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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1 (Jan 2016)

Issues

A result for O2-convergence to be topological in posets

Qingguo Li
  • Corresponding author
  • College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China
  • Email:
/ Zhiwei Zou
  • College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China
  • Email:
Published Online: 2016-04-23 | DOI: https://doi.org/10.1515/math-2016-0018

Abstract

In this paper, the α waybelow relation, which is determined by O2-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for O2-convergence to be topological is obtained.

Keywords: O2-convergence; O2-doubly continuous poset; Topological, α-doubly continuous poset; α-doubly continuous poset

MSC 2010: 54A20; 06B35

1. Introduction

The O2-convergence on a poset P, which is a generalization of O-convergence, is defined as follows [14]: Let P be a poset, a net (xi)iI is said to O2-converges to xP if there exist subsets M and N of P such that

  • 1.

    sup M = inf N = x;

  • 2.

    For each aM and bN, there exists kI such that axib holds for all ik.

By saying the O2-convergence on a poset P is topological, we mean that there exists a topology 𝒯 on P such that a net (xi)iI in P O2-converges to xP if and only if it converges to x with respect to the topology 𝒯. Otherwise, we say that O2-convergence is not topological. As is pointed in [5], in general, O2-convergence is not topological. The same problem exists for O-convergence and Lim-inf-convergence, Zhao and others have done some excellent work in [68].

In [5], Zhao and Li proved that if the O2-convergence on a poset P is topological, then P is an α-doubly continuous poset. Also they give a sufficient condition: If P is an α*-doubly continuous poset, then O2-convergence is topological. Moreover, if P is a poset satisfying condition (*), O2-convergence is topological if and only if P is an α-doubly continuous poset.

But as pointed in this paper, there exists poset which is α-doubly continuous but not α*-doubly continuous. Then for such a poset, is the O2-convergence topological? In other words, can we find a sufficient and necessary condition for O2-convergence to be topological for any poset?

In this paper, the α waybelow relation is characterized just by the order on a poset. The concept of O2-doubly continuous poset is given, which is strictly stronger than α-doubly continuous and strictly weaker than α*-doubly continuous. Most important, we prove that for a poset P, the O2-convergence on it is topological if and only if P is an O2-doubly continuous poset.

2. Preliminaries

This section will briefly mention some of the important results that Zhang and Li presented in [5]. For terms which are not defined here, please refer to [9, 10] and related references.

For a poset P, a, bP, we denote [a, b] = {xP : axb}. We use M0fin M to indicate M0 is a finite subset of M.

([5]).Let P be a poset. For x, y, zP, define xα y if for every net (xi)iI in P which O2-converges to y, xxi holds eventually; zα y if for every net (xi)iI in P which O2-converges to y, xiz holds eventually.

It follows from Definition 2.1 that xα yxy and zα yzy. If P has a bottom element ⊥ (top element ⊤), then ⊥ ≪α x (⊤ ⊳α x) for each xP

([5]). A poset P is called an α-doubly continuous poset if a = sup {xP : xα a} = inf {yP: zα a} for every aP.

  • (1)

    Every finite lattice is α-doubly continuous, every chain or antichain is α-doubly continuous.

  • (2)

    If P is α-doubly continuous, then for every aP, the set {xP : xα a} and {yP : zα a} are both nonempty.

([5]). Let P be a poset. For x, y, zP, define xα* y if for every net (xi)iI in P which O2-converges to some ωP with yω, xxi holds eventually. We define the order dual relation zα*y if zα*y in Pop, where Pop denotes P endowed with the reverse order.

([5]). A poset P is called an α *-doubly continuous poset if a= sup {xP : xα*a} = inf {yP: zα*a} for every aP.

([5]).

  • (1)

    Obviously, xα* yxα y and zα* yzα y, then an α*-doubly continuous poset is α-doubly continuous.

  • (2)

    If P is an α*-doubly continuous poset, then for each xP,

x = sup{aP : ∃a′ ∈ P, aα* a′ ≪α* x} = inf{bP : ∃b′P, bα* b′α* x}.

([5]). For any poset P, if O2-convergence is topological, then P is α-doubly continuous.

([5]). If P is an α*-doubly continuous poset, then O2-convergence is topological.

That is to say, α*-doubly continuous ⟹ O2-convergence is topological⟹ α-doubly continuous.

Condition (*). Let P be a poset and x, y, zP with xα yz, then xα z. Let w, s, tP with sα tw, then sα w.

([5]). For any poset P which satisfies condition (*), O2-convergence is topological if and only if P is α-doubly continuous.

3. O2-doubly continuous posets

In this section, we give a characterization of the relations ≪α and ⊳α, then introduce the concept of O2-doubly continuous poset which is strictly stronger than α-doubly continuous and strictly weaker than α*-doubly continuous.

Let P be a poset. yα x if and only if for every subset M and N such that sup M = inf N = x, there exist M0fin M and N0fin N such that aM0bN0[a,b]y.

Dually, zα x if and only if for every subset M and N such that sup M = inf N = x, there exist M0fin M and N0fin N such that aM0bN0[a,b]z.

Sufficiency, if there exists a net (xi)iI in P which O2-converges to x, then there exist subsets M and N of P such that sup M = inf N = x, and for each pair aM and bN, xi ∈ [a, b] holds eventually. By the condition there exist M0fin M and N0fin N such that aM0bN0[a,b]y. Since M0 and N0 are finite, xiaM0bN0[a,b]y holds eventually, thus yxi holds eventually, we conclude that yα x.

Necessary, suppose yα x and sup M = inf N = x, denote Bx={aAbB[a,b]:AfinM and 𝒟 = {(r, U) ∈ (⋃𝓑x) × 𝓑x: rU}, we define a preorder " ≤ " on 𝒟 by: (r1, U1) ≤(r2, U2) ⇔ U2U1. Arbitrarily take (r1, U1) and(r2, U2) ∈ 𝒟, suppose U1=aA1bB1[a,b] and U2=aA2b  B2[a,b] where A1, A2fin M and B1, B2fin N. Let A3 = A1A2, B3 = B1B2, U3=aA3bB3[a,b], then (x, U3) ∈ 𝒟 and (r1, U1) (r2, U2) ≤ (x, U3), hence (𝒟, ≤) is directed. We construct a net (xd)d∈𝒟 = x(r, U) = r. We know sup M = inf N = x, and for each mM and nN, [m, n] ∈ 𝓑x, when (r, U) ≥ (x, [m; n], we have U ⊆ [m, n] and x(r, U) = rU ⊆[m, n], so m ≤ (xd)d∈𝒟n holds eventually. This implies that (xd)d ∈ 𝒟 O2-converges to x. Then yα x implies that yx(r, U) holds eventually, i.e., there exists (ro, U0) ∈ 𝒟 such that yx(r, U) = r for every (r, U) ≥ (ro, U0). Since (ro, U0) ∈ 𝒟, there exist M0fin M and N0fin N such that U0=aM0b  N0[a,b], for each rU0, (r, U0) ≥ (ro, U0), so yx(r, U0) = r, we conclude that U0=aM0bN0[a,b]y.

A poset P is called an O2-doubly continuous poset if it satisfies the following condition:

  • (1)

    P is α-doubly continuous, i.e. x = sup {yP : yα*x} = inf {zP : zα*x} for each xP.

  • (2)

    If yα x and zα x, then there exist finite Afin {aP : aα x} and Bfin {bP : bα x} such that yα c and zα cmAmB[m,n].

Every finite lattice is O2-doubly continuous, every chain or antichain is O2-doubly continuous.

If P is an α*-doubly continuous poset, then P is an O2-doubly continuous poset.

By Remark 2.6 we know an α*-doubly continuous poset is α-doubly continuous. In an α*-doubly continuous poset P, we denote Mx = {aP : ∃a′ ∈ P; aα* a′ ≪α* x} and Ny = {bP : ∃ b′ ∈ P : ⊳α* b′ ⊳α* x} for each xP, then sup Mx = inf Nx = x. If yα x and zα x, from Proposition 3.1 there exist finite M0fin Mx and N0fin Nx such that aM0bN0[a,b][y,z]. Denote A0 = {a′P : aM0} and B0 = {b′P : bN0}, we have aA0aB0[a,b]aM0bM0[a,b][y,z]. Obviously A0fin {aP : aα x} and B0fin {bP : bα x}, so we only need to prove yα c and zα c hold for each caA0bB0[a,b].

If there exists a net (xi)iI O2-convergence to c, for each aM0 and bN0, aα c and bα c, then xi ∈ [a, b] holds eventually. Because M0 and N0 are both finite, thus xiaM0bN0[a,b][y,z] eventually, we conclude that yα c and zα c.

An α*-doubly continuous poset implies it is O2-doubly continuous. An O2-doubly continuous poset implies it is α-doubly continuous. But conversely both are untrue – see the following examples:

Let P1 = {1} ⋃ {aj i ∈ ℕ} ⋃ {bi : i ∈ ℕ} ⋃ {ci: i ∈ ℕ}, ⋃ {bij: i, j ∈ ℕ} ⋃ {cij: i, j ∈ ℕ} wheredenotes the set of all positive integers. The orderon P1 is constructed as follows (see Fig. 1 for the Hasse diagram of P1:

  • (1)

    aiai+1; bibi+1;cici+1; for all i ∈ ℕ.

  • (2)

    aibi; aici for all i ∈ ℕ

  • (3)

    bijbi,j+1bi and cijci,j+1, ≤ ci for all i, j ∈ ℕ.

  • (4)

    1 is the greatest element in P.

It is straightforward to verify that P1 is α-doubly continuous, {xP : xα 1 = {ai : i ∈ ℕ} and {xP : xα 1} = {1}. Notice a1α 1, for any finite Afin {xP : xα 1} = {ai : i ∈ ℕ}, there exists i0 ∈ ℕ such that ai0aiA[ai,1], then bi0aiA[ai,1], but a1α bi0 does not hold, so P1 is not an O2-doubly continuous poset.

Moreover, a1α a2b2 but a1α b2 does not hold, that means P1 does not satisfy the condition(*).

Let P2 = {1}⋃{a}⋃{b1, b2, b3, ...}. The order “≤” on P2 is defined as follows: (see Fig. 1 for the Hasse diagram): a ≤ 1; bi ≤ 1 for all i ∈ ℕ; bibj , whenever ij, for all i, j ∈ ℕ.

It is obvious that P is an α-doubly continuous poset and satisfies the condition (2) in Definition 3.2, then P is an O2-doubly continuous poset. But it is not α*-doubly continuous because {xP : xα* a} = ϕ.

4. A sufficient and necessary condition for O2-convergence to be topological

In this section, we obtain the main result that O2-convergence on a poset P is topological if and only if P is an O2-doubly continuous poset.

We recall that saying the O2-convergence on a poset P is topological means that there exists a topology 𝒯 on P such that a net (xi)iI in P O2-converges to xP if and only if it converges to x with respect to the topology 𝒯. Zhao and Li proved that for any poset P, if the O2-convergence is topological, then P is α-doubly continuous in [5]. We will prove it again using another method in the following theorem because some results from the process will be used next. The following theorem is a further result of Zhao and Li’s.

For any poset P, if the O2-convergence is topological, then P is an O2-doubly continuous poset.

For a poset P, if the O2-convergence is topological, then there exists a topology 𝒯 on P such that a net (xi)iI O2-converges to xP if and only if it converges to x with respect to the topology 𝒯. For every xP, let Ix = {(W,k,r) ∈ 𝒩(x)× ℕ× P : rW}, where 𝒩(x) consists of all open sets containing x in the topology 𝒯, i.e., 𝒩(x) = {W ∈ 𝒯: xW}. Now we define the lexicographic order on the first two coordinates on Ix, this means for (W1, k1, r1), (W2, k2, r2) ∈ Ix, (W1, k1, r1) ≤. (W2, k2, r2) if and only if W2 is a proper subset of W1 or W1 = W2 and k1k2. This is a preorder. Arbitrarily take (W1, k1, r1), (W2, k2, r2) ∈ Ix, then (W1W2, max{k1, k2}, x) ≤ (W1, k1, r1) and (W1W2, max{k1, k2},x) ≤ (W2, k2, r2). Hence Ix is directed. Let xi = r for every i = (W,k,r) ∈ Ix. Then it is straightforward to verify that the net (xi)iIx converges to x with respect to the topology 𝒯. Thus (xi)iIx O2-converges to x. By the definition of O2-convergence, there exist subsets M0 and N0 such that sup M0 = inf N0 = x, and for each mM0 and nN0 there exists i0 = (W0, k0, r0) ∈ Ix such that xi = r ∈ [m, n] holds for every i = (W,k,r) ≥ i0. In particular, we have (W0, k0 + 1; w) ≥ (W0, k0 + 1, w) ≥ (w0, k0, r0) for all wW0, hence x(W0, k0+1, w) = w ∈ [m, n] for all wW0, this means W0 ⊆ [m, n]. Since W0 is related to m and n, we denoted it by Wmn.

For arbitrary mM0 and nN0, if there exists a net (xj)jJ O2-convergence to x, then (xj)jJ converges to x with respect to the topology 𝒯, this means xj is eventually in Wmn ⊆ [m, n] (Wmn is an open neighborhood of x in the topology 𝒯), thus mα x and nα x. We conclude that M0 ⊆ {yP : yα x} and N0 ⊆ {zP : zα x}, since sup M0 = inf N0 = x, we have sup {yP : yα x} = inf{zP : zα x} = x, i.e., P is an α-doubly continuous poset.

Suppose yα x and zα x, by Proposition 3.1 we know there exist M1fin M0 and N1fin N0 such that mM1nN1[m,n][y,z]. For arbitrary mM1 and nN1, there exists Wmn ∈ 𝒩(x) such that Wmn ⊆ [m, n], denote W1=mM1nN1 Wmn, then W1 ∈ 𝒩(x) and W1mM1nN1[m,n][y,z] because M1 and N1 are both finite. We denote Ax = {aP : aα x} and Bx = {bP : bα x}. Let Dx={aA0bB0[a,b]:A0finAx and B0fin Bx}, 𝒟 = {r, U) ∈ (∪ Dx) × Dx : rU}. Define “≤” on 𝒟 by: (r1, U1) ≤ (r2, U2) ⇔ U2U1, then “≤” is a preorder on 𝒟 and 𝒟 is directed. Let x(r,U) = r for (r, U) ∈ 𝒟, we will prove (x(r,U) ∈ 𝒟 O2-convergence to x in the next paragraph.

Since P is an α-doubly continuous poset, we know sup Ax = inf Bx = x, for each aAx and bBx, [a, b] ∈ 𝒟, when (r, U) ≥ (x, [a, b]) i.e. U ⊆[a, b], x(r,U) = rU ⊆[a, b], so a ≤ (x(r,U))(r,U)∈𝒟 ≤ b holds eventually, by the definition of O2-convergence we know that (x(r,U))(r,U)∈𝒟 O2-convergence to x.

For poset P, the O2-convergence is topological, then (x(r,U))(r,U)∈𝒟 converges to x with respect to the topology 𝒯, hence x(r,U)W1 holds eventually. There exist finite subsets A1fin Ax and B1fin Bx such that when (r,U)(r0,aA1bB1[a,b]), we have x(r, U)W1. In particular for each taA1bB1[a,b],(t,aA1bB1[a,b](r0,aA1bB1[a,b]), then x(t,aA1bB1[a,b])=tW1, we conclude that aA1bB1[a,b]W1[y,z]. To prove P satisfies the condition (2) in Definition 3.2, it is sufficient to show yα c for each caA1bB1[a,b]. If there exists a net (xj)jJ O2-convergence to c, then (xj)jJ converges to c with respect to the topology 𝒯; this means xj is eventually in W1 since W1 ∈ 𝒯 and caA1bB1[a,b]W1, therefore xjW1mM1nN1[m,n][y,z] hold eventually, we conclude that yα c and zα c.

The relations among W1,mM1nN1[m,n] and aA1bB1[a,b] are revealed in Fig 2.

We review the general relation between convergence and topology. If on any set P a class 𝓛 of pairs ((xi)iI, x) is given, then 𝓞(𝓛) = {UX : whenever (xi)iI ∈ 𝓛 and xU, xiU eventually} is a topology on X. In this paper, we denote the class 𝓛 = {((xi)iI, x) : (xi)iI O2-converges to x}. Kelly gave a standard characterization for a class 𝓛 of convergent nets to be topological:

([11]). Given a poset P, the class 𝓛 is topological if and only if the following axioms are satisfied.

(Constant) If (xi)iI is a constant net with value xi = x for every iI, then ((xi)iI, x) ∈ 𝓛.

(Subnets) If ((xi)iI, x) ∈ 𝓛 and (yj)jJ is a subnet of (xi)iI, then ((yj)jJ ((yj)jJ, x) ∈ 𝓛.

(Divergence) If ((xi)iI,x) is not in 𝓛, then there exists a subnet (yj)jJ of (xi)iI which has no subnet (zk)kK so that ((zk)kK, x) belongs to 𝓛.

(Iterated Limits) If ((xi)iI, x), ∈ 𝓛, and if ((xi, j)jJ(i), xi) ∈ 𝓛 for all iI, then ((xi, f(i))i, f∈I×M,x) ∈ 𝓛, where M=iIJ(i) is a product of directed sets.

For an α-doubly continuous poset P, a net (xi)iI O2-converges to xP if and only if for any y, zP with yα x and zα x, yxiz holds eventually.

Necessary, if net (xi)iI O2-converges to xP and yα x, zα x, by the Definition of ≪α and ⊳α we know yxiz holds eventually. Conversely, just let M = {yP : yα x} and N = {zP : zα x}, then supM = infN = x. For each yM and zN, yxiz holds eventually, we have (xi)iI O2-converges to x.

For an O2-doubly continuous poset P, a net (xi)iI O2-converges to xP if and only if for any y, zP with yα x and zα x, yα xi and zα xi hold eventually.

Necessary, if net (xi)iI O2-converges to xP and yα x, zα x, by Definition 3.2 there exist finite Afin {aP : aα x} and Bfin {bP : bα x} such that yα c and zα c for each cccmAnB[m,n] Since A and B are both finite, by Lemma 4.4, xiximAnB[m,n] holds eventually, thus yα xi and zα xi hold eventually. Conversely, since yα xi and zα xi imply yxiz, by Lemma 4.4 we know (xi)iI O2-converges to xP.

For any poset P, the class 𝓛 satisfies axiom (Constant).

Suppose xi = x for every iI, just take M = N = {x}, sup M = inf N = x and xxi = xx holds for all iI. Therefore ((xi)iI, x) ∈ 𝓛.

For any poset P, the class 𝓛 satisfies axiom (Subnets).

Let ((xi)iI, x) ∈ 𝓛. There exist subsets M and N such that sup M = inf N = x, and for every mM and every nN there exists kI such that mxin for all ik. For any subnet (yj)jJ of net (xi)iI, there exists a mapping f : JI such that yj = xf(j) for every jJ, and for every iI there exists jiJ such that f(j) ≥ i for every jji. In particular, for kI, there exists jkJ such that f(j) ≥ k for all jJ such that f(j) ≤ i for every jji . In particular, for kI, there exists jkJ such that f(j) ≥ k for all jjk. Hence myj = xf(j)n for all jjk. Therefore ((yj)jJ, x) ∈ 𝓛.

For an α-doubly continuous poset P, the class 𝓛 satisfies axiom (Divergence).

Assume that ((xi)iI, x) is not in 𝓛. By lemma 4. 4, there exist y0, z0P with y0α x and z0α x such that y0xiz0 does not hold eventually. This means that for every iI we can choose a jiI such that y0xji or z0xji. Let J = {ji : iI} ⊆ I, then J is a cofinal subset of I. We now consider the subnet (xj)jJ of (xi)iI, since y0xj or z0xj for every jJ, by Lemma 4.4 again, (xj)jJ has no subnet (zk)kK such that ((zk)kK, x) belongs to 𝓛. Thus the class 𝓛 satisfies axiom (Divergence).

For an O2-doubly continuous poset P, the class 𝓛 satisfies axiom (Iterated Limits).

Let ((xi)iI, x) ∈ 𝓛 and ((xi, j)jJi, ∈ 𝓛 for every iI. For any y, zP with yα x and zα x, by Lemma 4.5 we know yα xi and zα xi hold eventually, i.e., there exists i0I such that yα xi and zα xi for all ii0. Since ((xi, j)jJi, xi) ∈ 𝓛, by Lemma 4.5 again, for every iI with ii0 there exists jiJi such that yα xi, j and zα xi, j for every jji. Define f0M=iIJ(i) as: f0(i) = ji for every ii0 and f0(i) is an arbitrarily fixed element in Ji for every ii0. Now we claim that ((xi, f(i))(i, f)∈I×M, x) ∈ 𝓛, because if (i, f) ∈ I × M and (i, f) ≥ (i0, f0) then yα xi, f(i) and zα xi, f(i), it means ((xi, f(i))(i, f)∈I×M, x) O2-converges to x by Lemma 4.4. This completes the proof.

From Propositions 4.6 –4.9 and Fact 4. 3, we have

For any poset P, if P is O2-doubly continuous, then O2-convergence is topological.

Combining Theorem 4.1 with Theorem 4.1 0, we obtain the following theorem.

Let P be a poset. Then the O2-convergence is topological if and only if P is O2-doubly continuous.

By Proposition 3.4 and Theorem 4.1 0, we have the following corollary.

([5]). If P is an α*-doubly continuous poset, then O2-convergence is topological.

  • (1)

    For the poset P1 showed in Example 3.5, the O2-convergence is not topological.

  • (2)

    In Example 3.6, the O2-convergence on the poset P2 is topological.

Acknowledgement

This work is supported by the Natural Science Foundation of China (Grant No: 11371130) and (Grant No: 2014GXNSFBA118015).

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About the article

Received: 2016-01-06

Accepted: 2016-03-09

Published Online: 2016-04-23

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0018.

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© 2016 Li and Zou, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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