𝓜𝓝-convergence and lim-inf_𝓜-convergence in partially ordered sets

Definition 1.1

([5]). LetPbe a poset andx, y, z ∈ P. We sayy≪_𝓞xif for every net (x_i)_i∈IinPwhichO-converges tox ∈ P, x_i ⩾ yholds eventually; dually, we sayz⊲_𝓞xif for every net (x_i)_i∈IinPwhichO-converges tox ∈ P, x_i ⩽ zholds eventually.

Definition 1.2

([5]). A posetPis said to be 𝓞-doubly continuous if for everyx ∈ P, the set {a ∈ P : a≪_𝓞x} is directed, the set {b ∈ P : b⊲_𝓞x} is filtered and sup{a ∈ P : a≪_𝓞x} = x = inf{b ∈ P : b⊲_𝓞x}.

Definition 1.3

([6]). LetPbe a poset andx, y, z ∈ P. We sayy≪_αxif for every net (x_i)_i∈IinPwhichO₂-converges tox ∈ P, x_i ⩾ yholds eventually; dually, we sayz⊲_αxif for every net (x_i)_i∈IinPwhichO₂-converges tox ∈ P, x_i ⩽ zholds eventually.

Definition 1.4

([7]). A posetPis said to beO₂-doubly continuous if for everyx ∈ P,

1. sup{a ∈ P : a≪_αx} = x = inf{b ∈ P : b⊲_αx};

2. for anyy, z ∈ Pwithy≪_αxandz⊲_αx, there existA ⊑ {a ∈ P : a≪_αx} andB ⊑ {b ∈ P : b⊲_αx} such thaty≪_αcandz⊲_αcfor eachc ∈ ⋂{↑a ∩ ↓b: a ∈ A & b ∈ B}.

Definition 2.1

Let (P, 𝓜,𝓝) be aPMN-space. Anet (x_i)_i∈IinPis said to 𝓜𝓝-converge to x ∈ Pif there existM ∈ 𝓜 andN ∈ 𝓝 satisfying:

1. sup M = x = inf N;

2. x_i ∈ ↑m ∩ ↓neventually for everym ∈ Mand everyn ∈ N.

In this case, we will write(xi)i∈I→MNx.

Remark 2.2

Let (P, 𝓜,𝓝) be aPMN-space.

1. If 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P), then a net(xi)i∈I→MNx ∈ Pif and only if itO-converges tox. That is to say, O-convergence is a particular case of 𝓜𝓝-convergence.

2. If 𝓜 = 𝓝 = 𝓟₀(P), then a net(xi)i∈I→MNx ∈ Pif and only if itO₂-converges tox. That is to say, O₂-convergence is a special case of 𝓜𝓝-convergence.

3. If 𝓜 = 𝓝 = 𝓛₀(P), then a net(xi)i∈I→MNx ∈ Pif and only ifx_i = xholds eventually.

4. The 𝓜𝓝-convergent point of a net (x_i)_i∈IinP, if exists, is unique.

   Indeed, suppose that(xi)i∈I→MNx1and(xi)i∈I→MNx2.Then there existA_k ∈ 𝓜 andB_k ∈ 𝓝 such that sup A_k = x_k = inf B_kanda_k ⩽ x_i ⩽ b_kholds eventually for everya_k ∈ A_kandb_k ∈ B_k (k = 1, 2). This implies that for anya₁ ∈ A₁, a₂ ∈ A₂, b₁ ∈ B₁andb₂ ∈ B₂, there existsi₀ ∈ Isuch thata₁ ⩽ x_i0 ⩽ b₂anda₂ ⩽ x_i0 ⩽ b₁. Thus we have sup A₁ = x₁ ⩽ inf B₂ = x₂and sup A₂ = x₂ ⩽ inf B₁ = x₁. Thereforex₁ = x₂.

5. For anyA ∈ 𝓜 andB ∈ 𝓝 with sup A = inf B = x ∈ P, we denoteF(A,B)x = {⋂{↑a ∩ ↓b : a ∈ A₀ & b ∈ B₀}: A₀ ⊑ A & B₀ ⊑ B}^[1]. LetD(A,B)x={(d,D)∈P×F(A,B)x:d∈D}and let the preorder ≤ onD(A,B)xbe defined by

   (∀(d1,D1),(d2,D2)∈D(A,B)x)(d1,D1)≤(d2,D2)⟺D2⊆D1.

   One can readily check that(D(A,B)x,≤)is directed. Now if we takex_(d,D) = dfor every(d,D)∈D(A,B)x,, then the net(x(d,D))(d,D)∈D(A,B)x→MNxbecause sup A = inf B = x, anda ⩽ x_(d,D) ⩽ bholds eventually for anya ∈ Aandb ∈ B.

6. Let(x(d,D))(d,D)∈D(A,B)xbe the net defined in (5) for anyA ∈ 𝓜 andB ∈ 𝓝 with sup A = inf B = x ∈ P. If(x(d,D))(d,D)∈D(A,B)xconverges top ∈ Pwith respect to some topology 𝓣 on the posetP, then for every open neighborhoodU_pofp, there existA₀ ⊑ AandB₀ ⊑ Bsuch that

   ⋂{↑a∩↓b:a∈A0&b∈B0}⊆Up.

   Indeed, suppose that(x(d,D))(d,D)∈D(A,B)x→Tp.Then for every open neighborhoodU_pofp, there exists (d₀,D₀) ∈ D(A,B)xsuch thatx_(d,D) = d ∈ U_pfor all (d, D) ≥ (d₀,D₀). Since (d, D₀) ≥ (d₀,D₀) for everyd ∈ D₀, x_(d,D) = d ∈ U_pfor everyd ∈ D₀. This showsD₀ ⊆ U_p. So, there existA₀ ⊑ AandB₀ ⊑ Bsuch that

   D0=⋂{↑a∩↓b:a∈A0&b∈B0}⊆Up.

Definition 2.3

Let (P, 𝓜,𝓝) be aPMN-space andx, y, z ∈ P.

1. We definey≪MNxif for anyA ∈ 𝓜 andB ∈ 𝓝 with sup A = x = inf B, there existA₀ ⊑ AandB₀ ⊑ Bsuch that

   ⋂{↑a∩↓b:a∈A0&b∈B0}⊆↑y.

2. Dually, we definez⊳MNxif for anyM ∈ 𝓜 andN ∈ 𝓝 with sup M = x = inf N, there existM₀ ⊑ MandN₀ ⊑ Nsuch that

   ⋂{↑m∩↓n:m∈M0&n∈N0}⊆↓z.

Remark 2.4

Let (P, 𝓜,𝓝) be aPMN-space andx, y, z ∈ P.

1. If there is noA ∈ 𝓜 such that sup A = x, thenp≪MNxandp⊳MNxfor allp ∈ P; similarly, if there is noB ∈ 𝓝 such that inf B = x, thenp≪MNxandp⊳MNxfor allp ∈ P.

2. By Definition 2.3, one can easily check that ifPhas the least element ⊥, then⊥≪MNpfor everyp ∈ P, and ifPhas the greatest element ⊤, then⊤⊳MNpfor everyp ∈ P.

3. The implicationsy≪MNx⇒x⩽yandz⊳MNx⇒z⩾xare not true necessarily. See the following example: let ℝ be the set of all real numbers, in its ordinal order, and 𝓜 = 𝓝 = {{n} : n ∈ ℤ}, where ℤ is the set of all integers. Then, by (1), we have1≪MN1/2and0⊳MN1/2.But 1⧸ ⩽ 1/2 and 0⧸ ⩾ 1/2.

4. Assume that sup A₀ = x = inf B₀for someA₀ ∈ 𝓜 andB₀ ∈ 𝓝. Then it follows from Definition 2.3 thaty≪MNximpliesy ⩽ xandz⊳MNximpliesz ⩾ x. In particular, if 𝓢₀(P) ⊆ 𝓜,𝓝, thenb≪MNaimpliesb ⩽ aandc⊳MNaimpliesc ⩾ afor anya, b, c ∈ P. More particularly, for anyp₁,p₂,p₃ ∈ P, we havep1≪S0S0p₂ ⟺ p₁ ⩽ p₂andp3⊳S0S0p2 ⟺ p₃ ⩾ p₂.

Proposition 2.5

Let (P, 𝓜,𝓝) be aPMN-space andx, y, z ∈ P. Then

1. y≪MNxif and only if for every net (x_i)_i∈Ithat 𝓜𝓝-converges tox, x_i ⩾ yholds eventually.

2. z⊳MNxif and only if for every net (x_i)_i∈Ithat 𝓜𝓝-converges tox, x_i ⩽ zholds eventually.

Proof

(1) Suppose y≪MNx. If a net (xi)i∈I→MNx, then there exist A ∈ 𝓜 and B ∈ 𝓝 such that sup A = x = inf B, and for any a ∈ A and b ∈ B, there exists iab∈I such that a ⩽ x_i ⩽ b for all i⩾iab. According to Definition 2.3 (1), it follows that there exist A₀ = {a₁,a₂, …,a_n} ⊑ A and B₀ = {b₁,b₂, …,b_m} ⊑ B such that x ∈ ⋂{↑a_k ∩ ↓b_j : 1 ≤ k ≤ n & 1 ≤ j ≤ m} ⊆ ↑y. Take i₀ ∈ I with that i0⩾iakbj for every k ∈ {1, 2, …, n} and every j ∈ {1, 2, …, m}. Then x_i ∈ ⋂{↑a_k ∩ ↓b_j : 1 ≤ k ≤ n & 1 ≤ j ≤ m} ⊆ ↑y for all i ⩾ i₀. This means x_i ⩾ y holds eventually.

Conversely, suppose that for every net (x_i)_i∈I that 𝓜𝓝-converges to x, x_i ⩾ y holds eventually. For every A ∈ 𝓜 and B ∈ 𝓝 with sup A = x = inf B, consider the net (x(d,D))(d,D)∈D(A,B)x defined in Remark 2.2 (5). By Remark 2.2 (5), the net (x(d,D))(d,D)∈D(A,B)x→MNx. So, there exists (d₀,D₀) ∈ D(A,B)x such that x_(d,D) = d ⩾ y for all (d, D) ≥ (d₀,D₀). Since (d, D₀) ≥ (d₀,D₀) for all d ∈ D₀, x_(d,D0) = d ⩾ y for all d ∈ D₀. Thus, we have D₀ ⊆ ↑y. It follows from the definition of D(A,B)x that there exist A₀ ⊑ A and B₀ ⊑ B such that D₀ = ⋂{↑a ∩ ↓b : a ∈ A₀ & b ∈ B₀} ⊆ ↑y. This shows y≪MNx.

The proof of (2) can be processed similarly. □

Remark 2.6

Let (P, 𝓜,𝓝) be aPMN-space.

1. If 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P), then≪DF=≪Oand⊳DF=⊳O.

2. If 𝓜 = 𝓝 = 𝓟₀(P), then≪P0P0=≪αand⊳P0P0=⊳α.

Definition 2.7

Let (P, 𝓜,𝓝) be aPMN-space. The posetPis called an 𝓜𝓝-doubly continuous poset if for everyx ∈ P, there existM_x ∈ 𝓜 andN_x ∈ 𝓝 such that

1. M_x ⊆ ▾MNx,Nx⊆△MNxand sup M_x = x = inf N_x.

2. For anyy∈▾MNxandz∈△MNx, ⋂{↑m ∩ ↓n : m ∈ M₀ & n ∈ N₀} ⊆ ▴MNy∩▽MNzfor someM₀ ⊑ M_xandN₀ ⊑ N_x.

Proposition 2.8

Let (P, 𝓜,𝓝) be aPMN-space andx, y, z ∈ P. If the posetPis an 𝓜𝓝-doubly continuous poset, theny≪MNximpliesy ⩽ xandz⊳MNximpliesz ⩾ x.

Example 2.9

Let (P, 𝓜,𝓝) be a PMN-space.

1. If 𝓜 = 𝓝 = 𝓢₀(P), then by Remark 2.4 (4), we have≪S0S0=⩽and⊳S0S0=⩾.By Definition 2.7, one can easily check thatPis an 𝓢₀𝓢₀-doubly continuous poset.

2. If 𝓜 = 𝓝 = 𝓛₀(P), then by Definition 2.3, we have≪L0L0=⩽and⊳L0L0=⩾.It can be easily checked from Definition 2.7 thatPis an 𝓛₀𝓛₀-doubly continuous poset.

3. Let 𝓜 = 𝓓(P) and 𝓝 = 𝓕(P). Then it is easy to check that ifPis an 𝓞-doubly continuous poset which satisfies Condition (△), then it is a 𝓓𝓕-doubly continuous poset. Particularly, finite posets, chains and anti-chains, completely distributive lattices are all 𝓓𝓕-doubly continuous posets.

4. Let 𝓜 = 𝓝 = 𝓟₀(P). Then the posetPis 𝓟₀𝓟₀-double continuous if and only if it isO₂-double continuous. Thus, chains and finite posets are all 𝓟₀𝓟₀-doubly continuous posets.

Definition 2.10

Given aPMN-space (P, 𝓜,𝓝), a subsetU of Pis called an 𝓜𝓝-open set if for every net (x_i)_i∈Iwith that(xi)i∈I→MNx ∈ U, x_i ∈ Uholds eventually.

Theorem 2.11

Let (P, 𝓜,𝓝) be aPMN-space. Then a subsetUofPis an 𝓜𝓝-open set if and only if for everyM ∈ 𝓜 andN ∈ 𝓝 with sup M = x = inf N ∈ U, we have

for someM₀ ⊑ MandN₀ ⊑ N.

Proof

Suppose that U is an 𝓜𝓝-open subset of P. For every M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf N ∈ U, let (x(d,D))(d,D)∈D(M,N)x be the net defined in Remark 2.2 (5). Then the net (x(d,D))(d,D)∈D(M,N)x→MNx. By the definition of 𝓜𝓝-open set, the exists (d₀,D₀) ∈ D(M,N)x such that x_(d,D) = d ∈ U for all (d, D) ≥ (d₀,D₀). Since (d, D₀) ≥ (d₀,D₀) for all d ∈ D₀, x_(d,D0) = d ∈ U for every d ∈ D₀, and thus D₀ ⊆ U. It follows from the definition of the directed set D(M,N)x that D₀ = ⋂{↑m ∩ ↓n : m ∈ M₀ & n ∈ N₀} ⊆ U for some M₀ ⊑ M and some N₀ ⊑ N.

Conversely, assume that U is a subset of P with the property that for any M ∈ 𝓜 and N ∈ 𝓝 with sup M = x = inf N ∈ U, there exist M₀ = {m₁,m₂, …,m_k} ⊑ M and N₀ = {n₁,n₂, …,n_l} ⊑ N such that ⋂{↑m_h ∩ ↓n_j : 1 ≤ h ≤ k & 1 ≤ j ≤ l} ⊆ U. Let (x_i)_i∈I be a net that 𝓜𝓝-converges to x ∈ U. Then there exist M ∈ 𝓜 and N ∈ 𝓝 such that sup M = x = inf N ∈ U, and for every m ∈ M and n ∈ N, m ⩽ x_i ⩽ n holds eventually. This means that for every m_h ∈ M₀ and n_j ∈ N₀, there exists i_h,j ∈ I such that m_h ⩽ x_i ⩽ n_j for all i ≥ i_h,j. Take i₀ ∈ I such that i₀ ≥ i_h,j for all h ∈ {1, 2, …, k} and j ∈ {1, 2, …, l}. Then x_i ∈ ⋂{↑m_h ∩ ↓n_j : 1 ≤ h ≤ k & 1 ≤ j ≤ l} ⊆ U for all i ≥ i₀. Therefore, U is an 𝓜𝓝-open subset of P. □

Proposition 2.12

Let (P, 𝓜,𝓝) be aPMN-space in whichPis an 𝓜𝓝-doubly continuous poset, andy, z ∈ P. Then▴MNy∩▽MNz∈OMN(P).

Proof

Suppose that M ∈ 𝓜 and N ∈ 𝓝 with sup M = inf N = x ∈ ▴MNy∩▽MNz. Since P is an 𝓜𝓝-doubly continuous poset, there exist M_x ∈ 𝓜 and N_x ∈ 𝓝 satisfying condition (A1) and (A2) in Definition 2.7. This means that there exist M₀ ⊑ M_x ⊆ ▾MNx and N₀ ⊑ N_x ⊆ △MNx such that ⋂{↑m₀ ∩ ↓n₀:m₀ ∈ M₀ & n₀ ∈ N₀} ⊆ ▴MNy∩▽MNz. As M₀ ⊑ M_x ⊆ ▾MNx and N₀ ⊑ N_x ⊆ △MNx, by Definition 2.3, there exist M_m0 ⊑ M and N_n0 ⊑ N such that ⋂{↑m ∩ ↓n : m ∈ M_m0 & n ∈ N_n0 } ⊆ ↑m₀ ∩ ↓n₀ for every m₀ ∈ M₀ and n₀ ∈ N₀. Take M_F = ⋃{M_m0:m₀ ∈ M₀} and N_F = ⋃{N_n0:n₀ ∈ N₀}. Then it is easy to check that M_F ⊑ M, N_F ⊑ N and

So, it follows from Theorem 2.11 that ▴MNy∩▽MNz∈OMN(P). □

Lemma 2.13

Let (P, 𝓜,𝓝) be aPMN-space in whichPis an 𝓜𝓝-doubly continuous poset. Then a net

Proof

From the definition of OMN(P), it is easy to see that a net

To prove the Lemma, it suffices to show that a net (xi)i∈I→OMN(P)x ∈ P implies (xi)i∈I→MNx. Suppose a net (xi)i∈I→OMN(P)x. Since P is an 𝓜𝓝-doubly continuous poset, there exist M_x ∈ 𝓜 and N_x ∈ 𝓝 such that M_x ⊆ ▾MNx, N_x ⊆ △MNx and sup M_x = x = inf N_x. By Proposition 2.12, x∈▴MNy∩▽MNz∈OMN(P) for every y ∈ M_x ⊆ ▾MNx and every z ∈ N_x ⊆ △MNx, and hence xi∈▴MNy∩▽MNz holds eventually for every y ∈ M_x ⊆ ▾MNx and every z ∈ N_x ⊆ △MNx. It follows from Proposition 2.8 that y ⩽ x_i ⩽ z holds eventually for every y ∈ M_x and z ∈ N_x. Thus (xi)i∈I→MNx □

Lemma 2.14

Let (P, 𝓜,𝓝) be aPMN-space. If the 𝓜𝓝-convergence inPis topological, thenPis 𝓜𝓝-doubly continuous.

Proof

Suppose that the 𝓜𝓝-convergence in P is topological. Then there exists a topology 𝓣 on P such that for every x ∈ P, a net (xi)i∈I→MNx if and only if (xi)i∈I→Tx. Define I_x = {(p, U) ∈ P × 𝓝(x) : p ∈ U}, where 𝓝(x) denotes the set of all open neighbourhoods of x in the topological space (P, 𝓣), i.e., 𝓝(x) = {U ∈ 𝓣 : x ∈ U}. Define the preorder ≼ on I_x as follows:

Now one can easily see that I_x is directed. Let x_(p,U) = p for every (p, U) ∈ I_x. Then it is straightforward to check that the net (x(p,U))(p,U)∈Ix→Tx, and thus (x(p,U))(p,U)∈Ix→MNx. By Definition 2.1, there exist M_x ∈ 𝓜 and N_x ∈ 𝓝 such that sup M_x = x = inf N_x, and for every m ∈ M_x and n ∈ N_x, there exists (pmn,Umn) ∈ I_x such that x_(p,U) = p ∈ ↑m ∩ ↓n for all (p,U)≽(pmn,Umn). Since (p,Umn)≽(pmn,Umn) for every p∈Umn,x(p,Umn)=p ∈ ↑m ∩ ↓n for every p∈Umn. This shows

For any A ∈ 𝓜 and B ∈ 𝓝 with sup A = x = inf B, let (x(d,D))(d,D)∈D(A,B)x be the net defined as in Remark 2.2 (5). Then (x(d,D))(d,D)∈D(A,B)x→MNx, and hence (x(d,D))(d,D)∈D(A,B)x→Tx. This implies, by Remark 2.2 (6), that there exist A₀ ⊑ A and B₀ ⊑ B satisfying

Therefore, m ∈ ▾MNx and n ∈ △MNx, and hence ▾MNx and N_x ⊆ △MNx.

Let y ∈ ▾MNx and z ∈ △MNx. Since sup M_x = x = inf N_x, by Definition 2.3, ⋂{↑m ∩ ↓n : m ∈ M₁ & n ∈ N₁} ⊆ ↑y ∩ ↓z for some M₁ ⊑ M_x and N₁ ⊑ N_x. This concludes by Condition (⋆) and the finiteness of sets M₁ and N₁ that ⋂{Umn:m∈M1&n∈N1} ∈ 𝓝(x) and

Considering the net (x(d,D))(d,D)∈D(Mx,Nx)x defined in Remark 2.2 (5), we have (x(d,D))(d,D)∈D(Mx,Nx)x→MNx, and hence (x(d,D))(d,D)∈D(Mx,Nx)x→Tx. So, by Remark 2.2 (6), there exist M₂ ⊑ M_x and N₂ ⊑ N_x such that

Finally, we show ⋂{↑m ∩ ↓n : m ∈ M₂ & n ∈ N₂} ⊆ ▴MNy∩▽MNz. Let (x(d,D))(d,D)∈D(M′,N′)x′ be the net defined in 2.2 (5) for any M′ ∈ 𝓜 and N′ ∈ 𝓝 with sup M′ = inf N′ = x′ ∈ ⋂{↑m ∩ ↓n : m ∈ M₂ & n ∈ N₂}. Then (x(d,D))(d,D)∈D(M′,N′)x′→MNx′, and thus (x(d,D))(d,D)∈D(M′,N′)x′→Tx′. This implies by Remark 2.2 (6) that there exist M0′⊑M′ and N0′⊑N′ satisfying

Hence, we have x′ ∈ ▴MNy∩▽MNz by Definition 2.3. This shows ⋂{↑m ∩ ↓n : m ∈ M₂ & n ∈ N₂} ⊆ ▴MNy∩▽MNz. Therefore, it follows from Definition 2.7 that P is 𝓜𝓝-doubly continuous. □

Theorem 2.15

Let (P, 𝓜,𝓝) be aPMN-space. Then the following statements are equivalent:

1. Pis an 𝓜𝓝-doubly continuous poset.

2. For any net (x_i)_i∈IinP, (xi)i∈I→MNxif and only if(xi)i∈I→OMN(P)x.

3. The 𝓜𝓝-convergence inPis topological.

Proof

(1) ⇒ (2): By Lemma 2.13.

(2) ⇒ (3): It is clear.

(3) ⇒ (1): By Lemma 2.14. □

Definition 3.1

([8]). Let (P, 𝓜) be aPM-space. Anet (x_i)_i∈IinPis said to lim-inf_𝓜-converge tox ∈ Pif there existsM ∈ 𝓜 such that

1. x ⩽ sup M;

2. for everym ∈ M, x_i ⩾ mholds eventually.

In this case, we write(xi)i∈I→Mx.

Remark 3.2

Let (P, 𝓜) be aPM-space andx, y ∈ P.

1. Suppose that a net(xi)i∈I→Mxandy ⩽ x. (xi)i∈I→Myby Definition 3.1. This concludes that the set of all lim-inf_𝓜-convergent points of the net (x_i)_i∈IinPis a lower subset ofP. Thus, the lim-inf_𝓜-convergent points of the net (x_i)_i∈Ineed not be unique.

2. IfPhas the least element ⊥ and ∅ ∈ 𝓜, then we have(xi)i∈I→M⊥for every net (x_i)_i∈IinP.

3. For everyM ∈ 𝓜 with sup M ⩾ x, we denoteFMx = {⋂{↑m : m ∈ M₀} : M₀ ⊑ M}^[2]. LetDMx = {(d, D) ∈ P × FMx : d ∈ D} be in the preorder ≤ defined by

   (∀(d1,D1),(d2,D2)∈DMx)(d1,D1)≦(d2,D2)⟺D2⊆D1.

   It is easy to see that the setDMxis directed. Takex_(d,D) = dfor every (d, D) ∈ DMx. Then, by Definition 3.1, one can straightforwardly check that the net(x(d,D))(d,D)∈DMx→Mfor everya ⩽ x.

4. If the net(x(d,D))(d,D)∈DMxdefined in (3) converges top ∈ Pwith respect to some topology 𝓣 onP, then for every open neighbourhoodU_pofp, there existsM₀ ⊑ Msuch that ⋂{↑m : m ∈ M₀} ⊆ U_p.

Definition 3.3

([8]). Let (P, 𝓜) be aPM-space.

1. Forx, y ∈ P, definey≪_α(𝓜)xif for every net (x_i)_i∈Ithat lim-inf_𝓜-converges tox, x_i ⩾ yholds eventually.

2. The posetPis said to beα(𝓜)-continuous if {x ∈ P : x≪_α(𝓜)a} ∈ 𝓜 anda = sup{x ∈ P : x≪_α(𝓜)a} holds for everya ∈ P.

Proposition 3.4

Let (P, 𝓜) be aPM-space andx, y ∈ P. Theny≪_α(𝓜)xif and only if for everyM ∈ 𝓜 with sup M ⩾ x, there existsM₀ ⊑ Msuch that

Proof

Suppose y≪_α(𝓜)x. Let (x(d,D))(d,D)∈DMx be the net defined in Remark 3.2 (3) for every M ∈ 𝓜 with sup M = p ⩾ x. Then the net (x(d,D))(d,D)∈DMx→Mx. By Definition 3.3 (1), there exists (d₀,D₀) ∈ DMx such that x_(d,D) = d ⩾ y for all (d, D) ≦ (d₀,D₀). Since (d, D₀) ≦ (d₀,D₀) for every d ∈ D₀, x_(d,D0) = d ⩾ y for every d ∈ D₀. So D₀ ⊆ ↑y. This shows that there exists M₀ ⊑ M such that D₀ = ⋂{↑m : m ∈ M₀} ⊆ ↑y.

Conversely, suppose that for every M ∈ 𝓜 with sup M ⩾ x, there exists M₀ ⊑ M such that ⋂{↑m : m ∈ M₀} ⊆ ↑y. Let (x_i)_i∈I be a net that lim-inf_𝓜-converges to x. Then, by Definition 3.1, there exists M ∈ 𝓜 such that sup M = p ⩾ x, and for every m ∈ M, there exists i_m ∈ I such that x_i ⩾ m for all i ≥ i_m. Take i₀ ∈ I with that i₀ ≥ i_m for every m ∈ M₀ ⊑ M, we have that x_i ∈ ⋂{↑m : m ∈ M₀} ⊆ ↑y for all i ≥ i₀. This shows that x_i ⩾ y holds eventually. Thus, by Definition 3.3 (1), we have y≪_α(𝓜)x. □

Remark 3.5

Let (P, 𝓜) be aPM-space andx, y ∈ P.

1. If there is noM ∈ 𝓜 such that sup M ⩾ x, thenp≪_α(𝓜)xfor everyp ∈ P. And, if the posetPhas the least element ⊥, then ⊥≪_α(𝓜)pfor everyp ∈ P.

2. The implicationy≪_α(𝓜)x ⟹ y ⩽ xmay not be true. For example, letP = {0,1, 2, …} be in the discrete order ⩽ defined by

   (∀i,j∈P)i⩽j⟺i=j.

   And let 𝓜 = {{2}}. Then, it is easy to see from Remark 3.5 (1) that 0≪_α(𝓜)1 and 0⧸ ⩽ 1.

3. Assume thePM-space (P, 𝓜) has the property that for everyp ∈ P, there existsM_p ∈ 𝓜 such that sup M_p = p. Then, by Proposition 3.4, we have

   (∀q,r∈P)q≪α(M)r⟹q⩽r.

Definition 3.6

Let (P, 𝓜) be aPM-space. The posetPis called anα^*(𝓜)-continuous poset if for everyx ∈ P, there existsM_x ∈ 𝓜 such that

1. sup M_x = xandM_x ⊆ ▾_𝓜x. And,

2. for everyy ∈ ▾_𝓜x, there existsF ⊑ M_xsuch that ⋂{↑f : f ∈ F} ⊆ ▴_𝓜y.

Proposition 3.7

Let (P, 𝓜) be aPM-space in which the posetPisα^*(𝓜)-continuous. Then

Example 3.8

Let (P, 𝓜) be aPM-space.

1. IfPis a finite poset, thenPis anα^*(𝓜)-continuous poset if and only if for everyx ∈ P, there existsM_x ∈ 𝓜 such that sup M_x = x.

2. Let 𝓜 = 𝓛(P). ThenPis anα^*(𝓛)-continuous poset. This means that every poset isα^*(𝓛)-continuous.

3. Let 𝓜 = 𝓓(P). Then we have ≪ = ≪_α(𝓓)(see Example 3.2 (1) in [8]). The posetPis a continuous poset if and only if it is anα^*(𝓓)-continuous poset. In particular, finite posets, chains, anti-chains and completely distributive lattices are allα^*(𝓓)-continuous.

4. Let 𝓜 = 𝓟(P). IfPis a finite poset (resp. chain, anti-chain), thenPis anα^*(𝓟)-continuous poset.

Proposition 3.9

Let (P, 𝓜) be aPM-space. IfPis anα(𝓜)-continuous poset, and {y ∈ P : (∃ z ∈ P) y≪_α(𝓜)z≪_α(𝓜)a} ∈ 𝓜 for everya ∈ P, thenPis anα^*(𝓜)-continuous poset.

Proof

Suppose that P is an α(𝓜)-continuous poset, and {y ∈ P : (∃ z ∈ P) y≪_α(𝓜)z≪_α(𝓜)a} ∈ 𝓜 for every a ∈ P. Take M_a = ▾_𝓜a. Then it is easy to see that sup M_a = a and M_a ⊆ ▾_𝓜a. By Remark 3.3 (4) in [8], we have sup{y ∈ P : (∃ z ∈ P) y≪_α(𝓜)z≪_α(𝓜)a} = a. This implies, by Proposition 3.4 and Remark 3.5 (2), that for every y ∈ ▾_𝓜a, there exist {y₁,y₂, …,y_n}, {z₁,z₂, …,z_n} ⊑ M_a = ▾_𝓜a such that

and y_i≪_α(𝓜)z_i≪_α(𝓜)a for every i ∈ {1, 2, …, n}. Next, we show ⋂{↑z_i : i ∈ {1, 2, …, n}} ⊆ ▴_𝓜y. For every M ∈ 𝓜 with sup M ⩾ b ∈ ⋂{↑z_i : i ∈ {1, 2, …, n}}, by Proposition 3.4, there exists M_i ⊑ M such that ⋂{↑m′:m′ ∈ M_i} ⊆ ↑y_i for every i ∈ {1, 2, …, n}. Take M₀ = ⋃{M_i : i ∈ {1, 2, …, n}}. Then M₀ ⊑ M and

This shows y≪_α(𝓜)b for every b ∈ ⋂{↑z_i : i ∈ {1, 2, …, n}}. Hence, ⋂{↑z_i : i ∈ {1, 2, …, n}} ⊆ ▴_𝓜y. Thus P is an α^*(𝓜)-continuous poset. □

Example 3.10

Let (P, 𝓜) be thePM-space in which the posetP = ℝ is the set of all real number with its usual order ⩽ and 𝓜 = 𝓢₀(ℝ). Then we have ≪_α(𝓢0) = ⩽ by Proposition 3.4. It is easy to check, by Definition 3.6, that ℝ is anα^*(𝓢₀)-continuous poset. But ℝ is not anα(𝓢₀)-continuous poset because ▾_𝓢0x = ↓x⧸ ∈ 𝓢₀(P) for everyx ∈ ℝ.

Definition 3.11

Let (P, 𝓜) be aPM-space. AsubsetVofPis said to be 𝓜-open if for every net(xi)i∈I→Mx∈V,x_i ∈ Vholds eventually.

Theorem 3.12

Let (P, 𝓜) be aPM-space. Then a subsetVofPis 𝓜-open if and only if it satisfies the following two conditions:

1. ↑V = V, i.e., Vis an upper set.

2. For everyM ∈ 𝓜 with sup M ∈ V, there existsM₀ ⊑ Msuch that ⋂{↑m : m ∈ M₀} ⊆ V.

Proof

Suppose that V is an 𝓜-open subset of P. By Remark 3.2 (1), it is easy to see that V is an upper set. Let (x(d,D))(d,D)∈DMx be the net defined in Remark 3.2 (3) for every M ∈ 𝓜 with sup M = x ∈ V. Then (x(d,D))(d,D)∈DMx→Mx∈V. This implies, by Definition 3.11, that there exists (d₀,D₀) ∈ DMx such that x_(d,D) = d ∈ V for all (d, D) ≥ q (d₀,D₀). Since (d, D₀) ≦ (d₀,D₀) for all d ∈ D₀, x_(d,D0) = d ∈ V for all d ∈ D₀. This shows D₀ ⊆ V. Thus there exists M₀ ⊑ M such that D₀ = ⋂{↑m : m ∈ M₀} ⊆ V.

Conversely, suppose V is a subset of P which satisfies Condition (V1) and (V2). Let (x_i)_i∈I be a net that lim-inf_𝓜-converges to x ∈ V. Then there exists M ∈ 𝓜 such that sup M = y ⩾ x ∈ V = ↑V (hence, y ∈ V), and for every m ∈ M, there exists i_m ∈ I such that x_i ⩾ m for all i ≥ i_m. By Condition (V2), we have that ⋂{↑m : m ∈ M₀} ⊆ V for some M₀ ⊑ M. Take i₀ ∈ I with that i₀ ≥ i_m for all m ∈ M₀. Then x_i ∈ ⋂{↑m : m ∈ M₀} ⊆ V for all i ≥ i₀. This shows that V is an 𝓜-open set. □

Proposition 3.13

Let (P, 𝓜) be aPM-space in which the posetPisα^*(𝓜)-continuous. Then ▴_𝓜x ∈ 𝓞_𝓜(P) for everyx ∈ P.

Proof

One can readily see, by Proposition 3.4, that {▴_𝓜}x is an upper subset of P for every x ∈ P. For every M ∈ 𝓜 with sup M = y ∈ {▴_𝓜}x, by Definition 3.6 (O1) there exists M_y ∈ 𝓜 such that M_y ⊆ {▾_𝓜}y and sup M_y = y. Since x≪_α(𝓜)y, by Definition 3.6 (O2), we have ⋂{↑m_i : i ∈ {1, 2, …, n}} ⊆ {▴_𝓜}x for some {m₁,m₂, …,m_n} ⊑ M_y. Observing {m₁,m₂, …,m_n} ⊑ M_y ⊆ {▾_𝓜}y, we can conclude that there exists M_i ⊑ M such that ⋂{↑a : a ∈ M_i} ⊆ ↑m_i for every i ∈ {1, 2, …, n}. Let M₀ = ⋃{M_i : i ∈ {1, 2, …, n}}. Then M₀ ⊑ M and