that ρ(w) ∈ ↑_{φ_ρ(X,Y)} 𝓕. Hence 𝓕 ≺_{φ_ρ(X,Y)}ρ(x).

(2) ⇒ (1) Suppose that 𝓕 ≺_{φ_ρ(X,Y)}ρ(x), then there exists {ρ(y₁), ρ(y₂), …, ρ(y_m)} ⊆ φ_ρ(X, Y) such that ρ(x) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)} {ρ(y₁), ρ(y₂), …, ρ(y_m)} ⊆ ↑_{φ_ρ(X,Y)} 𝓕. Now we have to show that ρ(x) ∈ Φ_ρ(X, Y) ∖ ↓_{Φ_ρ(X,Y)} {ρ(y₁), ρ(y₂), …, ρ(y_m)} ⊆ ↑_{Φ_ρ(X,Y)} 𝓕. Obviously, ρ(x) ∈ Φ_ρ(X, Y) ∖ ↓_{Φ_ρ(X,Y)} {ρ(y₁), ρ(y₂), …, ρ(y_m)}. Assume that there is a ρ(A) ∈ Φ_ρ(X, Y) ∖ ↓_{Φ_ρ(X,Y)} {ρ(y₁), ρ(y₂), …, ρ(y_m)} such that ρ(A) ∉ ↑_{Φ_ρ(X,Y)} 𝓕. Let 𝓕 = {ρ(u₁), ρ(u₂), …, ρ(u_n)}. Then ρ(A) ⊈ ρ(y_j)(j = 1, 2, …, m) and ρ(u_i) ⊈ ρ(A)(i = 1, 2, …, n). Thus there exist s_j ∈ A and v_j ∈ ρ(s_j) such that v_j ∉ ρ(y_j) (j = 1, 2, …, m), and t_i ∈ ρ(u_i) with t_i ∉ ρ(A)(i = 1, 2 …, n). We can conclude that there exist k₀ ∈ {1, 2, …, m} and l₀ ∈ {1, 2, …, n} such that t_l₀ ∈ ρ(s_k₀). If not, then for any k ∈ {1, 2, …, m} and l ∈ {1, 2, …, n}, t_l ∉ ρ(s_k). Let N_l be the greatest element of φ_{t_l}. Then ρ(s_k) ⊆ N_l, so for any k ∈ {1, 2, …, m}, ρ(s_k) is a lower bound of {N₁, N₂, …, N_n}. Since φ_ρ(X, Y) is an S-poset, there exists s ∈ X such that {ρ(s₁), ρ(s₂), …, ρ(s_m)} ⊆ ↓ ρ(s) ⊆ {N₁, N₂, …, N_n}^↓. Thus, v_j ∈ ρ(s) and ρ(s) ⊈ ρ(y_j)(j = 1, 2, …, m), that is ρ(s) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)} {ρ(y₁), ρ(y₂), …, ρ(y_n)}, so ρ(s) ∈ ↑_{φ_ρ(X,Y)} 𝓕. Thus there is a l ∈ {1, 2, …, n} such that ρ(u_l) ⊆ ρ(s). Notice that t_l ∈ ρ(u_l), we have t_l ∈ ρ(s). On the other side, since ρ(s) ∈ {N₁, N₂, …, N_n}^↓, ρ(s) ⊆ N_l. By the definition of N_l, t_l ∉ ρ(s), a contradiction. Therefore, there exist k₀ ∈ {1, 2, …, m} and l₀ ∈ {1, 2, …, n} such that t_l₀ ∈ ρ(s_k₀). Since ρ(s_{k_o}) ⊆ ρ(A), t_l₀ ∈ ρ(A), which contradicts t_l ∉ ρ(A) for any l ∈ {1, 2 …, n}. Hence 𝓕 ≺_{Φ_ρ(X,Y)}ρ(x).□

Theorem 3.11

For a binary relation ρ : X ⇀ Y with property M, consider the following conditions:

ρ is w-generalized finitely regular;
(φ_ρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset;
the interval topology on (φ_ρ(X, Y), ⊆) is split T₂;
(Φ_ρ(X, Y), ⊆) is a quasi-hypercontinuous lattice.

Then (1) ⇔ (2) ⇔ (3) ⇐ (4). If φ_ρ(X, Y) is an S-poset, then (1) – (4) are equivalent.

Proof

(1) ⇒ (2) For any ρ(x) ∈ φ_ρ(X, Y), if ρ(x) ⊈ ρ(u), then there is a y ∈ ρ(x) such that y ∉ ρ(u). Since ρ is w-generalized finitely regular, there are {u₁, u₂, …, u_n} ∈ X^(<ω) and {v₁, v₂, …, v_m} ∈ Y^(<ω) such that

(u_i, y) ∈ ρ, (x, v_j) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m), and
∀s ∈ X, T = {t₁, t₂, …, t_n} ⊆ Y, if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s, v_j) ∈ ρ (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ.

Let 𝓕 = {ρ(u₁), ρ(u₂), …ρ(u_n)}. Then 𝓕 ∈ φ_ρ(X, Y)^(<ω) and ρ(u_i) ⊈ ρ(u) (i = 1, 2, …, n), thus ρ(u) ∉ 𝓕. Let N_j be the greatest element of φ_{v_j} (if φ_{v_j} = ∅, let N_j = ∅). Then ρ(x) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)}{N₁, N₂, …, N_m} since v_j ∈ ρ(x) (j = 1, 2, … m). For any ρ(s) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)}{N₁, N₂, …, N_m}, ρ(s) ⊈ N_j(j = 1, 2, …, m). By the definition of N_j, v_j ∈ ρ(s). If ρ(s) ∉ ↑_{φ_ρ(X,Y)} 𝓕, then for any i ∈ {1, 2, …, n}, ρ(u_i) ⊈ ρ(s), so there is a t_i ∈ ρ(u_i) with t_i ∉ ρ(s). By the condition (b), there is a k ∈ {1, 2, …, m} such that (s, t_k) ∈ ρ, i.e., t_k ∈ ρ(s), which contradicts t_i ∉ ρ(s) for any i ∈ {1, 2, …, n}. Thus ρ(s) ∈ ↑_{φ_ρ(X,Y)} 𝓕.

All above show that ρ(x) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)} {N₁, N₂, …, N_m} ⊆ ↑_{φ_ρ(X,Y)} 𝓕, i.e., 𝓕 ≺_{φ_ρ(X,Y)}ρ(x). Note that ρ(u) ∉ ↑_{φ_ρ(X,Y)} 𝓕. Hence, for any ρ(x) ∈ φ_ρ(X, Y), ↑_{φ_ρ(X,Y)}ρ(x) = ⋂ {↑_{φ_ρ(X,Y)} 𝓕 : 𝓕 ∈ φ_ρ(X, Y)^(<ω) and 𝓕 ≺_{φ_ρ(X,Y)}ρ(x)}. Therefore, (φ_ρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset.

(2) ⇒ (3) For any ρ(x), ρ(y) ∈ φ_ρ(X, Y) with ρ(x) ⊈ ρ(y). By (2), there exists 𝓕 ∈ φ_ρ(X, Y)^(<ω) such that 𝓕 ≺_{φ_ρ(X,Y)}ρ(x) and ρ(y) ∉ ↑_{φ_ρ(X,Y)} 𝓕. By the definition of ≺, we have that ρ(x) ∈ int_{υ((φ_ρ(X,Y),⊆))} ↑_{φ_ρ(X,Y)} 𝓕 ⊆ ↑_{φ_ρ(X,Y)} 𝓕 ⊆ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)}ρ(y). Let 𝓤 = int_{υ((φ_ρ(X,Y),⊆))} ↑_{φ_ρ(X,Y)} 𝓕 and 𝓥 = φ_ρ(X, Y) ∖ ↑_{φ_ρ(X,Y)} 𝓕. Then ρ(x) ∈ 𝓤 ∈ υ((φ_ρ(X, Y), ⊆)), ρ(y) ∈ 𝓥 ∈ ω((φ_ρ(X, Y), ⊆)) and 𝓤 ∩ 𝓥 = ∅. Hence, the interval topology on (φ_ρ(X, Y), ⊆) is split T₂;

(3) ⇒ (1) For any (x, y) ∈ ρ, let N_y be the greatest element of φ_y (if φ_y = ∅, let N_y = ∅). Then ρ(x) ⊈ N_y. Since the interval topology on (φ_ρ(X, Y), ⊆) is split T₂, there exist {ρ(x₁), ρ(x₂), …, ρ(x_m)} ∈ φ_ρ(X, Y)^(<ω) and {ρ(u₁), ρ(u₂), …, ρ(u_n)}^(<ω) such that ρ(x) ∈ φ_ρ(X, Y) ∖ ↓_{φ_ρ(X,Y)} {ρ(x₁), ρ(x₂), …, ρ(x_m)}, N_y ∈ φ_ρ(X, Y) ∖ ↑_{φ_ρ(X,Y)} {ρ(u₁), ρ(u₂), …, ρ(u_n)} and ↓_{φ_ρ(X,Y)} {ρ(x₁), ρ(x₂) …, ρ(x_m)} ⋃ ↑_{φ_ρ(X,Y)} {ρ(u₁), ρ(u₂), …, ρ(u_n)} = φ_ρ(X, Y).

Since ρ(x) ⊈ ρ(x_j) (j = 1, 2, …, m), choose v_j ∈ ρ(x) and v_j ∉ ρ(x_j). On the other side, ρ(u_i) ⊈ N_y (i = 1, 2, …, n). By the definition of N_y, y ∈ ρ(u_i) (i = 1, 2, …, n). Thus {u₁, u₂, …, u_n} and {v₁, v₂, …, v_m} satisfy the condition (a) of Definition 3.2. ∀ s ∈ X, {t₁, t₂, …, t_n} ⊆ Y, if (u_i, t_i) ∈ ρ (i = 1, 2, …, n), (s, v_j) ∈ ρ (j = 1, 2, …, m), we have ρ(s) ⊈ ρ(x_j) for any j ∈ {1, 2, …, m}. Thus ρ(s) ∈ ↑_{φ_ρ(X,Y)} {ρ(u₁), ρ(u₂), …, ρ(u_n)}, i.e., there is a k ∈ {1, 2, …, n} such that ρ(u_k) ⊆ ρ(s). Notice that t_k ∈ ρ(u_k), thus t_k ∈ ρ(s), i.e., (s, t_k) ∈ ρ. Hence ρ is w-generalized finitely regular.

(4) ⇒ (1) By Theorem 2.4, ρ is generalized finitely regular. Hence (1) holds.

(2) ⇒ (4) For any ρ(A) ∈ (Φ_ρ(X, Y), ⊆), let ρ(B) ∈ Φ_ρ(X, Y) with ρ(A) ⊈ ρ(B). Then there is a y ∈ ρ(A) such that y ∉ ρ(B). Choose x ∈ A with y ∈ ρ(x). Let N_y be the greatest element of φ_y. Then ρ(x) ⊈ N_y and ρ(B) ⊆ N_y. Since (φ_ρ(X, Y), ⊆) is weak quasi-hypercontinuous, there exists 𝓕 ∈ φ_ρ(X, Y)^(<ω) such that 𝓕 ≺_{φ_ρ(X,Y)}ρ(x) with N_y ∉ ↑_{φ_ρ(X,Y)} 𝓕. By Lemma 3.10 and ρ(B) ⊆ N_y, 𝓕 ≺_{Φ_ρ(X,Y)}ρ(x) ⊆ ρ(A) and ρ(B) ∉ ↑_{Φ_ρ(X,Y)} 𝓕. Hence (Φ_ρ(X, Y), ⊆) is a quasi-hypercontinuous lattice.□

For any poset P, let ρ = ≰ on P. Then δ(P) is order isomorphism to (Φ_≰(P), ⊆) (define δ(P) → (Φ_≰(P), ⊆) by A^δ ↦ P ∖ A^↑) (see [24]). Furthermore, we can check that P is order isomorphism to (φ_≰(P), ⊆). In deed, define f : P → (φ_≰(P), ⊆) by f(x) = P ∖ ↑ x and g : (φ_≰(P), ⊆) → P by g(ρ(x)) = x. One can easily check that f, g are order preserving and f ∘ g = id_{(φ_≰(P),⊆)}, g ∘ f = id_P. Therefore, using Corollary 3.6 and Theorem 3.11, we have the following.

Theorem 3.12

Let P be a poset. Consider the following conditions.

P is a weak quasi-hypercontinuous poset;
the relation ≰ is w-generalized finitely regular;
for any x, y ∈ P with x ≰ y, there are {u₁, u₂, …, u_n}, {v₁, v₂, …, v_m} ∈ P^(<ω) such that
1. u_i ≰ y, x ≰ v_j (i = 1, 2, …, n; j = 1, 2, …, m), and
2. ∀z ∈ P, ∃k ∈ {1, 2, …, n} such that u_k ≤ z or ∃l ∈ {1, 2, …, m} such that z ≤ v_l;
for any x, y ∈ P with x ≰ y, there exist F, G ∈ P^(<ω) such that
1. x ∉ ↓ G, y ∉ ↑ F, and
2. ↓ G ⋃ ↑ F = P;
θ(P) is split T₂;
there is a w-generalized finitely regular relation ρ : X ⇀ Y satisfying property M such that P ≅ (φ_ρ(X, Y), ⊆);
there is a w-generalized finitely regular relation ρ : X ⇀ X satisfying property M such that P ≅ (φ_ρ(X), ⊆);
(δ(P), ⊆) is a quasi-hypercontinuous lattice.

Then (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (6) ⇔ (7)⇐ (8). If P is an S-poset, then (1) – (8) are equivalent.

Proof

(1) ⇒ (2) Let ρ = ≰ on P. By Theorem 3.11 and P ≅ (φ_≰(P), ⊆).

(2) ⇒ (3) Let x, y ∈ P with x ≰ y. By the definition of w-generalized finitely regular, there are {u₁, u₂, …, u_n}, {v₁, v₂, …, v_m} ∈ P^(<ω) such that

u_i ≰ y, x ≰ v_j (i = 1, 2, …, n; j = 1, 2, …, m), and
∀s ∈ X, {t₁, t₂, …, t_n} ⊆ Y, if u_i ≰ t_i (i = 1, 2, …, n), s ≰ v_j (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that s ≰ t_k.

For any z ∈ P, let s = z and t_i = z (i = 1, 2, …, n). Then by (ii), ∃k ∈ {1, 2, …, n} such that u_k ≤ z or ∃l ∈ {1, 2, …, m} such that z ≤ v_l.

(3) ⇒ (1) Let x, y ∈ P with x ≰ y. By (3), there are {u₁, u₂, …, u_n}, {v₁, v₂, …, v_m} ∈ P^(<ω) such that

u_i ≰ y, x ≰ v_j (i = 1, 2, …, n; j = 1, 2, …, m), and
∀z ∈ P, ∃k ∈ {1, 2, …, n} such that u_k ≤ z or ∃l ∈ {1, 2, …, m} such that z ≤ v_l.

Let F = {u₁, u₂, …, u_n}. Then we have y ∉ ↑ F and x ∈ P ∖ ↓ {v₁, v₂, …, v_m}. Now we show that P ∖ ↓ {v₁, v₂, …, v_m} ⊆ ↑ F. For any z ∈ P ∖ ↓ {v₁, v₂, …, v_m}, z ≰ v_l(l = 1, 2, …, m), by (ii), there is a k ∈ {1, 2, …, n} such that u_k ≤ z. Thus z ∈ ↑ F, and thus F ≺ x. Therefore P is a weak quasi-hypercontinuous poset.

(3) ⇔ (4) See [24].

(4) ⇒ (5) Let x, y ∈ P with x ≰ y. By (4) there exist F, G ∈ P^(<ω) such that

x ∉ ↓ G, y ∉ ↑ F;
↓ G ⋃ ↑ F = P.

Let U = P ∖ ↓ G, V = P ∖ ↑ F. Then x ∈ U ∈ υ(P), y ∈ V ∈ ω(P) and U ∩ V = ∅, thus θ(P) is split T₂.

(5) ⇒ (4) Let x, y ∈ P with x ≰ y. By (5), there are U ∈ υ(P), V ∈ ω(P) such that x ∈ U, y ∈ V with U ∩ V = ∅. Thus there exist F, G ∈ P^(<ω) such that x ∈ P ∖ ↓ G ⊆ U and y ∈ P ∖ ↑ F ⊆ V. It is easy to see that F, G satisfy the conditions (a) and (b) of (4).

(2) ⇒ (6) Let X = Y = P and ρ = ≰. By Corollary 3.6.

(6) ⇒ (7) Obviously.

(7) ⇒ (1) Follows from Theorem 3.11.

(8) ⇒ (1) By P ≅ (φ_≰(P), ⊆), δ(P) ≅ (Φ_≰(P), ⊆) and Theorem 3.11.

(1) ⇒ (8) Let P be an S-poset. Since P ≅ (φ_≰(P), ⊆) and δ(P) ≅ (Φ_≰(P), ⊆), by Theorem 3.11, the condition (8) holds.□

Corollary 3.13

Let P be a poset. Then P is weak quasi-hypercontinuous if and only if P^op is weak quasi-hypercontinuous.

Corollary 3.14

Let P be a sup-semilattice. Then the following two conditions are equivalent:

P is a quasi-hypercontinuous poset;
(δ(P), ⊆) is a quasi-hypercontinuous lattice.

Follows from Corollary 3.14, we establish a necessary and sufficient condition for a quasi-hypercontinuous poset to have a normal completion which is a quasi-hypercontinuous lattice, that is, P only need to be a sup-semilattice. This condition is weaker than that appears in Theorem 5.2 of [29].

Acknowledgements

The authors would like to thank the referees for their very careful reading of the manuscript and valuable comments. This research is supported by the National Natural Science Foundation of China (No. 11661057, 11661040), the Ganpo 555 project for leading talents of Jiangxi Province and the Natural Science Foundation of Jiangxi Province (No. 20192ACBL20045), the Foundation of Education Department of Jiangxi Province (No. GJJ160660).

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Received: 2019-01-23

Accepted: 2019-11-10

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0136

Keywords for this article

split T2; T2 property; weak generalized finitely regular; weak quasi-hypercontinuous poset

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