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

In this paper, we rst introduce the notion of MN-convergence in posets as an uni ed form of Oconvergence and O2-convergence. Then, by studying the fundamental properties of MN-topology which is determined by MN-convergence according to the standard topological approach, an equivalent characterization to the MN-convergence being topological is established. Finally, the lim-infM-convergence in posets is further investigated, and a su cient and necessary condition for lim-infM-convergence to be topological is obtained.


Introduction, Notations and Preliminaries
The concept of O-convergence in partially ordered sets (posets, for short) was introduced by Birkho [1], Frink [2] and Mcshane [3]. It is de ned as follows: a net (x i ) i∈I in a poset P is said to O-converge to x ∈ P if there exist subsets D and F of P such that (1) D is directed and F is ltered; (2) sup D = x = inf F; (3) for every d ∈ D and e ∈ F, d x i e holds eventually, i.e., there exists i ∈ I such that d x i e for all i i .
As what has been showed in [4], the O-convergence (Note: in [4], the O-convergence is called orderconvergence) in a general poset P may not be topological, i.e., it is possible that P can not be endowed with a topology such that the O-convergence and the associated topological convergence are consistent. Hence, much work has been done to characterize those special posets in which the O-convergence is topological. The most recent result in [5] shows that the O-convergence in a poset which satis es Condition ( ) is topological if and only if the poset is O-doubly continuous. This means that for a special class of posets, a su cient and necessary condition for O-convergence being topological is obtained. As a direct generalization of O-convergence, O -convergence in posets has been discussed in [11] from the order-theoretical point of view. It is de ned as follows: a net (x i ) i∈I in a poset P is said to O -converge to x ∈ P if there exist subsets A and B of P such that (1) sup A = x = inf B; (2) for every a ∈ A and b ∈ B, a x i b holds eventually.
In fact, the O -convergence is also not topological generally. To clarify those special posets in which the O -convergence is topological, Zhao and Li [6] showed that for any poset P satisfying Condition (*), Oconvergence is topological if and only if P is α-doubly continuous. As a further result, Li and Zou [7] proved that the O -convergence in a poset P is topological if and only if P is O -doubly continuous. This result demonstrates the equivalence between the O -convergence being topological and the O -double continuity of a given poset.
On the other hand, Zhou and Zhao [8] have de ned the lim-inf M -convergence in posets to generalize lim-inf-convergence and lim-inf -convergence [4]. They also found that the lim-inf M -convergence in a poset is topological if and only if the poset is α(M)-continuous when some additional conditions are satis ed (see [8], Theorem 3.1). This result clari ed some special conditions of posets under which the lim-inf Mconvergence is topological. However, to the best of our knowledge, the equivalent characterization to the lim-inf M -convergence in general posets being topological is still unknown.
One goal of this paper is to propose the notion of MN-convergence in posets which can unify Oconvergence and O -convergence and search the equivalent characterization to the MN-convergence being topological. More precisely, (G11)Given a general poset P, we hope to clarify the order-theoretical condition of P which is su cient and necessary for the MN-convergence being topological. (G12)Given a poset P satisfying such condition, we hope to provide a topology that can be equipped on P such that the MN-convergence and the associated topological convergence agree.
Another goal is to look for the equivalent characterization to the lim-inf M -convergence being topological. More precisely, (G21)Given a general poset P, we expect to present a su cient and necessary condition of P which can precisely serve as an order-theoretical condition for the lim-inf M -convergence being topological. (G22)Given a poset P satisfying such condition, we expect to give a topology on P such that the lim-inf Mconvergence and the associated topological convergence are consistent.
To accomplish those goals, motivated by the ideal of introducing the Z-subsets system [9] for de ning Zcontinuous posets , we propose the notion of MN-doubly continuous posets and de ne the MN-topology on posets in Section 2. Based on the study of the basic properties of the MN-topology, it is proved that the MNconvergence in a poset P is topological if and only if P is an MN-doubly continuous poset if and only if the MN-convergence and the topological convergence with respect to MN-topology are consistent. In Section 3, by introducing the notion of α * (M)-continuous posets and presenting the fundamental properties of Mtopology which is induced by the lim-inf M -convergence, we show that the lim-inf M -convergence in a poset P is topological if and only if P is an α * (M)-continuous poset if and only if the lim-inf M -convergence and the topological convergence with respect to M-topology are consistent. Some conventional notations will be used in the paper. Given a setX, F X means that F is a nite subset of X. Given a topological space (X, T) and a net (x i ) i∈I in X, we take (x i ) i∈I T −→ x to mean the net (x i ) i∈I converges to x ∈ P with respect to the topology T.
Let P be a poset and x ∈ P. ↑x and ↓x are always used to denote the principal lter {y ∈ P : y x} and the principal ideal {z ∈ P : z x} of P, respectively. Given a poset P and A ⊆ P, by writing sup A we mean that the least upper bound of A in P exists and equals to sup A ∈ P; dually, by writing inf A we mean that the greatest lower bound of A in P exists and equals to inf A ∈ P. And the set A is called an upper set if A = ↑A = {b ∈ P; (∃a ∈ A) a b}, the lower set is de ned dually. To make this paper self-contained, we brie y review the following notions: De nition 1.1 ([5]). Let P be a poset and x, y, z ∈ P. We say y O x if for every net (x i ) i∈I in P which O-converges to x ∈ P, x i y holds eventually; dually, we say z£ O x if for every net (x i ) i∈I in P which O-converges to x ∈ P, x i z holds eventually.
De nition 1.2 ([5]). A poset P is said to be O-doubly continuous if for every x ∈ P, the set {a ∈ P : a O x} is directed, the set {b ∈ P : b£ O x} is ltered and sup{a ∈ P : a O x} = x = inf{b ∈ P : b£ O x}.
Condition ( ). A poset P is said to satisfy Condition( ) if (1) for any x, y, z ∈ P, x O y z implies x O z; (2) for any w, s, t ∈ P, w£ O s t implies w£ O t.

De nition 1.3 ([6]
). Let P be a poset and x, y, z ∈ P. We say y α x if for every net (x i ) i∈I in P which Oconverges to x ∈ P, x i y holds eventually; dually, we say z£αx if for every net (x i ) i∈I in P which O -converges to x ∈ P, x i z holds eventually.

De nition 1.4 ([7]). A poset P is said to be O -doubly continuous if for every x ∈ P,
(1) sup{a ∈ P : a α x} = x = inf{b ∈ P : b£αx}; (2) for any y, z ∈ P with y α x and z£αx, there exist A {a ∈ P : a α x} and B {b ∈ P : b£αx} such that y α c and z£αc for each c ∈ {↑a ∩ ↓b :

MN-topology on posets
Based on the introduction of MN-convergence in posets, the MN-topology can be de ned on posets. In this section, we rst de ne the MN-double continuity for posets. Then, we show the equivalence between the MN-convergence being topological and the MN-double continuity of a given poset. A PMN-space is a triplet (P, M, N) which consists of a poset P and two subfamily M, N ⊆ P(P).
All PMN-spaces (P, M, N) considered in this section are assumed to satisfy the following conditions: (C1) If P has the least element ⊥, then {⊥} ∈ M; (C2) If P has the greatest element , then { } ∈ N; (C3)∅ / ∈ M and ∅ / ∈ N.
(MN2)x i ∈ ↑m ∩ ↓n eventually for every m ∈ M and every n ∈ N.
In this case, we will write (x i ) i∈I Then there exist A k ∈ M and B k ∈ N such that sup A k = x k = inf B k and a k x i b k holds eventually for every a k ∈ A k and b k ∈ B k (k = , ). This implies that for any a ∈ A , a ∈ A , b ∈ B and b ∈ B , there exists i ∈ I such that a One can readily check that  For convenience, given a PMN-space (P, M, N) and x ∈ P, we will brie y denote - From the logical point of view, we stipulate {↑a ∩ ↓b :

P). Then it is easy to check that if P is an O-doubly continuous poset which satis es Condition ( ), then it is a DF-doubly continuous poset. Particularly, nite posets, chains and anti-chains, completely distributive lattices are all DF-doubly continuous posets. (4) Let M = N = P (P). Then the poset P is P P -double continuous if and only if it is O -double continuous.
Thus, chains and nite posets are all P P -doubly continuous posets.
Next, we are going to consider the MN-topology on posets, which is induced by the MN-convergence.
Proof. From the de nition of O N M (P), it is easy to see that a net To prove the Lemma, it su ces to show that a net (x i ) i∈I De ne the preorder on Ix as follows: Now one can easily see that Ix is directed. Let x (p,U) = p for every (p, U) ∈ Ix. Then it is straightforward to check that the net (x (p,U) ) (p,U)∈Ix T −→ x, and thus (x (p,U) ) (p,U)∈Ix MN −→ x. By De nition 2.1, there exist Mx ∈ M and Nx ∈ N such that sup Mx = x = inf Nx, and for every m ∈ Mx and n ∈ Nx, there exists (p n m , U n m ) ∈ Ix such that x (p,U) = p ∈ ↑m ∩ ↓n for all (p, U) (p n m , U n m ). Since (p, U n m ) (p n m , U n m ) for every p ∈ U n m , x (p,U n m ) = p ∈ ↑m ∩ ↓n for every p ∈ U n m . This shows (∀m ∈ Mx, n ∈ Nx) (∃U n m ∈ N(x)) x ∈ U n m ⊆ ↑m ∩ ↓n. Combining Lemma 2.13 and Lemma 2.14, we obtain the following theorem. In this case, we write ( It is worth noting that both lim-inf-convergence and lim-inf -convergence [4] in posets are particular cases of lim-inf M -convergence.