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BY 4.0 license Open Access Published by De Gruyter Open Access December 31, 2019

Split Hausdorff internal topologies on posets

  • Shuzhen Luo and Xiaoquan Xu EMAIL logo
From the journal Open Mathematics

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 : XY satisfies a certain condition, ρ is weak generalized finitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(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.

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 (Φρ(X), ⊆) is completely distributive, where Φρ(X) = {ρ(A) : AX}, ρ(A) = {yX : ∃aA 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 (Φρ(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 : XY satisfies property M, ρ is weak generalized finitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T2, where φρ(X, Y) = {ρ(x): xX}. 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 xP, AP, let ↑ x = {yP : xy} and ↑ A = ⋃aAa; ↓ x and ↓ A are defined dually. A and A denote the sets of all upper and lower bounds of A, respectively. Let Aδ = (A) and δ(P) = {Aδ : AP}. To avoid ambiguities, we also denote A, A and Aδ 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) = {FX : F is nonempty and finite}.

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

Definition 2.1

Let ρ : XY, τ : YZ be two binary relations. Define

  1. τρ = {(x, z) : ∃yY, (x, y) ∈ ρ, (y, z) ∈ τ}. The relation τρ : XZ is called the composition of ρ and τ.

  2. ρ–1 = {(y, x) ∈ Y × X : (x, y) ∈ ρ}.

  3. ρc = X × Yρ.

  4. ρ(A) = {yY : ∃xA 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.

  5. Φρ(X, Y) = {ρ(A) : AX}.

  6. φρ(X, Y) = {ρ(x) : xX}.

  7. φy = {ρ(u) ∈ φρ(X, Y) : yρ(u)}.

Clearly, φρ(X, Y) ⊆ Φρ(X, Y), and (Φρ(X, Y), ⊆) is a complete lattice in which the join operation ∨ is the set union operator ∪. But in general (φρ(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 (φρ(X, Y), ⊆).

Definition 2.2

[12] Let P be a poset and xP, AP.

  1. Define a relationon P by AP x iff xintυ(P)A. Without causing confusion, we write Ax for short.

  2. P is called quasi-hypercontinuous if for all xP, ↑ x = ⋂ {↑ F : F is finite and Fx} and {FP(<ω) : Fx} 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 xL, and Uυ(L) with xU, there exists FL(<ω) such that x ∈ intυ(L)F ⊆ ↑ FU.

Definition 2.3

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

  1. (ui, y) ∈ ρ, (x, vj) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m), and

  2. ∀ {s1, s2, …, sm} ∈ X(<ω), {t1, t2, …, tn} ∈ Y(<ω), if (ui, ti) ∈ ρ (i = 1, 2, …, n), (sj, vj) ∈ ρ (j = 1, 2, …, m), then ∃ (l, k) ∈ {1, 2, …, m} × {1, 2, …, n} such that (sl, tk) ∈ ρ.

Theorem 2.4

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

  1. ρ is generalized finitely regular;

  2. (Φρ(X, Y), ⊆) is a quasi-hypercontinuous lattice.

Definition 2.5

[12] Let τ and δ be two topologies on a poset P. α = τδ is called split T2 or split Hausdorff about τ and δ, if for any x, y with xy, there exists (U, V) ∈ τ × δ such that xU, yV with UV = ∅. 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, ifx = ⋂ {↑ F : FP(<ω), Fx} for all xP.

In contrast to quasi-hypercontinuous posets, a weak quasi-hypercontinuous poset need not be the case that the set {FP(<ω) : Fx} 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 : XY is called weak generalized finitely regular, w-generalized finitely regular for short, if for any (x, y) ∈ ρ, there are {u1, u2, …, un} ∈ X(<ω) and {v1, v2, …, vm} ∈ Y(<ω) such that

  1. (ui, y) ∈ ρ, (x, vj) ∈ ρ(i = 1, 2, …, n; j = 1, 2, …, m), and

  2. sX, {t1, t2, …, tn} ⊆ Y, if (ui, ti) ∈ ρ (i = 1, 2, …, n), (s, vj) ∈ ρ (j = 1, 2, …, m), then there is a k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ.

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

Proposition 3.3

For a binary relation ρ : XY, the following conditions are equivalent:

  1. ρ is w-generalized finitely regular;

  2. ∀(x, y) ∈ ρ, ∃(U, V) ∈ X(<ω) × Y(<ω) such that

    1. Uρ–1(y), Vρ(x);

    2. ∀(s, T) ∈ X × Y(<ω), if Uρ–1(T) and Vρ(s), then Tρ(s) ≠ ∅.

Proof

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

  1. (ui, y) ∈ ρ, (x, vj) ∈ ρ(i = 1, 2, …, n; j = 1, 2, …, m), and

  2. sX, {t1, t2, …, tn} ⊆ Y, if (ui, ti) ∈ ρ (i = 1, 2, …, n), (s, vj) ∈ ρ (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ.

Let U = {u1, u2, …, un}, V = {v1, v2, …, vm}. Then (U, V) ∈ X(<ω) × Y(<ω). By the condition (a), we have that Uρ–1(y), Vρ(x), i.e., the condition (i) in (2) is satisfied. Now we check the condition (ii) in (2). ∀(s, T) ∈ X × Y(<ω), if Uρ–1(T) and Vρ(s), then ∀ i ∈ {1, 2, …, n}, ∃tiT such that (ui, ti) ∈ ρ, and ∀ j ∈ {1, 2, …, m}, (s, vj) ∈ ρ, by the condition (b), ∃k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ. Thus Tρ(s) ≠ ∅.

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

  1. Uρ–1(y), Vρ(x), and

  2. ∀(s, T) ∈ X × Y(<ω), if Uρ–1(T) and Vρ(s), then Tρ(s) ≠ ∅.

Let U = {u1, u2, …, un}, V = {v1, v2, …, vm}. Then by condition (i), we have that (ui, y) ∈ ρ, (x, vj) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m). For any sX, {t1, t2, …, tn} ⊆ Y, if (ui, ti) ∈ ρ (i = 1, 2, …, n), (s, vj) ∈ ρ (j = 1, 2, …, m), let T = {t1, t2, …, tn}. Then Uρ–1(T) and Vρ(s). By the condition (ii), Tρ(s) ≠ ∅, i.e., ∃k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ. Thus (1) holds.□

Definition 3.4

Let ρ : XY be a binary relation. We call ρ satisfies property M if for any yY, φy = ∅ or φy has the greatest element, where φy = {ρ(u) ∈ φρ(X, Y) : yρ(u)}.

Example 3.5

  1. Let E be a binary relation on a set X with reflexive and transitive. Then the relation Ec = X2E satisfies property M.

    In fact, for any yX, since E is reflexive, yEc(y). Thus φy ≠ ∅. Let uX with yEc(u), i.e., (u, y) ∈ E. Suppose that Ec(u) ⊈ Ec(y), then there is a tEc(u) such that tEc(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 Ec(y) is the greatest element of φy. Hence, the relation Ec satisfies property M.

  2. Let X be a set and Y = {y}. Define a function f : XY by f(x) = y for any xX. Then f satisfies property M, since φy = ∅ for any yY.

  3. Let X, Y be two sets and g : XY a injective function. If |X| > 2, then g is not satisfy property M, since for any x1, x2X, 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 relationon P satisfies property M.

Lemma 3.7

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

Proof

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

1 η is order preserving. Let {ρ(x):xA}φρ(X,Y)δ {ρ(y):yB}φρ(X,Y)δ . Then {ρ(y):yB}φρ(X,Y) {ρ(x):xA}φρ(X,Y) . Now we have to show that ρ(A) ⊆ ρ(B). For any wρ(A), there is a xwA such that wρ(xw). Since ρ satisfies property M, let Nw be the greatest element of φw (if φw = ∅, let Nw = ∅). Then ρ(xw) ⊈ Nw. Thus Nw {ρ(x):xA}φρ(X,Y) . Note that {ρ(y):yB}φρ(X,Y) {ρ(x):xA}φρ(X,Y) , we have Nw {ρ(y):xB}φρ(X,Y) , it follows from that there is a bB such that ρ(b) ⊈ Nw. By the definition of Nw, wρ(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):yB}φρ(X,Y) {ρ(x):xA}φρ(X,Y) . Suppose not, there is a ρ(w) ∈ {ρ(y):yB}φρ(X,Y) such that ρ(w) ∉ {ρ(x):xA}φρ(X,Y) . Thus for any yB, ρ(y) ⊆ ρ(w) and ρ(x0) ⊈ ρ(w) for some x0A, it follows that there is a z0ρ(x0) with z0ρ(w). Since ρ(x0) ⊆ ρ(A) ⊆ ρ(B), there exists y0B such that z0ρ(y0). Note that ρ(y) ⊆ ρ(w) for any yB. Thus z0ρ(w), which contradicts z0ρ(w). Thus {ρ(y):yB}φρ(X,Y) {ρ(x):xA}φρ(X,Y) . Therefore, ψ(ρ(A)) ⊆ ψ(ρ(B)).

Obviously, ηψ = id(Φρ(X,Y),⊆) and ψη = idδ((φρ(X,Y),⊆)). All there show that δ((φρ(X, Y), ⊆)) ≅ (Φρ(X, Y), ⊆).□

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

Definition 3.8

[24] A poset P is called S-poset, if for any F, GP(<ω) ∖ {∅}, FG, there exists uP such that F ⊆ ↓ uG.

Lemma 3.9

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

Lemma 3.10

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

  1. 𝓕 ≺Φρ(X,Y) ρ(x);

  2. 𝓕 ≺φρ(X,Y)ρ(x).

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

Proof

(1) ⇒ (2) Let 𝓕 ≺Φρ(X,Y)ρ(x). Then there exist ρ(A1), ρ(A2), …, ρ(Am) ⊆ Φρ(X, Y) such that ρ(x) ∈ Φρ(X, Y) ∖ ↓Φρ(X,Y) {ρ(A1), ρ(A2), …, ρ(Am)} ⊆ ↑Φρ(X,Y) 𝓕. Thus for any j ∈ {1, 2, …, m}, ρ(x) ⊈ ρ(Aj), and thus there is a zjρ(x) with zjρ(Aj). Obviously, φzj ≠ ∅. Let Nj be the greatest of φzj. Then ρ(x) ⊈ Nj for any j ∈ {1, 2, …, m}, and thus ρ(x) ∈ φρ(X) ∖ ↓φρ(X){N1, N2, …, Nj}. Now we show that φρ(X, Y) ∖ ↓φρ(X,Y){N1, N2, …, Nj} ⊆ ↑φρ(X,Y) 𝓕. Let ρ(w) ∈ φρ(X, Y) ∖ ↓φρ(X,Y){N1, N2, …, Nj}. Then for any j ∈ {1, 2, …, m}, ρ(w) ⊈ Nj. By the definition of Nj, we have zjρ(w), and thus ρ(w) ⊈ρ(Aj) (since zjρ(Aj)). Hence ρ(w) ∈ Φρ(X, Y) ∖ ↓Φρ(X,Y) {ρ(A1), ρ(A2), …, ρ(Am)}, it follows that ρ(w) ∈ ↑φρ(X,Y) 𝓕. Hence 𝓕 ≺φρ(X,Y)ρ(x).

(2) ⇒ (1) Suppose that 𝓕 ≺φρ(X,Y)ρ(x), then there exists {ρ(y1), ρ(y2), …, ρ(ym)} ⊆ φρ(X, Y) such that ρ(x) ∈ φρ(X, Y) ∖ ↓φρ(X,Y) {ρ(y1), ρ(y2), …, ρ(ym)} ⊆ ↑φρ(X,Y) 𝓕. Now we have to show that ρ(x) ∈ Φρ(X, Y) ∖ ↓Φρ(X,Y) {ρ(y1), ρ(y2), …, ρ(ym)} ⊆ ↑Φρ(X,Y) 𝓕. Obviously, ρ(x) ∈ Φρ(X, Y) ∖ ↓Φρ(X,Y) {ρ(y1), ρ(y2), …, ρ(ym)}. Assume that there is a ρ(A) ∈ Φρ(X, Y) ∖ ↓Φρ(X,Y) {ρ(y1), ρ(y2), …, ρ(ym)} such that ρ(A) ∉ ↑Φρ(X,Y) 𝓕. Let 𝓕 = {ρ(u1), ρ(u2), …, ρ(un)}. Then ρ(A) ⊈ ρ(yj)(j = 1, 2, …, m) and ρ(ui) ⊈ ρ(A)(i = 1, 2, …, n). Thus there exist sjA and vjρ(sj) such that vjρ(yj) (j = 1, 2, …, m), and tiρ(ui) with tiρ(A)(i = 1, 2 …, n). We can conclude that there exist k0 ∈ {1, 2, …, m} and l0 ∈ {1, 2, …, n} such that tl0ρ(sk0). If not, then for any k ∈ {1, 2, …, m} and l ∈ {1, 2, …, n}, tlρ(sk). Let Nl be the greatest element of φtl. Then ρ(sk) ⊆ Nl, so for any k ∈ {1, 2, …, m}, ρ(sk) is a lower bound of {N1, N2, …, Nn}. Since φρ(X, Y) is an S-poset, there exists sX such that {ρ(s1), ρ(s2), …, ρ(sm)} ⊆ ↓ ρ(s) ⊆ {N1, N2, …, Nn}. Thus, vjρ(s) and ρ(s) ⊈ ρ(yj)(j = 1, 2, …, m), that is ρ(s) ∈ φρ(X, Y) ∖ ↓φρ(X,Y) {ρ(y1), ρ(y2), …, ρ(yn)}, so ρ(s) ∈ ↑φρ(X,Y) 𝓕. Thus there is a l ∈ {1, 2, …, n} such that ρ(ul) ⊆ ρ(s). Notice that tlρ(ul), we have tlρ(s). On the other side, since ρ(s) ∈ {N1, N2, …, Nn}, ρ(s) ⊆ Nl. By the definition of Nl, tlρ(s), a contradiction. Therefore, there exist k0 ∈ {1, 2, …, m} and l0 ∈ {1, 2, …, n} such that tl0ρ(sk0). Since ρ(sko) ⊆ ρ(A), tl0ρ(A), which contradicts tlρ(A) for any l ∈ {1, 2 …, n}. Hence 𝓕 ≺Φρ(X,Y)ρ(x).□

Theorem 3.11

For a binary relation ρ : XY with property M, consider the following conditions:

  1. ρ is w-generalized finitely regular;

  2. (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset;

  3. the interval topology on (φρ(X, Y), ⊆) is split T2;

  4. (Φρ(X, Y), ⊆) is a quasi-hypercontinuous lattice.

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

Proof

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

  1. (ui, y) ∈ ρ, (x, vj) ∈ ρ (i = 1, 2, …, n; j = 1, 2, …, m), and

  2. sX, T = {t1, t2, …, tn} ⊆ Y, if (ui, ti) ∈ ρ (i = 1, 2, …, n), (s, vj) ∈ ρ (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ.

Let 𝓕 = {ρ(u1), ρ(u2), …ρ(un)}. Then 𝓕 ∈ φρ(X, Y)(<ω) and ρ(ui) ⊈ ρ(u) (i = 1, 2, …, n), thus ρ(u) ∉ 𝓕. Let Nj be the greatest element of φvj (if φvj = ∅, let Nj = ∅). Then ρ(x) ∈ φρ(X, Y) ∖ ↓φρ(X,Y){N1, N2, …, Nm} since vjρ(x) (j = 1, 2, … m). For any ρ(s) ∈ φρ(X, Y) ∖ ↓φρ(X,Y){N1, N2, …, Nm}, ρ(s) ⊈ Nj(j = 1, 2, …, m). By the definition of Nj, vjρ(s). If ρ(s) ∉ ↑φρ(X,Y) 𝓕, then for any i ∈ {1, 2, …, n}, ρ(ui) ⊈ ρ(s), so there is a tiρ(ui) with tiρ(s). By the condition (b), there is a k ∈ {1, 2, …, m} such that (s, tk) ∈ ρ, i.e., tkρ(s), which contradicts tiρ(s) for any i ∈ {1, 2, …, n}. Thus ρ(s) ∈ ↑φρ(X,Y) 𝓕.

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

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

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

Since ρ(x) ⊈ ρ(xj) (j = 1, 2, …, m), choose vjρ(x) and vjρ(xj). On the other side, ρ(ui) ⊈ Ny (i = 1, 2, …, n). By the definition of Ny, yρ(ui) (i = 1, 2, …, n). Thus {u1, u2, …, un} and {v1, v2, …, vm} satisfy the condition (a) of Definition 3.2. ∀ sX, {t1, t2, …, tn} ⊆ Y, if (ui, ti) ∈ ρ (i = 1, 2, …, n), (s, vj) ∈ ρ (j = 1, 2, …, m), we have ρ(s) ⊈ ρ(xj) for any j ∈ {1, 2, …, m}. Thus ρ(s) ∈ ↑φρ(X,Y) {ρ(u1), ρ(u2), …, ρ(un)}, i.e., there is a k ∈ {1, 2, …, n} such that ρ(uk) ⊆ ρ(s). Notice that tkρ(uk), thus tkρ(s), i.e., (s, tk) ∈ 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) ∈ (Φρ(X, Y), ⊆), let ρ(B) ∈ Φρ(X, Y) with ρ(A) ⊈ ρ(B). Then there is a yρ(A) such that yρ(B). Choose xA with yρ(x). Let Ny be the greatest element of φy. Then ρ(x) ⊈ Ny and ρ(B) ⊆ Ny. Since (φρ(X, Y), ⊆) is weak quasi-hypercontinuous, there exists 𝓕 ∈ φρ(X, Y)(<ω) such that 𝓕 ≺φρ(X,Y)ρ(x) with Ny ∉ ↑φρ(X,Y) 𝓕. By Lemma 3.10 and ρ(B) ⊆ Ny, 𝓕 ≺Φρ(X,Y)ρ(x) ⊆ ρ(A) and ρ(B) ∉ ↑Φρ(X,Y) 𝓕. Hence (Φρ(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 AδPA) (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 fg = id(φ(P),⊆), gf = idP. 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.

  1. P is a weak quasi-hypercontinuous poset;

  2. the relationis w-generalized finitely regular;

  3. for any x, yP with xy, there are {u1, u2, …, un}, {v1, v2, …, vm} ∈ P(<ω) such that

    1. uiy, xvj (i = 1, 2, …, n; j = 1, 2, …, m), and

    2. zP, ∃k ∈ {1, 2, …, n} such that ukz orl ∈ {1, 2, …, m} such that zvl;

  4. for any x, yP with xy, there exist F, GP(<ω) such that

    1. x ∉ ↓ G, y ∉ ↑ F, and

    2. G ⋃ ↑ F = P;

  5. θ(P) is split T2;

  6. there is a w-generalized finitely regular relation ρ : XY satisfying property M such that P ≅ (φρ(X, Y), ⊆);

  7. there is a w-generalized finitely regular relation ρ : XX satisfying property M such that P ≅ (φρ(X), ⊆);

  8. (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, yP with xy. By the definition of w-generalized finitely regular, there are {u1, u2, …, un}, {v1, v2, …, vm} ∈ P(<ω) such that

  1. uiy, xvj (i = 1, 2, …, n; j = 1, 2, …, m), and

  2. sX, {t1, t2, …, tn} ⊆ Y, if uiti (i = 1, 2, …, n), svj (j = 1, 2, …, m), then ∃k ∈ {1, 2, …, m} such that stk.

For any zP, let s = z and ti = z (i = 1, 2, …, n). Then by (ii), ∃k ∈ {1, 2, …, n} such that ukz or ∃l ∈ {1, 2, …, m} such that zvl.

(3) ⇒ (1) Let x, yP with xy. By (3), there are {u1, u2, …, un}, {v1, v2, …, vm} ∈ P(<ω) such that

  1. uiy, xvj (i = 1, 2, …, n; j = 1, 2, …, m), and

  2. zP, ∃k ∈ {1, 2, …, n} such that ukz or ∃l ∈ {1, 2, …, m} such that zvl.

Let F = {u1, u2, …, un}. Then we have y ∉ ↑ F and xP ∖ ↓ {v1, v2, …, vm}. Now we show that P ∖ ↓ {v1, v2, …, vm} ⊆ ↑ F. For any zP ∖ ↓ {v1, v2, …, vm}, zvl(l = 1, 2, …, m), by (ii), there is a k ∈ {1, 2, …, n} such that ukz. Thus z ∈ ↑ F, and thus Fx. Therefore P is a weak quasi-hypercontinuous poset.

(3) ⇔ (4) See [24].

(4) ⇒ (5) Let x, yP with xy. By (4) there exist F, GP(<ω) such that

  1. x ∉ ↓ G, y ∉ ↑ F;

  2. G ⋃ ↑ F = P.

Let U = P ∖ ↓ G, V = P ∖ ↑ F. Then xUυ(P), yVω(P) and UV = ∅, thus θ(P) is split T2.

(5) ⇒ (4) Let x, yP with xy. By (5), there are Uυ(P), Vω(P) such that xU, yV with UV = ∅. Thus there exist F, GP(<ω) such that xP ∖ ↓ GU and yP ∖ ↑ FV. 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 Pop is weak quasi-hypercontinuous.

Corollary 3.14

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

  1. P is a quasi-hypercontinuous poset;

  2. (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).

References

[1] Erne M., Separation axioms for interval topologies, Proc. Amer. Math. Soc., 1980, 79, 185–190.10.1090/S0002-9939-1980-0565335-8Search in Google Scholar

[2] Frink O., Topology in lattices, Trans. Amer. Math. Soc., 1942, 51, 569–582.10.1090/S0002-9947-1942-0006496-XSearch in Google Scholar

[3] Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D., A Compendium of Continuous Lattices, Springer Verlag, 1980.10.1007/978-3-642-67678-9Search in Google Scholar

[4] Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.10.1017/CBO9780511542725Search in Google Scholar

[5] Gierz G., Lawson J.D., Generalized continuous and hypercontinuous lattices, Rocky Mountain J. Math., 1981, 11, 271–296.10.1216/RMJ-1981-11-2-271Search in Google Scholar

[6] Northam F.S., The interval topology of a lattice, Proc. Amer. Math. Soc., 1953, 4, 824–827.10.1090/S0002-9939-1953-0057534-2Search in Google Scholar

[7] Kogan S.A., Solution of three problems in lattice theory, Uspehi Math. Nauk., 1956, 11, 185–190.Search in Google Scholar

[8] Wolk E.S., Topologies on a partially ordered set, Proc. Amer. Math. Soc., 1958, 9, 524–529.10.1090/S0002-9939-1958-0096596-8Search in Google Scholar

[9] Matsushima Y., Hausdorff interval topology on a partially ordered set, Proc. Amer. Math. Soc., 1960, 11, 233–235.10.1090/S0002-9939-1960-0111705-9Search in Google Scholar

[10] Kolibiar M., Bemerkungen über Intervalltopologie in halbgeordneten Mengen, in: General topology and its Relations to Modern Analysis and Algebra (Proc. Sympos, Prague, 1961), Academic Press, New York; Publ. Hause Czech. Acad. Sci., Prague, 1962, 252–253.Search in Google Scholar

[11] Birkhoff G., Lattice Theory, vol. 25 of AMS Colloquium Publications, revised edition, RhodeIsland, 1967.Search in Google Scholar

[12] Xu X.Q., Relational representations of complete lattices and their applications, PhD thesis, Sichuan University, 2004.10.1007/978-94-017-1291-0_3Search in Google Scholar

[13] Xu X.Q., Liu Y.M., Regular relations and strictly completely regular ordered spaces, Topology Appl., 2004, 135, 1–12.10.1016/S0166-8641(03)00108-1Search in Google Scholar

[14] Xu X.Q., Liu Y.M., Regular relations and monotone normal ordered spaces, Chinese Ann. Math., 2004, 25B, 157–164.10.1142/S0252959904000160Search in Google Scholar

[15] Xu X.Q., Luo M.K., Regular relations and normality of topologies, Semigroup Forum, 2006, 72, 477–480.10.1007/s00233-004-0179-0Search in Google Scholar

[16] Xu X.Q., Liu Y.M., Regular relations and completely regular spaces, Chinese Ann. Math., 2008, 29A, 819–828.Search in Google Scholar

[17] Xu X.Q., Luo M.K., Regular relations and normal spaces, Acta Math. Sinica (Chin. Ser.), 2009, 52, 393–402.Search in Google Scholar

[18] Zareckiǐ A., The semigroup of binary relations, Mat.Sbornik, 1963, 61, 291–305.Search in Google Scholar

[19] Markowsky G., Idempotents and product representations with applications to the semigroup of binary relations, Semigroup Forum, 1972, 5, 95–119.10.1007/BF02572880Search in Google Scholar

[20] Schein B.M., Regular elements of the semigroup of all binary relations, Semigroup Forum, 1976, 13, 95–102.10.1007/BF02194925Search in Google Scholar

[21] Bandelt H.J., Regularity and complete distributivity, Semigroup Forum, 1980, 19, 123–126.10.1007/BF02572509Search in Google Scholar

[22] Bandelt H.J., On regularity classes of binary relations, In: Universal Algebra and Applications, Banach Center Publications, PWN, Warsaw, 1982, 9, 329–333.10.4064/-9-1-329-333Search in Google Scholar

[23] Xu X.Q., Liu Y.M., Relational representations of hypercontinuous lattices, In: Domain Theory, Logic, and Computation, Kluwer Academic, Dordrecht, 2003, 65–74.10.1007/978-94-017-1291-0_3Search in Google Scholar

[24] Xu X.Q., Order and topology (in chinese), Science Press, Beijin, 2016.Search in Google Scholar

[25] Erné M., The Dedekind-MacNeille completion as a reflector, Order, 1991, 8, 159–173.10.1007/BF00383401Search in Google Scholar

[26] Jiang G.H., Xu L.S., Conjugative relations and applications, Semigroup Forum, 2010, 1, 85–91.10.1007/s00233-009-9185-6Search in Google Scholar

[27] Jiang G.H., Xu L.S., Cai C., Han G.W., Normal relations on sets and applications, Int. J. Contemp. Math. Sci, 2011, 6, 721–726.Search in Google Scholar

[28] Jiang G.H., Xu L.S., Dually normal relations on sets, Semigroup Forum, 2012, 85, 75–80.10.1007/s00233-011-9364-0Search in Google Scholar

[29] Xu X.Q., Zhang W.F., The upper topology and interval topology on quasi-hypercontinuous posets, Topology and its Applications, 2017, 230, 539–549.10.1016/j.topol.2017.08.048Search in Google Scholar

Received: 2019-01-23
Accepted: 2019-11-10
Published Online: 2019-12-31

© 2019 Shuzhen Luo and Xiaoquan Xu, published by De Gruyter

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

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