The Equivalence of Two Modes of Order Convergence

: It is well known that if a poset satisfies Property A and its dual form, then the o -convergence and o 2 -convergence in the poset are equivalent. In this paper, we supply an example to illustrate that a poset in which the o -convergence and o 2 -convergence are equivalent may not satisfy Property A or its dual form, and carry out some further investigations on the equivalence of the o -convergence and o 2 -convergence. By introducing the concept of the local Frink ideals (the dually local Frink ideals) and establishing the correspondence between ID-pairs and nets in a poset, we prove that the o -convergence and o 2 -convergence of nets in a poset are equivalent if and only if the poset is ID-doubly continuous. This result gives a complete solution to the problem of E.S. Wolk in two modes of order convergence, which states under what conditions for a poset the o -convergence and o 2 -convergence in the poset are equivalent.


Introduction
Let P be a poset and (x i ) i∈I a net on an up-directed set I with value in the poset P. The concept of order convergence of nets in a poset P was introduced by Birkhoff [1], Mcshane [2], Frink [3], Rennie [4] and Ward [5].It is worth noting that the authors may have attached different meanings to the order convergence.Following the formulation of Wolk [6], we correspond to the following two modes of order convergence: Definition 1 ([1-3]).A net (x i ) i∈I in a poset P is said to o-converge to an element x ∈ P (in symbol (x i ) i∈I o − → x) if there exist subsets M and N of P such that (A0) M is up-directed and N is down-directed; (B0) sup M = x = inf N; (C0) For every m ∈ M and n ∈ N, m ⩽ x i ⩽ n holds eventually, i.e., there is i 0 ∈ I such that m ⩽ x i ⩽ n for all i ≧ i 0 .
Definition 2 ([4-6]).A net (x i ) i∈I in a poset P is said to o 2 -converge to an element x ∈ P (in symbol (x i ) i∈I o 2 − → x) if there exist subsets M and N of P such that In [7], based on the introduction of Condition(*) and the double continuity for posets, Zhou and Zhao proved that, for a double continuous poset P with Condition(*), the oconvergence in the poset P is topological.As a further result, Condition (△), a weaker condition than Condition(*), and the O-doubly continuous posets were defined in [8].It was shown that, for a poset P with Condition (△), the o-convergence in the poset P is topological if and only if the poset P is O-doubly continuous.Following the ideal in [8], Sun and Li [9] studied the B-topology on posets and found that the o-convergence in a poset P is topological if and only if the poset P is S * -doubly continuous, which demonstrates the equivalence between the o-convergence being topological and the S * -double continuity of a poset.Moreover, the ideal-o-convergence, a generalized form of o-convergence established via ideals, was defined in posets by Georgiou et al. [10,11].Also, the authors obtained that the ideal-o-convergence in a poset P is topological if and only if the poset P is S * -doubly continuous.This generalized the previous results on the o-convergence.
On the other hand, the o 2 -convergence is also not topological generally.To characterize these posets so that the o 2 -convergence is topological, Zhao and Li [12] studied the notions of α-double continuous posets and α * -double continuous posets.Under some additional conditions, the o 2 -convergence in these posets is topological.Ulteriorly, Li and Zou [13] proposed the concept of O 2 -doubly continuous posets and showed that the o 2 -convergence in a poset P is topological if and only if the poset P is O 2 -doubly continuous, meaning that they gave a sufficient and necessary condition for the o 2 -convergence to be topological.Further, Georgiou et al. [14] extended the o 2 -convergence to be the ideal-o 2 -convergence via ideals, and showed that the O 2 -double continuity can equivalently characterize such a convergence to be topological.
From the order-theoretical aspect, by the definitions, one can readily verify that the o-convergence implies the o 2 -convergence, i.e., if a net (x i ) i∈I in a poset P o-converges to an element x ∈ P, then it o 2 -converges to x.However, the converse implication is not true.This fact can be demonstrated by the example in [6].Hence, in [6], Wolk posed the following fundamental problem: Under what conditions for a poset P do the o-convergence and o 2 -convergence in P agree?
A well-known result on this problem is that the o-convergence and o 2 -convergence in a lattice are equivalent.Then, Wolk [6] obtained a result on the characterization of posets for the associated o-inf convergence (a counterpart of o-convergence) and o 2 -inf convergence (a counterpart of o 2 -convergence) being equivalent, which provides an approximate solution to the fundamental problem, using the concepts of Frink ideals and dual Frank ideals [15].
Motivated by these results toward the problem mentioned above, in this paper, we continue to make some further investigations on the o-convergence and o 2 -convergence, hoping to clarify the order-theoretical condition of a poset P, which is sufficient and necessary for the o-convergence and o 2 -convergence to be equivalent.
To this end, in Section 2, following the Frink ideal (the dual Frink ideal), the concepts of local Frink ideals (dually local Frink ideals) and ID-pairs in posets are further proposed, and then the relationship between ID-pairs and nets is presented.Section 3 is devoted to the order-theoretical characterization of the local Frink ideal (the dually local Frink ideal) generated by a general set.Using this characterization, we prove that the IDdouble continuity is the precise feature for those posets for which the two modes of order convergence are equivalent.
For the unexplained notions and concepts, one can refer to [6,16,17].

Local Frink Ideal (Dually Local Frink Ideal) in Posets
We appoint some conventional notations to be used in the sequel.Let X be a set.We take F ⊑ X to mean that F is a finite subset of the set X, including the empty set ∅. Given a poset P and K, L ⊆ P. The notations K u and L l are used to denote the set of all upper bounds of K and the set of all lower bounds of L, respectively, i.e., K u = {y ∈ P : (∀p ∈ K) y ⩾ p} and L l = {z ∈ P : (∀p ∈ L) z ⩽ p}.Particularly, if the sets K and L are all reduced to be a singleton {y}, then the notations ↑y and ↓y are reserved to denote the sets {y} u and {y} l , respectively.
Since the Frink ideal (the dual Frink ideal) in posets plays a fundamental role in the discussion of this section, we first review its definition.Definition 3 ([15]).Let P be a poset.
(1) A subset K of the poset P is called a Frink ideal if, for every F ⊑ K, we have (F u ) l ⊆ K.
Furthermore, a Frink ideal K is said to be normal if (K u ) l = K. (2) A subset L of the poset P is called a dual Frink ideal if, for every S ⊑ L, we have (S l ) u ⊆ L.
Furthermore, a dual Frink ideal L is said to be normal if (L l ) u = L.
Based on the Frink ideal (the dual Frink ideal), we further define the local Frink ideal (the dually local Frink ideal) in posets.Definition 4. Let P be a poset and K, L ⊆ P.
(1) The subset K is called a local Frink ideal in L if, for every F ⊑ K and every S ⊑ L, we have (F u S l ) l ⊆ K. (2) The subset L is called a dually local Frink ideal in K if, for every F ⊑ K and every S ⊑ L, we have (F u S l ) u ⊆ L.
Example 1.Let R be the set of all real numbers, in its usual order, and let a ∈ R. If we take K = (−∞, a] and L = [a, +∞), then, by Definition 4, the interval K is a local Frink ideal in L and the interval L is a dually local Frink ideal in K.
Given a poset P and K, L ⊆ P. We simply denote by L(L) the family of all local Frink ideals in L and, by D(K), the family of all dually local Frink ideals in K.
Remark 1.Let P be a poset and K, L ⊆ P.Then, (1) From the logic viewpoint, it is reasonable to stipulate that ∅ u = ∅ l = P. Thus, for every L ⊆ P and every K ∈ L(L), we have ⊥ ∈ K if the poset P has the least element ⊥.Dually, for every K ⊆ P and every L ∈ D(K), we have ⊤ ∈ L if the greatest element ⊤ exists in the poset P. (2) If K ∈ L(L), then K ∈ L(L 0 ) for every L 0 ⊆ L. And, dually, if L ∈ D(K), then L ∈ D(K 0 ) for every K 0 ⊆ K. (3) The subset K is a Frink ideal if and only if K ∈ L(∅).And, dually, the subset L is a dual Frink ideal if and only if L ∈ D(∅).
Proposition 1.Let P be a poset and K, L ⊆ P.
(1) If K ∈ L(L), then the subset K is a Frink ideal.
(2) If L ∈ D(K), then the subset L is a dual Frink ideal.
Proof.(1): Suppose that K ∈ L(L).Then, we have (F u S l ) l ⊆ K for every F ⊑ K and S ⊑ L .This implies that (F u ) l ⊆ (F u S l ) l ⊆ K. Thus, we conclude that (F u ) l ⊆ K for every F ⊑ K.This shows that the subset K is a Frink ideal.
(2): The proof is similar to that of (1).
However, the converse implications of Proposition 1 may not be true.This fact can be clarified in Example 7.
Definition 5. Let P be a poset.A pair (K, L) consisting of subsets K and L of P is called an ID-pair in P if K ∈ L(L) and L ∈ D(K).Moreover, an ID-pair (K, L) in P is said to be nontrivial if one of the following conditions is exactly satisfied: (1) |P| = 1, where |P| denotes the cardinal of the poset P; (2) |P| ≥ 2 and (K, L) ̸ = (P, P).
Example 2. Let P = {a, b} {⊥, ⊤}, with the partial order ⩽ defined by Take K = {⊥} and L = {⊤}.Then, it is easy to see from Definitions 4 and 5 that the pair (K, L) = ({⊥}, {⊤}) is a nontrivial ID-pair.Proposition 2. Let (K, L) be an ID-pair in a poset P.Then, the ID-pair (K, L) is nontrivial if and only if F u S l ̸ = ∅ for every F ⊑ K and every S ⊑ L.
Proof.(⇒): Let (K, L) be a nontrivial ID-pair in a poset P. We consider the following cases: (i) |P| = 1, i.e., the poset P = {p} contains only one element p.
It is easy to check that F u S l = {p} ̸ = ∅ for every F ⊑ K and every S ⊑ L.
Suppose that (F 0 ) u (S 0 ) l = ∅ for some F 0 ⊑ K and S 0 ⊑ L.Then, we have ) is an ID-pair in the poset P.This implies that (K, L) = (P, P), which is a contradiction to the assumption that the ID-pair (K, L) is nontrivial.Hence, we have that F u S l ̸ = ∅ for every F ⊑ K and every S ⊑ L. By (i) and (ii), we conclude that F u S l ̸ = ∅ for every F ⊑ K and every S ⊑ L.
(⇐): Suppose that (K, L) is an ID-pair such that F u S l ̸ = ∅ for every F ⊑ K and every S ⊑ L. If (K, L) ̸ = (P, P), then the ID-pair (K, L) is nontrivial by Definition 5.If (K, L) = (P, P), i.e., K = L = P, then, by the assumption, we have {p} u {q} l = ↑p ↓q ̸ = ∅ and {q} u {p} l = ↑q ↓p ̸ = ∅ for all p, q ∈ P. It follows that p = q for all p, q ∈ P. Hence, we conclude that |P| = 1.This shows, by Definition 5, that the ID-pair (K, L) is nontrivial.
In fact, given a poset P and a Frink ideal K (resp.a dual Frink ideal L) of the poset P, we can select a subset L (resp.a subset K) of P such that the pair (K, L) is a nontrivial ID-pair.
Theorem 1.Let P be a poset.
(1) If K is a Frink ideal of the poset P, then the pair (K, L) is a nontrivial ID-pair for some subset L of the poset P; (2) If L is a dual Frink ideal of the poset P, then the pair (K, L) is a nontrivial ID-pair for some subset K of the poset P.
Proof.(1): Suppose that K is a Frink ideal of P. Set L = {(F u ) u : F ⊑ K}.Now, we process to show that the pair (K, L) is an ID-pair.Let F 0 ⊑ K and S 0 ⊑ L. We consider the following two cases: Since K is a Frink ideal, by the definition of L, we have By the definition of L, there exists . ., m}.This means that (F i ) u ⊆ ↓s i for every i ∈ {1, 2, . . ., m}.Thus, we have and The combination of (i) and (ii) shows that the pair (K, L) is an ID-pair in P. Finally, we prove that the ID-pair (K, L) is nontrivial.Assume that (K, L) = (P, P).Let x, y ∈ L = P.Then, by the definition of L, there exists F y ⊑ P such that y ∈ [(F y ) u ] u , which implies that (F y ) u ⊆ ↓y.Since ({x} ∪ F y ) u ⊆ (F y ) u ⊆ ↓y, we have x ∈ ↓y, i.e., x ⩽ y.Similarly, we can prove that y ⩽ x.This means that x = y, and thus we have |P| = 1.By Definition 5, it follows that the ID-pair (K, L) is nontrivial.
(2): By a similar verification to that of (1).
Example 3. Let P be a chain, i.e., for all x, y ∈ P, either x ⩽ y or y ⩽ x.For every x ∈ P, by Definition 4 we have that the set ↓x is a Frink ideal.Obviously, by Definitions 4 and 5, the set ↑x can be selected such that the pair (↓x, ↑x) is a nontrivial ID-pair in P.
Given a poset P and a net (x i ) i∈I in the poset P, an element p ∈ P is called an eventually lower bound of the net (x i ) i∈I provided that there exists i 0 ∈ I such that x i ⩾ p for all i ≧ i 0 .An eventually upper bound of the net (x i ) i∈I is defined dually.Following the notations of Wolk [6], we also take the symbols P x and Q x to mean the set of all eventually lower bounds of the net (x i ) i∈I and the set of all eventually upper bounds of the net (x i ) i∈I , respectively.If we denote For a set X, the symbol Y ⊂ X means that Y is a proper subset of the set X, i.e., Y ⊆ X and Y ̸ = X.In the following, we always take ≥ o to represent the ordinary order on N, the set of all positive integers.Now, we can establish a correspondence between the nets and the ID-pairs: Theorem 2. Let P be a poset.Then, a pair (K, L) in P is a nontrivial ID-pair if and only if there exists a net (x i ) i∈I in P such that P x = K and Q x = L.
Proof.(⇐): Let (K, L) be a pair of subsets of the poset P. Suppose also that (x i ) i∈I is a net in the P such that P x = K and Q x = L.For every F ⊑ P x = K and every S ⊑ Q x = L, we consider the following cases: By a similar verification to that of (ii), we can also prove that F u S l ̸ = ∅, (F u S l ) l ⊆ P x and (F u S l ) u ⊆ Q x .(iv) F = {e 1 , e 2 , . . ., e m } ⊑ P x and S = {s 1 , s 2 , . . ., s n } ⊑ Q x .
Since F = {e 1 , e 2 , . . ., e m } ⊑ P x and S = {s 1 , s 2 , . . ., s n } ⊑ Q x , there exist i r , i t ∈ I such that E x (i r ) ⊆ ↑e r and E x (i t ) ⊆ ↓s t for all 1 ≤ o r ≤ o m and 1 ≤ o t ≤ o n.Take i 0 ∈ I such that i 0 ≧ i r , i t for all 1 ≤ o r ≤ o m and 1 ≤ o t ≤ o n.Then, we have By (i)-(iv), Definition 4 and Proposition 2, we conclude that the pair (P x , Q x ) = (K, L) is a nontrivial ID-pair in the poset P.
(⇒): Assume that the pair (K, L) is a nontrivial ID-pair in the poset P. We take the following cases into consideration: (v) Either the set K or the set L is infinite.
Without loss of generality, we can assume that the set K is infinite.As the ID-pair (K, L) is nontrivial, we have that F u S l ̸ = ∅ for every F ⊑ K and every S ⊑ L by Proposition 2. Let κ S F be the cardinal, linearly ordered by ≥ S F , of the set F u S l , and a S F : κ S F → F u S l be a one-to-one function from κ S F onto F u S l for every F ⊑ K and every if and only if one of the following conditions is satisfied: Now, one can readily check that the ordered set I is up-directed.Let the net (x i ) i∈I in the poset P be defined by x (F,S,λ) = a S F (λ) for every (F, S, λ) ∈ I. Next, we proceed to prove that P x = K and Q x = L. Let p ∈ P x .Then, there exists (F 1 , S 1 , λ 1 ) ∈ I such that p ∈ [E x ((F 1 , S 1 , λ 1 ))] l .Take S 2 = S 1 and F 2 ⊑ K with F 1 ⊂ F 2 .Then, we have [(F 2 ) u (S 2 ) l ] l ⊆ K since the pair (K, L) is a nontrivial ID-pair.According to the definition of I, it follows that (F 2 , S 2 , λ 2 ) ≧ (F 1 , S 1 , λ 1 ) for every Then, by the definition of I, it is easy to see that (F 0 , S 0 , λ 0 ) ∈ I for all λ 0 ∈ κ S 0 F 0 .For every (F, S, λ) ∈ I with (F, S, λ) ≧ (F 0 , S 0 , λ 0 ), by the definition of I, we have F 0 ⊆ F and S 0 ⊆ S, which implies that F u S l ⊆ (F 0 ) u (S 0 ) l = ↑q.It follows that x (F,S,λ) = a S F (λ) ∈ F u S l ⊆ ↑q for every (F, S, λ) ≧ (F 0 , S 0 , λ 0 ).This means that q ∈ P x .Hence, we conclude that K ⊆ P x .This shows that P x = K.It can be similarly proved that Q x = L. (vi) Both the sets K and L are finite.
Since the pair (K, L) is a nontrivial ID-pair in the poset P, it follows that K u L l ̸ = ∅, (K u L l ) l ⊆ K and (K u L l ) u ⊆ L. Let κ L K , well ordered by ≥ L K , denote the cardinal of the set K u L l , and a L K : κ L K → K u L l be a one-to-one function from the cardinal κ L K onto the set K u L l .Set I = {(n, λ) : n ∈ N, λ ∈ κ L K }.For any (n 1 , λ 1 ), (n 2 , λ 2 ) ∈ I, we define (n 2 , λ 2 ) ≧ (n 1 , λ 1 ) if and only if one of the following conditions is satisfied: It can easily be checked that the ordered I is up-directed.Let (x i ) i∈I be the net in the poset P by defining x (n,λ) = a L K (λ) ∈ K u L l for all λ ∈ κ L K .Now, it remains to show that K = P x and L = Q x .Let q ∈ K.Then, we have K u L l ⊆ ↑q.By the definition of the net (x i ) i∈I , it follows that x (n,λ) = a L K (λ) ∈ K u L l ⊆ ↑q for all (n, λ) ∈ I.This means that q ∈ [E x ((n, λ))] l for all (n, λ) ∈ I. Hence, we conclude that q ∈ P x , which shows that K ⊆ P x .Conversely, let p ∈ P x .Then, there exists (n 0 , λ 0 ) ∈ I such that p ∈ [E x ((n 0 , λ 0 ))] l .Since (n 0 + 1, λ) ≧ (n 0 , λ 0 ), for all λ ∈ κ L K , it follows that x (n 0 +1,λ) = a L K (λ) ∈ K u L l for all λ ∈ κ L K .This implies that K u L l ⊆ E x ((n 0 , λ 0 )).Hence, we have p ∈ [E x ((n 0 , λ 0 ))] l ⊆ (K u L l ) l ⊆ K.This shows that P x ⊆ K. Therefore, P x = K.A similar verification can show that Q x = L.By (v) and (vi), we can conclude that there exists a net (x i ) i∈I in the poset P such that P x = K and Q x = L. Thus, the proof is completed.
Consider the net (x n ) n∈N defined by where the up-directed set N is the set of all positive integers in its usual order.By the definition of the net (x n ) n∈N , we have P x = ∅ and Q x = {⊤}.On the other hand, it follows from Definition 4 and Definition 5 that the pair (∅, {⊤}) is a nontrivial ID-pair.This demonstrates Theorem 2 in the case.
The combination of Proposition 1 and Theorems 1 and 2 indicates that the eventually lower bounds P x and eventually upper bounds Q x of a net (x i ) i∈I are precisely a Frink ideal and a dual Frink ideal, respectively (see Corollary 1).However, they are not independent.Theorem 2 clarifies the correlation between the Frink ideal P x and the dual Frink ideal Q x from the point of view of order; that is, the Frink ideal P x and the dual Frink ideal Q x must be matched as a nontrivial ID-pair.Also, this is the initial motivation of introducing the local Frink ideal (the dually local Frink ideal) and ID-pair for posets in the sequel.

Corollary 1 ([6]
).Let P be a poset and K, L ⊆ P.Then, (1) The subset K is a Frink ideal if and only if P x = K for some net (x i ) i∈I in the poset P; (2) The subset L is a dual Frink ideal if and only if Q y = L for some net (y j ) j∈J in the poset P.

ID-Doubly Continuous Posets
Then, it is easy to see that M ⊆ M N .Now, we proceed to prove that M N ∈ L(N).Let F ⊑ M N and S ⊑ N. We should consider the following cases: Since F = ∅, it follows that (M a ) u S l ⊆ F u S l = S l for all M a ⊑ M, which implies that (F u S l ) l ⊆ [(M a ) u S l ] l for all M a ⊑ M.This means that (M a ) u S l ⊆ ↑p ′ for all p ′ ∈ (F u S l ) l .Hence, we infer that (F u S l ) l ⊆ M L .(ii) F = {e 1 , e 2 , . . ., e m } ̸ = ∅.
This implies that (F u S l ) l ⊆ [(M 0 ) u (N 0 ) l ] l , which concludes that (M 0 ) u (N 0 ) l ⊆ ↑p for all p ∈ (F u S l ) l .Hence, by Proposition 3, we have (F u S l ) l ⊆ IG N (M).
According to (i)-(iv) and Definition 4, we infer that IG N (M) ∈ L(DG M (N)).We turn to define the ID-double continuity for posets.Since the ID-double continuity has a close relationship to Property A, proposed by Wolk, we review Property A and its dual form for posets in the following:

Definition 6 ([6]
).A poset P has Property A if, for every non-normal Frink ideal K with sup K = x ∈ P, there exists an up-directed subset K U ⊆ K such that sup K U = x.Dually, a poset P has Property DA if, for every non-normal dual Frink ideal L with inf L = y ∈ P, there exists a down-directed subset L D ⊆ L such that inf L D = y.Definition 7. A poset P is called an ID-doubly continuous poset if, for every ID-pair (K, L) in the poset P with sup K = x = inf L ∈ P, there exist an up-directed subset K U ⊆ K and a down-directed subset L D ⊆ L such that sup K U = x = inf L D .

Example 5. (1) Every finite poset is ID-doubly continuous;
(2) Every lattice is ID-doubly continuous.Suppose that P is a finite poset and (K, L) is an ID-pair with sup K = x = inf L ∈ P.Then, we have that K, L ⊑ P and K u L l = {x}.Since the pair (K, L) is an ID-pair, by Definition 4 and Definition 5, it follows that (K u L l ) l = ↓x ⊆ K and (K u L l ) u = ↑x ⊆ L, which implies that x ∈ K and x ∈ L. This means that the singleton {x} is an up-directed subset of K and also a down-directed subset of L such that sup{x} = x = inf{x}.So, by Definition 7, the finite poset P is ID-doubly continuous.
The fact that every lattice is ID-doubly continuous can also be readily checked by Definition 7.
Proposition 4. Let P be a poset.If the poset P has Property A and Property DA, then it is an ID-doubly continuous poset.
Proof.Let (K, L) be an ID-pair in the poset P with sup K = x = inf L ∈ P.Then, by Proposition 1, the set K is a Frink ideal.If x ∈ K, then we have that {x} is an up-directed subset of K and sup{x} = x.If x / ∈ K, then K is a non-normal Frink ideal since x ∈ (K u ) l = ↓x ̸ = K.By Property A, it follows that there exists an up-directed subset K U ⊆ K such that sup K U = x.A similar verification can prove that there exists a down-directed subset L D ⊆ L such that inf L D = x.Hence, the poset P is ID-doubly continuous.
In general, an ID-doubly continuous poset may not possess Property A and Property DA.For such an example, one can refer to Example 7 in Section 4. Now, we arrive at the main result: Proof.(⇒): Suppose that a poset P is ID-doubly continuous.To prove the equivalence between the o-convergence and o 2 -convergence, it suffices to show that, for every net (x i ) i∈I in the poset P, we have Then, by Definition 2, there exist subsets M, N ⊆ P such that sup M = x = inf N, and, for every m ∈ M and every n ∈ N, m ⩽ x i ⩽ n holds eventually.This means that M ⊆ P x and N ⊆ Q x , which implies that IG N (M) ⊆ P x and DG M (N) ⊆ Q x by Remark 1 and Theorem 2. According to Lemma 1 and 2, it follows that (IG N (M), DG M (N)) is an ID-pair with sup IG N (M) = x = inf DG M (N).Since the poset P is ID-doubly continuous, we have that sup (⇐): Assume that the o-convergence and o 2 -convergence in a poset P are equivalent.Let (K, L) be an ID-pair in the poset P with sup K = x = inf L ∈ P. Since x ∈ F u S l ̸ = ∅ for all F ⊑ K and S ⊑ L, the pair (K, L) is a nontrivial ID-pair by Proposition 2. According to Theorem 2, there exists a net (x i ) i∈I in the poset P such that K = P x and L = Q x .Thus, we have (x i ) i∈I o 2 − → x.By the hypothesis, it follows that (x i ) i∈I o − → x.This means that sup K U = x = inf L D for some up-directed subset K U ⊆ K = P x and some down-directed subset L D ⊆ L = Q x .So, the poset P is an ID-doubly continuous poset.

Example
In this section, we mainly give an example to clarify the following facts: (1) A Frink ideal K of a poset P may not be a local Frink ideal in every nonempty subset L of P; Dually, a dual Frink ideal K need not be a dually local Frink ideal in every nonempty subset K of P. (2) An ID-doubly continuous poset fails to satisfy Property A and Property DA.Let K = {b 1 , b 2 , . . ., b n , . . .}.Then, the set K is a non-normal Frink ideal by Definition 3 and the definition of the poset P.However, the poset P does not process Property A since we can easily see that sup K = x, and sup K U ̸ = x for every up-directed subset K U ⊆ K.We next show that K / ∈ L(L) for any nonempty subset L of the poset P by analyzing the following cases: The combination of (i) and (ii) shows that the set K is not a local Frink ideal in any nonempty subset L of the poset P. Now, we are going to verify that P is an ID-doubly continuous poset.Let (K ′ , L ′ ) be an ID-pair in the poset P with sup K ′ = p = inf L ′ .We consider the following cases: (iii) p = a i (resp.p = b i , c i , d i ) for some i ∈ N.
It is easy to see, by the definition of the poset P, that there exist K 0 ⊑ K ′ and L 0 ⊑ L ′ such that sup K 0 = inf L 0 = p = a i .Since the pair (K ′ , L ′ ) is an ID-pair, we have Since inf L ′ = x, one can readily check that L ′ = {x}.Take K ′ U = L ′ D = {x}.Then, we have that the set K ′ U is an up-directed subset of the set K ′ , the set L ′ D is a down-directed subset of the set L ′ and sup K If the set K ′ {a 1 , a 2 , . . .} is finite, then we have that the set K ′ {b 1 , b 2 , . . .} is also finite.Otherwise, suppose that the set K ′ {b 1 , b 2 , . . .} is infinite.Then, there exists {b i 1 , b i 2 , . . .} ⊆ K ′ .Since the pair (K ′ , L ′ ) is an ID-pair in the poset P, we have that a i k ∈ ({b i 1 , b i k } u {x} l ) l for every k ∈ N with k ≥ o 2. This means that {a i 2 , a i 3 , . . .} ⊆ K ′ {a 1 , a 2 , . . .}, contradicting the hypothesis that the set K ′ {a 1 , a 2 , . . .} is finite.Let {a j 1 , a j 2 , . . ., a j m } = K ′ {a 1 , a 2 , . . .} and {b , . . .}, and let j 0 = max{j 1 , j 2 , . . ., j m } and i 0 = max{i 1 , i 2 , . . ., i n }.Since sup K ′ = x, we also take the following cases into consideration: In this case, we can return the verification to Case (iv).(v2) c i 0 ∈ K ′ for some i 0 ∈ N with i 0 < o j 0 .
In this case, if we take K 0 = {a j 0 , c i 0 } and L 0 = {x}, then we have K 0 ⊑ K ′ and L 0 ⊑ L ′ with sup K 0 = x = inf L 0 .By a similar verification to that of Case (iii), there exist an up-directed subset K ′ U of the set K ′ and a down-directed subset L ′ D of the set L ′ such that sup K ′ U = x = inf L ′ D .(v3) c i 1 ∈ K ′ for some i 1 ∈ N with i 1 < o i 0 .
In this case, if we take K 0 = {b i 0 , c i 1 } and L 0 = {x}, then we have K 0 ⊑ K ′ and L 0 ⊑ L ′ with sup K 0 = x = inf L 0 .By a similar verification to that of (iii), there exist an up-directed subset K ′ U of the set K ′ and a down-directed subset L ′ D of the set L ′ such that sup K ′ U = x = inf L ′ D .(v4) c i 2 , c i 3 ∈ K ′ for some i 2 , i 3 ∈ N.
In this case, if we take K 0 = {c i 2 , c i 3 } and L 0 = {x}, then we have K 0 ⊑ K ′ and L 0 ⊑ L ′ with sup K 0 = x = inf L 0 .By a similar verification to (iii), there exist an up-directed subset K ′ U of the set K ′ and a down-directed subset L ′ D of the set L ′ such that sup K ′ U = x = inf L ′ D .(vi) p = x and c i ∈ K ′ for some i ∈ N.
Since the pair (K ′ , L ′ ) is an ID-pair, we have a i ∈ ({c i } u {x} l ) l ⊆ K ′ .So, we can return the verification to Case (v).(vii) p = x and b i ∈ K ′ for some i ∈ N.
Since sup K ′ = x, there exists j ∈ N such that a j ∈ K ′ (resp.c j ∈ K ′ , x ∈ K ′ ).So, we can return the verification to Case (v) (resp.Case (vi), Case (iv)).
By Definition 7 and the combination of Cases (iii)-(vii), we conclude that the poset P is an ID-doubly continuous poset.

Discussion
This paper introduced the notion of ID-pairs in posets.It was shown that the set of all eventually lower bounds and the set of all eventually upper bounds of a net in a given poset can be precisely paired to be an ID-pair.This result provides a potential approach for dealing with the general nets in posets, since some kinds of order convergent nets, such as the o-convergent nets and o 2 -convergent nets, are uniquely determined by their eventually lower bounds sets and eventually upper bounds sets.
Furthermore, in order to characterize these posets in which the o-convergence and o 2 -convergence are equivalent, the concept of ID-doubly continuous posets is proposed.It is proved that the equivalence of the o-convergence and o 2 -convergence in a poset is equivalent to the ID-double continuity of the poset.This result provides a sufficient and necessary condition for the o-convergence and o 2 -convergence to be equivalent.
However, it may be complicated to verify the ID-double continuity for some posets, such as the poset in Example 7. On the contrary, the lattices, a special kind of poset, can be easily proved to be ID-double continuous.This indicates that the ID-double continuity has some close relationships with some special kinds of posets.These relationships deserve further investigation.

Proposition 3 .
Given a poset P and M, N ⊆ P, let L M (N) = {K ∈ L(N) : M ⊆ K}.Then, one can readily verify by Definition 4 that the intersection L M (N) contains the set M and is again a local Frink ideal in the set N. This local Frink ideal is called the local Frink ideal generated by the set M and denoted by IG N (M).The dually local Frink ideal generated by the set N is defined dually, and denoted by DG M (N).Next, we clarify the structure of IG N (M) and DG M (N): Let P be a poset and M, N ⊆ P.Then, (1) IG N (M) = {p ∈ P :

Lemma 2 .
Let P be a poset and M, N ⊆ P. If sup M = x = inf N ∈ P, then we have sup IG N (M) = x = inf DG M (N).Proof.Let sup M = x = inf N ∈ P.Then, one can readily check, by Proposition 3, that M ⊆ IG N (M) ⊆ ↓x and N ⊆ DG M (N) ⊆ ↑x.It follows that sup IG N (M) = x = inf DG M (N).

Theorem 3 .
A poset P is ID-doubly continuous if and only if the o-convergence and o 2 -convergence in the poset P are equivalent.

By Example 5 andCorollary 2 .
Theorem 3, we immediately have the following: Example 6. (1) In every finite poset, the o-convergence and the o 2 -convergence are equivalent; (2) In every lattice, the o-convergence and the o 2 -convergence are equivalent.By Proposition 4 and Theorem 3, or by Definition 2 and Theorem 2 and 5 in [6], we readily have the following: If a poset P has Property A and Property DA, then the o-convergence and o 2convergence in the poset P are equivalent.

Example 7 .Figure 1 .
Figure 1.The diagram for the poset in Example 7.