 Research
 Open Access
Convergence in \(s_{2}\)quasicontinuous posets
 Xiaojun Ruan^{1, 2}Email author and
 Xiaoquan Xu^{3}
 Received: 11 August 2015
 Accepted: 16 February 2016
 Published: 29 February 2016
Abstract
In this paper, we present one way to generalize \({\mathcal {S}}\)convergence and \({\mathcal {GS}}\)convergence of nets for arbitrary posets by use of the cut operator instead of joins. Some convergence theoretical characterizations of \(s_2\)continuity and \(s_2\)quasicontinuity of posets are given. The main results are: (1) a poset P is \(s_2\)continuous if and only if the \({\mathcal {S}}\)convergence in P is topological; (2) P is \(s_2\)quasicontinuous if and only if the \({\mathcal {GS}}\)convergence in P is topological.
Keywords
 \(s_2\)Continuous poset
 \(s_2\)Quasicontinuous poset
 Weak Scott topology
 \({\mathcal {S}}\)Convergence
 \({\mathcal {GS}}\)Convergence
Mathematics Subject Classification
 06B35
 06B75
 54F05
Background
The theory of continuous domains, due to its strong background in computer science, general topology and logic has been extensively studied by people from various areas (see Abramsky and Jung 1994; Gierz et al. 1980, 2003). Since many models may not be dcpos, an important direction in the study of continuous domains is to extend the theory of continuous domains to that of posets as much as possible (see Huang et al. 2009; Lawson and Xu 2004; Mislove 1999; Markowsky 1981; Mao and Xu 2006; Venugopalan 1990; Zhang 1993; Zhang and Xu 2015). It has turned out to be very fruitful for many categorical and topological developments generalizing the theory of continuous domains, but it is still rather restrictive, taking into consideration only the case of existing a join. Furthermore, it fails to be completioninvariant, that is, the normal completion of a continuous poset is not always a continuous lattice, which means some useful information of subsets whose joins do not exist has been thrown away in some sense. In 1981, Erné introduced the concept of \(s_2\)continuous posets in terms of the cut operator instead of joins. The notion of \(s_2\)continuity admits to generalize most important characterizations of continuity from dcpos to arbitrary posets and has the advantage that not even the existence of directed joins has to be required. As a generalization of \(s_2\)continuity, the concept of \(s_2\)quasicontinuity was introduced by Zhang and Xu (2015), their basic idea is to generalize the way below relation between the points to the case of sets. It was proved that \(s_2\)quasicontinuous posets equipped with the weak Scott topologies are precisely the hypercontinuous lattices.
Various kinds of convergent classes in posets were studied in Gierz et al. (2003), Zhao and Zhao (2005), Zhou and Zhao (2007), Wang and Zhao (2013), Zhao and Li (2006), Zhou and Li (2013), Chen and Kou (2014). By different convergence, not only are many notions of continuity characterized, but also they make order and topology across each other. In Gierz et al. (2003), the concept of \({\mathcal {S}}\)convergence for dcpos was introduced by Scott to characterize continuous domains. It was proved that for a dcpo, the \({\mathcal {S}}\)convergence is topological if and only if it is a continuous domain. In this paper, making a slight modification of \({\mathcal {S}}\)convergence, we generalize the concept of \({\mathcal {S}}\)convergence to the setting of arbitrary poset by means of the cut operator instead of joins. It is proved that the \({\mathcal {S}}\)convergence in a poset is topological if and only if the poset is \(s_2\)continuous. Although Erné investigated the \({\mathcal {S}}\)convergence through filter, we would give a satisfactory sufficient and necessary condition for the \({\mathcal {S}}\)convergence to be topological by the net, which is more simple and direct than the filter. In order to characterize the \(s_2\)quasicontinuity we shall also consider another type of \({\mathcal {GS}}\)convergence in a poset, and get the desired result that the \({\mathcal {GS}}\)convergence in a poset is topological if and only if the poset is \(s_2\)quasicontinuous.
Preliminaries
Let P be a partially ordered set (poset, for short). We put \(P^{(<\omega )}=\{F\subseteq P : F\ \text{ is } \text{ finite }\}\). For all \(x\in P\), \(A\subseteq P\), define \(\downarrow x=\{y\in P: y\le x\}\) and \(\downarrow A=\{x\in P: x\le a\) for some \(a\in A\}\); \(\uparrow x\) and \(\uparrow A\) are defined dually. \(A^\uparrow \) and \(A^\downarrow \) denote the sets of all upper and lower bounds of A, respectively. A cut operator \(\delta \) is defined by \(A^{\delta } =(A^\uparrow )^\downarrow \) for every \(A\subseteq P\). Notice that whenever A has a join (supremum) then \(x \in A^{\delta }\) means \(x\le \vee A\).
For a poset P, a subset U of P is called Scott open if (i) \(U=\uparrow U\), and (ii) if D is a directed set of P and \(\vee D\in U\) whenever \(\vee D\) exists, then there is some \(d\in D\) with \(d\in U\). It is easy to see that all the Scott open subsets of P form a topology, which we shall call the Scott topology, denoted by \(\sigma (P)\).
Let P be a poset. We order the collection of nonempty subsets of P by \(G\le H\) if \(\uparrow H\subseteq \uparrow G\). We say that a nonempty family of sets is directed if given \(F_1\), \(F_2\) in the family, there exists F in the family such that \(F_1\), \(F_2\le F\), i.e., \(F\subseteq \uparrow F_1\cap \uparrow F_2\). For nonempty subsets F and G of P, we say F approximates G if for every directed subset \(D\subseteq P\), whenever \(\vee D\) exists, \(\vee D\in \uparrow G\) implies \(d\in \uparrow F\) for some \(d\in D\). A dcpo P is called a quasicontinuous domain if for all \(x\in P\), \(\uparrow x\) is the directed (with respect to reverse inclusion) intersection of sets of the form \(\uparrow F\), where F approximates \(\{x\}\) and F is finite. In particular, a poset P is called a continuous poset if for all \(x\in P, x\) is the directed supremum of sets of the form y, where \(\{y\}\) approximates \(\{x\}\).
Definition 1
 (1)
For any \(x, y\in P\), we say that x is way below y, written \(x\ll y\) if for all directed sets \(D\subseteq P\) with \(y\in D^\delta \), there exists \(d\in D\) such that \(x\le d\). The set \(\{y\in P:y\ll x\}\) will be denoted by \(\Downarrow x\) and \(\{y\in P:x\ll y\}\) denoted by \(\Uparrow x\).
 (2)
P is called \(s_2\)continuous if for all \(x\in P\), \(x\in (\Downarrow x)^\delta \) and \(\Downarrow x\) is directed.
Indeed, we have \(x=\vee \Downarrow x\) iff \(x\in (\Downarrow x)^\delta \) by \(\Downarrow x\subseteq \downarrow x\).
Let us note that an \(s_2\)continuous poset is continuous, but the converse may not be true:
Example 1
The following lemma shows that the \(s_2\)continuous poset has the interpolation property.
Lemma 1
(Erné 1981) Let P be an \(s_2\)continuous poset and \(x, y\in P\). If \(x\ll y\), then there is some \(z\in P\) such that \(x\ll z\ll y\).
Definition 2
The collection of all weak Scott open subsets of P forms a topology, it will be called the weak Scott topology of P and will be denoted by \(\sigma _2(P)\).
Remark 1
\(\sigma _2(P)\) is always coarser than \(\sigma (P)\), and both topologies coincide on dcpos.
Example 2

\(\downarrow a_{0}=\{a_{0}\}\cup B\),

\(\downarrow a_{n}=\{b_{m}: m<n\}(n\in {\mathbf {N}}, n\ne 2)\),

\(\downarrow a_{2}=\{b_{0}, b_{1}\}\cup C\),

\(\downarrow b_{n}=\{b_{n}\}(n\in \mathbf {N_{0}})\),

\(\downarrow c_{n}=\{c_{m}: m\le n\}(n\in {\mathbf {N}})\),

\(x\le y\Leftrightarrow x\in \downarrow y\).
Then \(\uparrow b_{0}\) is open in \(\sigma (P)\) but not in \(\sigma _2(P)\) since \(C=\{c_{n}: n\in {\mathbf {N}}\}\) is a directed lower set with \(b_{0}\in C^{\delta }\cap \uparrow b_{0}\ne \emptyset \) while \(C\cap \uparrow b_{0}=\emptyset \). Hence in this example, we have \(\sigma _2(P)\) is proper contained in \(\sigma (P)\).
Definition 3
(Zhang and Xu 2015) Let P be a poset and G, \(H\subseteq P\), we say that G is way below H and write \(G\ll H\) if for all directed sets \(D\subseteq P\), \(\uparrow H\cap D^\delta \ne \emptyset \) implies \(\uparrow G\cap D\ne \emptyset \). We write \(G\ll x\) for \(G\ll \{x\}\) and \(y\ll H\) for \(\{y\}\ll H\). The set \(\{x\in P:F\ll x\}\) will be denoted \(\Uparrow F\).
Definition 4
(Zhang and Xu 2015) Let P be a poset. P is called \(s_2\)quasicontinuous if for each \(x\in P\), \(w(x)=\{F\subseteq P:F\in P^{(<\omega )}\) and \(F\ll x\}\) is directed and \(\uparrow x=\bigcap \{\uparrow F:F\in w(x)\) }.
Obviously, the \(s_2\)continuous is \(s_2\)quasicontinuous, but the converse may not be true.
Example 3
(Zhang and Xu 2015) Let \(P=\{a\}\cup \{a_n:n\in {\mathbf {N}}\}\). The partial order on P is defined by setting \(a_n<a_{n+1}\) for all \(n\in {\mathbf {N}}\), and \(a_1< a\). Then P is an \(s_2\)quasicontinuous poset which is not \(s_2\)continuous.
The following theorem shows that the \(s_2\)quasicontinuous poset has the interpolation property.
Theorem 1
(Zhang and Xu 2015) Let P be an \(s_2\)quasicontinuous poset and \(K\in P^{(<\omega )}\), \(H\subseteq P\). If \(H\ll K\), then there exists a finite set F such that \(H\ll F\ll K\).
Lemma 2
(Zhang and Xu 2015) Let \({\mathcal{F}}\) be a directed family of nonempty finite sets in a poset. If \(G\ll x\) and \(\bigcap \nolimits _{F\in {\mathcal {F}}}\uparrow F\subseteq \uparrow x\), then \(F\subseteq \uparrow G\) for some \(F\in {\mathcal {F}}\).
Lemma 3
 (1)
For any nonempty set H in P, \(\Uparrow H=int_{\sigma _2(P)}\uparrow H\).
 (2)
A subset U of P is weak Scott open iff for each \(x\in U\) there exists a finite \(F\ll x\) such that \(\uparrow F\subseteq U\). The sets \(\{\Uparrow F:F\in P^{(<\omega )}\}\) form a basis for the weak Scott topology \(\sigma _2(P)\).
The following lemma is wellknown Rudin Lemma.
Lemma 4
(Gierz et al. 2003) Let \({\mathcal{F}}\) be a directed family of nonempty finite subsets of a poset P. Then there exists a directed set \(D\subseteq \bigcup \nolimits _{F\in {\mathcal {F}}}F\) such that \(D\cap F\ne \emptyset \) for all \(F\in {\mathcal {F}}\).
\({\mathcal {S}}\)Convergence in \(s_2\)continuous posets
In this section, the concept of \({\mathcal {S}}\)convergence in a poset is introduced. It is proved that the poset P is \(s_2\)continuous if and only if the \({\mathcal {S}}\)convergence in P is topological.
Definition 5
 (1)
A point \(y\in P\) is called an eventual lower bound of a net \((x_{j})_{j\in J}\) in P, if there exists \(k\in J\) such that \(y\le x_{j_{}}\) for all \(j\ge k\);
 (2)
A point \(x\in P\) is called an \({\mathcal {S}}\)limit of the net \((x_{j})_{j\in J}\) if there exists some directed set D of eventual lower bounds of a net \((x_{j})_{j\in J}\) such that \(x\in D^{\delta }\). We also say \((x_{j})_{j\in J}\) \({\mathcal {S}}\) converges to x and write \(x\equiv _{{\mathcal {S}}}\)lim \(x_{j}\).
Let \({\mathcal {S}}\) denote the class of those pairs \(((x_{j})_{j\in J}, x)\) with \(x\equiv _{{\mathcal {S}}}\)lim \(x_{j}\), then \(\mathcal {O({\mathcal {S}})}=\{U\subseteq P:\) whenever \(((x_{j})_{j\in J}, x)\in {\mathcal {S}}\) and \(x\in U\), then eventually \(x_{j}\in U\}\) is a topology.
Remark 2
For dcpos the preceding definition of \({\mathcal {S}}\)limit is equivalent to the standard one (Gierz et al. 2003, Definition II1.1) (as in a dcpo, \(x\in D^{\delta }\) means \(x\in \downarrow \vee D\)).
Lemma 5
Let P be a poset, then \(\mathcal {O({\mathcal {S}})}=\sigma _2(P)\).
Proof
First, suppose that \(U\in \mathcal {O({\mathcal {S}})}\). To prove \(U=\uparrow U\), assume that \(u\in U\) and \(u\le x\). Then \(u\le x\equiv _{S}\) lim x with the constant net (x) with value x. So by the definition \(((x),u)\in {\mathcal {S}}\). Since we have \(u\in U\in \mathcal {O({\mathcal {S}})}\), we conclude from the definition of \(\mathcal {O({\mathcal {S}})}\) that the net (x) must be eventually in U. This means \(x\in U\). In order to show that \(D^{\delta }\cap U\ne \emptyset \Rightarrow U\cap D\ne \emptyset \) for each directed set \(D\subseteq P\), let \(x\in D^{\delta }\cap U\ne \emptyset \) . Consider the net \((x_{d})_{d\in D}\) with \(x_{d}=d\). Now since \(((x_{d})_{d\in D}, x)\in {\mathcal {S}}\), we conclude that \(d=x_{d}\) is eventually in U; whence \(D\cap U\ne \emptyset \).
Conversely, suppose that \(U\in \sigma _{2}(P)\). For any \(((x_{j})_{j\in J}, x)\in {\mathcal {S}}\) with \(x\in U\), by the definition of \({\mathcal {S}}\),we have \(x\in D^{\delta }\) for some directed set D of eventual lower bounds of the net \((x_{j})_{j\in J}\). Now \(x\in D^{\delta }\cap U\), and then \(u\in D\) for some \(u\in U\) by the definition of \(\sigma _{2}(P)\). By definition \(u\le x_{j}\) for all \(k\le j\) for some \(k\in J\). By \(U=\uparrow U\), \(x_{j}\in U\) holds eventually. Hence \(U\in \mathcal {O({\mathcal {S}})}\).\(\square \)
Lemma 6
Let P be an \(s_2\)continuous poset. Then for any \(x\in P\), \(\Uparrow x\in \sigma _{2}(P)\).
Proof
It follows from Lemma 1.\(\square \)
Lemma 7
Let P be a poset and \(y\in int_{\sigma _{2}(P)}\uparrow x\). Then \(x\ll y\), where \(int_{\sigma _{2}(P)}\uparrow x\) denotes the interior of \(\uparrow x\) with respect to the weak Scott topology \(\sigma _{2}(P)\).
Proof
Let \(y\in int_{\sigma _{2}(P)}\uparrow x\). For any directed set D with \(y\in D^{\delta }\), we have \(D^{\delta }\cap int_{\sigma _{2}(P)}\uparrow x\ne \emptyset \), and whence \(int_{\sigma _{2}(P)}\uparrow x\cap D\ne \emptyset \). Thus there is \(d\in int_{\sigma _{2}(P)}\uparrow x\cap D\). Now we have \(x\le d\) and \(d\in D\). Therefore \(x\ll y\).\(\square \)
Proposition 1
Let P be an \(s_2\)continuous poset. Then \(x\equiv _{{\mathcal {S}}}lim\) \(x_{j}\) if and only if the net \((x_{j})_{j\in J}\) converges to the element x with respect to the weak Scott topology \(\sigma _{2}(P)\). That is, the \({\mathcal {S}}\)convergence is topological.
Proof
The necessity follows from Lemma 5. Now suppose that the net \((x_{j})_{j\in J}\) converges to an element x with respect to the weak Scott topology. For all \(y\in \Downarrow x\), we have \(x\in \Uparrow y\in \sigma _{2}(P)\) by Lemma 6. Thus there is \(k\in J\) such that \(x_{j}\in \Uparrow y\) for all \(j\ge k\). Since P is \(s_2\)continuous, \(x\in (\Downarrow x)^{\delta }\) and \(\Downarrow x\) is directed. Hence we have \(((x_{j})_{j\in J}, x)\in {\mathcal {S}}\), that is, \(x\equiv _{\mathcal {S}}\)lim \(x_{j}\).\(\square \)
Proposition 2
Let P be a poset. If the \({\mathcal {S}}\)convergence is topological, then P is \(s_2\)continuous.
Proof
By Lemma 5, the topology induced by \({\mathcal {S}}\)convergence is the weak Scott topology. So if the \({\mathcal {S}}\)convergence is topological, then we must have \(x\equiv _{\mathcal {S}}\)lim \(x_{j}\) if and only if the net \((x_{j})_{j\in J}\) converges to the element x in the weak Scott topology. For any \(x\in P\), let \(J=\{(U, n, a)\in N(x)\times {\mathbb {N}}\times P: a\in U\}\), where N(x) consists of all weak Scott open sets containing x, and define an order on J to be the lexicographic order on the first two coordinates, i.e., \((U, m, a)\le (V, n, b)\) if and only if V is proper subset of U or \(U=V\) and \(m\le n\). Put \(x_{j}=a\) for each \(j=(U, m, a)\in J\). Then it is not difficult to check that the net \((x_{j})_{j\in J}\) converges to x with respect to the weak Scott topology, and hence \(x\equiv _{\mathcal {S}}\)lim \(x_{j}\). Thus there is a directed set D of eventual lower bounds of the net \((x_{j})_{j\in J}\) such that \(x\in D^{\delta }\). If \(d\in D\), then there is \(k=(U, m, a)\in J\) such that \((V, n, b)=j\ge k\) implies \(d\le x_{j}=b\). Specially we have \((U, m+1, b)\ge (U, m, a)=k\) for all \(b\in U\). Therefore \(x\in U\subseteq \uparrow d\). It follows that \(D\subseteq \downarrow x\) and \(x\in int_{\sigma _{2}(P)}\uparrow d\). By Lemma 7 \(d\ll x\), and then \(D\subseteq \Downarrow x\). Thus \(x\in D^{\delta }\subseteq (\Downarrow x)^{\delta }\). Obviously, \(\Downarrow x\) is directed. Hence P is \(s_2\)continuous.\(\square \)
From Propositions 1 and 2, we immediately have:
Theorem 2
 (1)
P is \(s_2\)continuous;
 (2)
The \({\mathcal {S}}\)convergence in P is topological for the weak Scott topology, that is, for all \(x\in P\) and all nets \((x_{j})_{j\in J}\) in P, \(x\equiv _{\mathcal {S}}\)lim \(x_{j}\) if and only if \((x_{j})_{j\in J}\) converges to the element x with respect to the weak Scott topology.
Corollary 1
 (1)
P is a domain;
 (2)
The \({\mathcal {S}}\)convergence in P is topological for the Scott topology, that is, for all \(x\in P\) and all nets \((x_{j})_{j\in J}\) in P, \(x\equiv _{\mathcal {S}}lim\) \(x_{j}\) if and only if \((x_{j})_{j\in J}\) converges to the element x with respect to the Scott topology.
\({\mathcal {GS}}\)Convergence in \(s_2\)quasicontinuous posets
In this section, the concept of \({\mathcal {GS}}\)convergence in a poset is introduced. It is proved that the poset P is \(s_2\)quasicontinuous if and only if the \({\mathcal {GS}}\)convergence in P is topological.
Definition 6
Let P be a poset and \((x_{j})_{j\in J}\) a net in P. \(F\subseteq P\) is called a quasieventual lower bound of a net \((x_{j})_{j\in J}\) in P, if F is finite and there exists \(k\in J\) such that \(x_{j}\in \uparrow F\) for all \(j\ge k\).
Obviously, an eventual lower bound is the quasieventual lower bound.
Definition 7
Let P be a poset and \((x_{j})_{j\in J}\) a net in P. x is called a \({\mathcal {GS}}\)limit of the net \((x_{j})_{j\in J}\) if there exists a directed family \({\mathcal {F}}=\{F\subseteq P: F\) is finite} of quasieventual lower bounds of the net \((x_{j})_{j\in J}\) in P such that \(\bigcap _{F\in {\mathcal {F}}}\uparrow F\subseteq \uparrow x\). We also say \((x_{j})_{j\in J}\) quasi \({\mathcal {S}}\) converges to x and write \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\).
Lemma 8
An \({\mathcal {S}}\)limit of the net \((x_{j})_{j\in J}\) must be a \({\mathcal {GS}}\)limit of the net \((x_{j})_{j\in J}\).
Proof
Let P be a poset and \((x_{j})_{j\in J}\) a net with \(x\equiv _{\mathcal {S}}\)lim \(x_{j}\). Then there is a directed set D of eventual lower bounds of the net \((x_{j})_{j\in J}\) with \(x\in D^{\delta }\). Let \({\mathcal {F}}=\{\{d\}: d\in D\}\), then \({\mathcal {F}}\) is a directed family of quasieventual lower bounds of the net \((x_{j})_{j\in J}\) and \(D^{\uparrow }=\bigcap \{\uparrow d: d\in D\}\subseteq \uparrow x\). Thus \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\).\(\square \)
Remark 3
A \({\mathcal {GS}}\)limit of the net \((x_{j})_{j\in J}\) may not be an \({\mathcal {S}}\)limit of the net \((x_{j})_{j\in J}\).
Example 4
 (1)
\(\forall x\in P, x\le \top \);
 (2)
\(\forall x, y\in {\mathbf {N}}, x\le y \) if x is less than or equal to y according to the usual order on natural numbers.
Then P is \(s_2\)quasicontinuous but not \(s_2\)continuous. Also for all \(n\in {\mathbf {N}}, \{z, n\}\ll z\) and \(\uparrow z=\bigcap _{n\in {\mathbf {N}}}\uparrow \{z, n\}\). Let \(x_{2n}=n, x_{2n+1}=z\), then \((x_{j})_{j\in {\mathbf {N}}}\) is a net and \(\{z, n\}\) is a quasieventual lower bound of it. Hence \(z\equiv _{{\mathcal {GS}}}\)lim \(x_{n}\). It is not difficult to check that \(z\le x_{n}\) does not hold eventually. Thus z is not an \({\mathcal {S}}\)limit of the net \((x_{n})_{n\in {\mathbf {N}}}\).
Proposition 3
Let \({\mathcal {F}}\) be a directed family of nonempty finite sets in a poset P. If \(x\in U\in \sigma _2(P)\) and \(\bigcap _{F\in {\mathcal {F}}}\uparrow F\subseteq \uparrow x\), then \(F\subseteq U\) for some \(F\in {\mathcal {F}}\).
Proof
Suppose not, then the collection \(\{F\backslash U: F\in {\mathcal {F}}\}\) is a directed family of nonempty finite sets. By Lemma 4, there is some directed set \(D\subseteq \bigcup \{F\backslash U: F\in {\mathcal {F}}\}\) such that \(D\cap (F\backslash U)\ne \emptyset \) for all \(F\in {\mathcal {F}}\). Then \(D^{\uparrow }=\bigcap _{d\in D}\uparrow d\subseteq \bigcap _{F\in {\mathcal {F}}}\uparrow (F\backslash U)\subseteq \bigcap _{F\in {\mathcal {F}}}\uparrow F\subseteq \uparrow x\). Thus \(x\in (D^{\uparrow })^{\downarrow }=D^{\delta }\). Now we have \(x\in D^{\delta }\cap U\ne \emptyset \), and hence \(D\cap U\ne \emptyset \) by the definition of the weak Scott open set, that is, there is some \(d\in D\) with \(d\in U\). But this contradicts \(d\in F\setminus U\) for some \(F\in {\mathcal {F}}\).\(\square \)
Let \({\mathcal {GS}}\) denote the class of those pairs \(((x_{j})_{j\in J}, x)\) with \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\), then \(\mathcal {O({\mathcal {GS}})}=\{U\subseteq P:\) whenever \(((x_{j})_{j\in J}, x)\in {\mathcal {GS}}\) and \(x\in U\), then eventually \(x_{j}\in U\}\) is also a topology.
Though \({\mathcal {S}}\)limit and \({\mathcal {GS}}\)limit of the net \((x_{j})_{j\in J}\) are different, they may generate the same topology.
Proposition 4
Let P be a poset, then \(\mathcal {O({\mathcal {GS}})}=\mathcal {O(S)}=\sigma _2(P)\).
Proof
By Lemma 5, we only need to show that \(\mathcal {O({\mathcal {GS}})}=\sigma _2(P)\). By Lemma 8, we have \({\mathcal {S}}\subseteq {\mathcal {GS}}\), so \(\mathcal {O({\mathcal {GS}})}\subseteq \sigma _2(P)\). Conversely, let \(U\in \sigma _2(P)\) and \(((x_{j})_{j\in J}, x)\in {\mathcal {GS}}\) with \(x\in U\). Since \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\), there is a directed family \({\mathcal {F}}=\{F\subseteq P: F\) is finite} of quasieventual lower bounds of a net \((x_{j})_{j\in J}\) in P such that \(\bigcap _{F\in {\mathcal {F}}}\uparrow F\subseteq \uparrow x\). By Proposition 3 there is \(F\in {\mathcal {F}}\) such that \(\uparrow F\subseteq U\). Notice that F is a quasieventual lower bound of a net \((x_{j})_{j\in J}\), there is some \(j_{0}\in J\) such that \(x_{j}\in \uparrow F\subseteq U\) for all \(j\ge j_{0}\). Thus \(U\in \mathcal {O({\mathcal {GS}})}\).\(\square \)
Now we derive the \({\mathcal {GS}}\)convergence in the \(s_2\)quasicontinuous poset is topological.
Proposition 5
Let P be an \(s_2\)quasicontinuous poset. Then \(x\equiv _{{\mathcal {GS}}}lim\) \(x_{j}\) if and only if the net \((x_{j})_{j\in J}\) converges to the element x with respect to the weak Scott topology.
Proof
The necessity follows from Proposition 4. Now suppose that the net \((x_{j})_{j\in J}\) converges to an element x with respect to the weak Scott topology. Since P is \(s_2\)quasicontinuous, there exists a directed family \(w(x)=\{F\subseteq P:F\in P^{(<\omega )}\) and \(F\ll x\}\) and \(\uparrow x=\bigcap \{\uparrow F:F\in w(x)\) }. For all \(F\in w(x)\), let \(U_{F}=\{y\in P: F\ll y\}\). Then \(U_{F}\in \sigma _{2}(P)\) and \(x\in U_{F}\) by Lemma 3, and hence \(x_{j}\in U_{F}\) eventually holds. Thus F is a quasieventual lower bound of the net \((x_{j})_{j\in J}\) and \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\).\(\square \)
The converse is also true.
Proposition 6
Let P be a poset. If the \({\mathcal {GS}}\)convergence is topological, then P is \(s_2\)quasicontinuous.
Proof
Suppose that the \({\mathcal {GS}}\)convergence is topological. Then \(x\equiv _{{\mathcal {GS}}}\)lim \(x_{j}\) if and only if the net \((x_{j})_{j\in J}\) converges to the element x with respect to the weak Scott topology \(\sigma _{2}(P)\) by Proposition 4.
 (1)
Let \(D\subseteq P\) be directed with \(x\in D^{\delta }\). Since F is a quasieventual lower bound of the net \((x_{j})_{j\in J}\), there is \(j_{0}=(U, m, a)\in J\) such that \(x_{j}\in \uparrow F\) for all \(j=(V, n, b)>j_{0}\). Notice \(x\in U\), so \(D\cap U\ne \emptyset \). Pick \(d\in D\cap U\). Set \(i=(U, m+1, d)\), then \(i>(U, m, a)=j_{0}\). Thus \(d=x_{i}\in \uparrow F\), that is, \(F\ll x\).
 (2)
We only need to show that \(\uparrow x\subseteq \bigcap _{F\in {\mathcal {F}}}\uparrow F\). Suppose not, then there exists \(y\ge x\) but \(y\notin \bigcap _{F\in {\mathcal {F}}}\uparrow F\), that is, there exists \(F\in {\mathcal {F}}\) with \(y\notin \uparrow F\). And then \(\uparrow F\subseteq P\backslash \downarrow x\). Again since F is a quasieventual lower bound of the net \((x_{j})_{j\in J}\), there exists \(j_{0}=(U, m, a)\in J\) such that \(x_{j}\in \uparrow F\) for all \(j=(V, n, b)>j_{0}\). Now we have \(x\in U\). Set \(i=(U, m+1, x)\), then \(i>(U, m, x)=j_{0}\). Thus \(x=x_{i}\in \uparrow F\subseteq P\backslash \downarrow x\), a contradiction.
From Propositions 5 and 6 we have:
Theorem 3
 (1)
P is \(s_2\)quasicontinuous;
 (2)
The \({\mathcal {GS}}\)convergence in P is topological for the weak Scott topology \(\sigma _2(P)\), that is, for all \(x\in P\) and all nets \((x_{j})_{j\in J}\) in P, \(x\equiv _{{\mathcal {GS}}}lim\) \(x_{j}\) if and only if \((x_{j})_{j\in J}\) converges to x with respect to the weak Scott topology.
Corollary 2
 (1)
P is a quasicontinuous domain;
 (2)
\(S^{*}\)convergence in P is topological for the Scott topology \(\sigma (P)\), that is, for all \(x\in P\) and all nets \((x_{j})_{j\in J}\) in P, \((x_{j})_{j\in J}\) \(S^{*}\) converges to x if and only if \((x_{j})_{j\in J}\) converges to x with respect to the Scott topology.
Conclusions
In this paper, we present one way to generalize \({\mathcal {S}}\)convergence and \({\mathcal {GS}}\)convergence of nets for arbitrary posets by use of the cut operator instead of joins and come to the main conclusions are: (1) A poset P is \(s_2\)continuous if and only if the \({\mathcal {S}}\)convergence in P is topological; (2) P is \(s_2\)quasicontinuous if and only if the \({\mathcal {GS}}\)convergence in P is topological.
Declarations
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
Xiao‑jun Ruan was supported by the National Natural Science Foundation of China (11201216, 61175127, 11501281, 11561046) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB2010031, 20151BAB201020). Xiao‑quan Xu was supported by the National Natural Science Foundation of China (11161023) and the Fund for the Author of National Excellent Doctoral Dissertation of China (2007B14).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Abramsky S, Jung A (1994) Domain theory. In: Abramsky S, Gabbay DM, Maibaum TSE (eds) Handbook of logic in computer science, vol 3. Oxford University Press, OxfordGoogle Scholar
 Chen QY, Kou H (2014) A characterization of quasicontinuous domains by nets. J Sichuan Univ (Nat Sci Ed) 51(3):433–435Google Scholar
 Erné M (1981) Scott convergence and Scott topology on partially ordered sets II. In: Banaschewski B, Hoffman RE (eds) Continuous Lattices, Bremen 1979, vol 871., Lecture notes in mathematicsSpringer, Berlin, Heidelberg, New York, pp 61–96Google Scholar
 Erné M (1991) The Dedekind–MacNeille completion as a reflector. Order 8(2):159–173View ArticleGoogle Scholar
 Erné M (2009) Infinite distributive laws versus local connectedness and compactness properties. Topol Appl 156(12):2054–2069View ArticleGoogle Scholar
 Gierz G, Lawson JD (1981) Generalized continuous and hypercontinuous lattices. Rocky Mt J Math 11(2):271–296View ArticleGoogle Scholar
 Gierz G, Lawson JD, Stralka AR (1983) Quasicontinuous posets. Houston J Math 9(2):191–208Google Scholar
 Gierz G, Hofmann K, Keimel K, Lawson JD, Mislove M, Scott D (1980) A compendium of continuous lattices. Springer, Berlin, Heidelberg, New YorkView ArticleGoogle Scholar
 Gierz G, Hofmann K, Keimel K, Lawson JD, Mislove M, Scott D (2003) Continuous lattices and domains, encyclopedia of mathematics and its applications 93. Cambridge University Press, CambridgeView ArticleGoogle Scholar
 Huang MQ, Li QG, Li JB (2009) Generalized continuous posets and a new cartesian closed category. Appl Categ Struct 17(1):29–42View ArticleGoogle Scholar
 Lawson JD, Xu LS (2004) Posets having continuous intervals. Theor Comput Sci 316(1–3):89–103View ArticleGoogle Scholar
 Mao XX, Xu LS (2006) Quasicontinuity of posets via Scott topology and sobrification. Order 23(4):359–369View ArticleGoogle Scholar
 Markowsky G (1981) A motivation and generalization of Scott’s notion of a continuous lattice. In: Banaschewski B, Hoffman RE (eds) Lecture notes in mathematics, vol 871., Continuous lattices, Bremen 1979Springer, Berlin, Heidelberg, New York, pp 298–307Google Scholar
 Mislove M (1999) Local DCPOs, local CPOs and local completions. Electron Notes Theor Comput Sci 20:1–14View ArticleGoogle Scholar
 Venugopalan P (1990) Quasicontinuous posets. Semigroup Forum 41(1):193–200View ArticleGoogle Scholar
 Wang KY, Zhao B (2013) Some further results on orderconvergence in posets. Topol Appl 160(1):82–86View ArticleGoogle Scholar
 Zhang H (1993) A note on continuous partially ordered sets. Semigroup Forum 47(1):101–104View ArticleGoogle Scholar
 Zhang WF, Xu XQ (2015) \(s_2\)Quasicontinuous posets. Theor Comput Sci 574:78–85View ArticleGoogle Scholar
 Zhao B, Zhao DS (2005) Liminf convergence in partially ordered sets. J Math Anal Appl 309(2):701–708View ArticleGoogle Scholar
 Zhao B, Li J (2006) O\(_{2}\)convergence in posets. Topol Appl 153(15):2971–2975View ArticleGoogle Scholar
 Zhou YH, Zhao B (2007) Orderconvergebce and liminf\(_{M}\)convergence in posets. J Math Anal Appl 325(1):655–664View ArticleGoogle Scholar
 Zhou LJ, Li QG (2013) Convergence on quasicontinuous domain. J Comput Anal Appl 15(1):381–390Google Scholar