Quasiexact Posets and the Moderate Meet-continuity

Scott [11] proposed a model for information systems using the Scott topology and a binary relation $\prec$ in connection with the information models. For continuous lattices, the relation $\prec$ coincides with the way-below relation. The class of continuous complete lattices was introduced by Scott [12]. However, for general complete lattices, the aforementioned two relations may be distinct.

One of the notable features of continuous lattices is that they admit a unique compact Hausdorff topology for which the meet operation is continuous. This topology, referred to as the CL-topology [4], turns out to be ‘order intrinsic’ - it can be defined merely using the lattice structure. Gierz and Lawson [4] characterized those complete lattices for which the CL-topology is Hausdorff and called them generalized continuous lattices. Gierz et al. [5] introduced the quasicontinuous posets and showed that a complete lattice is generalized continuous if and only if it is quasicontinuous. The key result for establishing the major properties of quasicontinuous dcpos is the Rudin’s Lemma [10].

Coecke and Martin [2] introduced two orders: the Bayesian order on classical states and the spectral order on quantum states. They revealed that the corresponding sets are dcpos with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain-like theoretic structure. Mushburn [14] called the approximation in [2] the weak way-below relation, and defined two topologies on posets: the way-below topology and the weak way-below topology. These topologies coincide with the Scott topology for continuous posets, but are very different for non-continuous posets. Mushburn also showed that while domain representable spaces must be Baire, this is not the case with respect to the new topologies. Thus, Mushburn defined the weak domains and weak domain representable spaces and constructed an example to show that weak domain representable spaces need not be Baire [15].

The class of meet continuous lattices was first introduced by Birkhoff [1]. Later, much investigations on meet continuity for lattices and semilattices sprang up. One can refer to Isbell [8], Hofmann and Stralka [7] and [3]. Kou et al. [9] extended the notion of meet continuity to general dcpos and proved that a dcpo is continuous iff it is meet continuous and quasicontinuous. See [17] and [18] for the investigation of the more general meet-continuous posets and quasicontinuous posets.

As a generalization of the way-below relation between two subsets of posets and the weak way-below relation between elements, we define the weak way-below relation between two subsets of a poset and use this relation to define the quasiexact posets. We show some connections among quasiexact posets, quasicontinuous domains and weak domains. Furthermore, we introduce the weak way-below finitely determined topology (briefly, wf topology) and study its properties as well as its links to Scott topology and weak way-below topology in some special posets. In addition, we prove that a poset is a weak domain if it is moderately meet continuous and quasiexact dcpo with the relation $\ll_{w}$ weakly increasing, a result similar to the characterization of domains in terms of meet continuity and quasicontinuity. Finally, employing a result by Shen et al. [16], it is deduced that a dcpo is a domain if and only if it is quasiexact and moderately meet continuous with the weak way-below relation weakly increasing.

In this section we recall some notations, definitions and results to be used later.

For any subset $A$ of a poset $P$, we write $\uparrow\nobreak\!A=\{x\in P:y\leq x~{}\mbox{for some}~{}y\in A\}$. A subset $A\subseteq P$ is called an upper set if $\uparrow\nobreak\!A=A$.

A nonempty subset $D$ of a poset $(P,\leq)$ is directed if every two elements in $D$ have an upper bound in $D$. A poset $P$ is called a directed complete poset (dcpo, for short) if every directed subset $D$ of $P$ has a supremum in $P$, denoted by $\bigvee D$.

For any poset $P$, the way-below relation $\ll$ on $P$ is defined as follows: for every directed subset $D$ with $\bigvee D$ existing, $y\leq\bigvee D$ implies $x\leq d$ for some $d\in D$. A poset $P$ is continuous if for every $x\in P$, the set $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x=\{y\in P:y\ll x\}$ is directed and $x=\bigvee\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x$. For any $x\in P$, one writs $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}x=\{y\in P:x\ll y\}$.

A subset $U$ of a poset $P$ is Scott open if $U$ is an upper set and for any directed subset $D$ of $P$ for which $\bigvee D$ exists, $\bigvee D\in U$ implies $D\cap U\neq\varnothing$. All Scott open subsets of $P$ form a topology, called the Scott topology on $P$ and denoted by $\sigma(P)$. The space $(P,\sigma(P))$ is called the Scott space of $P$, and is denoted by $\Sigma P$.

A poset $P$ is said to be meet continuous¹¹1In general, the meet continuity means for dcpos. In the current paper, we slightly misuse this notion. if for any $x\in P$ and any directed subset $D$, $x\leq\bigvee D$ implies $x\in\operatorname{cl}_{\sigma}(\downarrow\nobreak\!D\cap\downarrow\nobreak\!x)$.

For any topological space $(X,\tau)$ and a subset $A\subseteq X$, the closure and the interior of $A$ are denoted by $\operatorname{cl}_{\tau}(A)$ and $\operatorname{int}_{\tau}(A)$, respectively.

Let $(X,\tau)$ be a topological space. A nonempty subset $A$ of $X$ is called irreducible if for any closed sets $B$ and $C$, $A\subseteq B\cup C$ implies $A\subseteq B$ or $A\subseteq C$. The space $X$ is sober if for every irreducible closed set $A$, there exists a unique $x\in X$ such that $\operatorname{cl}_{\tau}\{x\}=A$.

The weak way-below relation on a poset $P$ is defined as follows [14]: for any $x,y\in P$, $x$ is called weakly way below $y$, denoted by $x\ll_{w}y$ if for any directed subset $D$ of $P$, $\bigvee D=y$ implies $D\cap\uparrow\nobreak\!x\neq\varnothing$.

Note that for continuous posets, the relations $\ll_{w}$ and $\ll$ are the same. For any $x\in P$, we write $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}_{w}x=\{y\in P:y\ll_{w}x\}$. Mushburn [14] pointed out the following properties.

It is well-known that the way-below relation $\ll$ is transitive: $x\ll y\leq z$ implies $x\ll z$. Whereas, this property may not be true for $\ll_{w}$. In fact, Coecke and Martin [2] showed that the transitivity is true for $\ll_{w}$ if and only if $\ll_{w}=\ll$.

A poset $P$ is called exact if for any $x\in P$, the set $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}_{w}x$ is directed and $\bigvee\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}_{w}x=x$ [14]. The relation $\leq_{w}$ on a poset $P$ is said to be weakly increasing if for any $x,y,z,u\in P$, $x\ll_{w}y\leq z\ll_{w}u$ implies $x\ll_{w}z$. A poset $P$ is called a weak domain if it is an exact dcpo with the relation $\ll_{w}$ weakly increasing.

Shen et al. [16] proved the following characterization of exact dcpos.

For a poset $P$, the collection of all nonempty subsets of $P$ is denoted by $\mathcal{P}^{*}(P)$. A preorder $\leq$ on $\mathcal{P}^{*}(P)$ is defined by $G\leq H$ if $\uparrow\nobreak\!H\subseteq\uparrow\nobreak\!G$. This preorder is sometimes called the Smyth order, see [13].

A nonempty family $\mathcal{F}$ of subsets of a poset $P$ is said to be directed if given $F_{1}$, $F_{2}\in\mathcal{F}$, there exists $F_{3}\in\mathcal{F}$ such that $F_{1},~{}F_{2}\leq F_{3}$, that is $F_{3}\subseteq\uparrow\nobreak\!F_{1}\cap\uparrow\nobreak\!F_{2}$, or equivalently, $\uparrow\nobreak\!F_{3}\subseteq\uparrow\nobreak\!F_{1}\cap\uparrow\nobreak\!F_{2}$. For any $G,H\subseteq P$, $G$ is way below $H$, written as $G\ll H$, if for every directed subset $D\subseteq P$, $\sup D\in\uparrow\nobreak\!H$ implies $d\in\uparrow\nobreak\!G$ for some $d\in D$. Sometimes, one writes $G\ll x$ instead of $G\ll\{x\}$ and $y\ll H$ instead of $\{y\}\ll H$. For more details one can refer to [3].

Recall that a dcpo $P$ is called a quasicontinuous domain if for each $x\in P$ the family

is directed and whenever $x\nleq y$, then there exists $F\in\operatorname{fin}(x)$ with $y\notin\uparrow\nobreak\!F.$ This statement is equivalent to for each $x\in P$ the family

is directed and $\bigcap\{\uparrow\nobreak\!F:F\in\operatorname{fin}(x)\}=\uparrow\nobreak\!x.$

For more about posets and related notions and results, we refer to [3] and [6].

We first introduce the following definition, as the generalization of both the way-below relation between two nonempty subsets and the weak way-below relation between two points.

The following properties can be verified straightforwardly.

Thereafter, we write $G\ll_{w}x$ instead of $G\ll_{w}\{x\}$ and $y\ll_{w}H$ instead of $\{y\}\ll_{w}H$. By Lemma 4, $y\ll_{w}x$ is unambiguously defined and this fact is similar with the one for the way-below relation.

For any poset $P$ and $x\in P$, write

Sometimes it is difficult to characterize the weak way-below relation on a poset. To show a poset $P$ is quasiexact, it is sufficient to know for any $x\in P$, there are ‘enough’ elements that are weakly way below $x$. In order to elaborate this fact, we first show the following lemmas.

The proofs of the following Lemmas 7 and 8 are straightforward.

The following lemma given by Rudin [10] is crucial in proving a number of major properties of quasicontinuous domains, one can also refer to Lemma III-3.3 in [3].

Similar to Proposition 2 for exact posets, we have the following result.

We write $\operatorname{fin}_{(bl)w}(x)=\{F:F\in\operatorname{fin}_{w}(x),F\subseteq\downarrow\nobreak\!x\}$. For any $F\in\operatorname{fin}_{w}(x)$, we have $\downarrow\nobreak\!x\cap F\ll_{w}x$ by Lemma 7. It follows that $\downarrow\nobreak\!x\cap F\in\operatorname{fin}_{(bl)w}(x)$. Note that $F=\downarrow\nobreak\!x\cap F$ for any $F\in\operatorname{fin}_{(bl)w}(x)$, so $\operatorname{fin}_{(bl)w}(x)=\{\downarrow\nobreak\!x\cap F:F\in\operatorname{fin}_{(bl)w}(x)\}\subseteq\{\downarrow\nobreak\!x\cap F:F\in\operatorname{fin}_{w}(x)\}$. Therefore, $\operatorname{fin}_{(bl)w}(x)=\{\downarrow\nobreak\!x\cap F:F\in\operatorname{fin}_{w}(x)\}$.

Now, we give another characterization of quasiexact dcpos.

Applying Propositions 11 and 12, we can derive the following result.

Some relationships among quasiexact dcpos, weak domains and quasicontinuous domains are shown in the following result.

Quasiexact posets have some weak and quasicontinuous domain-like features, so this terminology seems appropriate.

Shen et al. [16] used two examples to show that quasicontinuous domains need not be weak domains, and that weak domains need not be quasicontinuous domains. Precisely, Example 3.11 in [16] shows that quasicontinuous domains are not necessarily exact, and Example 3.12 in [16] shows that weak domains are not necessarily quasicontinuous.

Here, we shall elaborate some more detailed relationships than those in [16].

The following example shows that quasicontinuous domains are not necessarily weakly increasing. Since every quasicontinuous domain is quasiexact, this example also shows that quasiexact posets need not to be exact or weakly increasing.

The following example shows that quasicontinuous domains need not be exact, even when the relation $\ll_{w}$ is weakly increasing.

Examples 3.6 and 16 in the current paper also shows that quasiexacnesst does not imply stronger properties of $\ll_{w}$.

We take Johnstone’s dcpo to illustrate that weak domains are not necessarily quasicontinuous domain, hence quasiexact dcpos with the relation $\ll_{w}$ weakly increasing are not necessarily quasicontinuous.

In this section, we investigate the links between the quasiexactness and some topologies on posets.

In [14], when the family $\{\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}_{w}x:x\in P\}$ generates a topology on a poset $P$, then this topology is called the weak way-below topology (wwb topology, for short), denoted by $\tau_{wwb}(P)$. The topological space $(P,\tau_{wwb}(P))$ is simply written as $(P,\tau_{wwb})$.

For the following result, we refer the reader to the Theorems 3.8 and 3.10 in [14].

For any subset $F\subseteq P$, we write $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}_{w}F=\{a\in P:F\ll_{w}a\}$. Whenever $F=\{x\}$ for some $x\in P$, we replace $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}_{w}\{x\}$ with $\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}_{w}x$.

Whenever the family $\{\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}_{w}F:F\subseteq P,F~{}\mbox{is finite}\}$ generates a topology on $P$, we call it the weak way-below finitely determined topology (briefly, wf topology) on $P$, denoted by $\tau_{wf}(P)$. The topological space $(P,\tau_{wf}(P))$ will be simply written as $(P,\tau_{wf})$.

By Lemma 18 and Remark 20 (2), we have the following result.

Note that every quasicontinuous dcpo admits the wf topology. However, the following example shows that quasicontinuous dcpos do not necessarily admit the wwb topology.