2. Preliminaries

For brevity, unexplained notations and concepts can be found in Gierz et al. (Reference Gierz, Hofmann and Keimel2003).

Let $P$ be a poset, $A\subseteq P$ , and $p\in P$ . As usual, we define $\downarrow _{P} A=\{x\in P \mid \exists a\in A,\ \mathrm{s.\ t.}\ x\leq a\}$ . $\downarrow _{P}\{p\}$ is denoted by $\downarrow _{P} p$ . Dually, we can define $\uparrow _{P} A$ and $\uparrow _{P} x.$ If no confusion arises, the subscript will be omitted. If $A=\downarrow A$ , then $A$ is called a lower set. We say $x$ is covered by $y$ , written as $x-\llap {\lt }y$ , if $x\lt y$ and $x\leq z\lt y$ implies $x=z$ . The set of all minimal upper bounds of $A$ is denoted by $\mathrm{mub}(A)$ . A subset $D\subseteq P$ is called directed if for any $a, b\in D$ , there exists a $c\in D$ such that $a, b \leq c.$ A directed lower set is called an ideal. The set of compact elements of $P$ is denoted by $K(P).$

A mapping $f: P\rightarrow Q$ between posets is Scott continuous if for any directed subset $D\subseteq P$ for which $\bigvee D$ exists in $P$ , we have $f(\bigvee D) = \bigvee f(D)$ . It is well known that if $P$ and $Q$ are dcpos, then $f$ is Scott continuous if and only if it is continuous with respect to the Scott topology Gierz et al. (Reference Gierz, Hofmann and Keimel2003).

A retraction-embedding pair $(r, e)$ for dcpos is a pair of Scott-continuous functions $Q{\stackrel {r}{\longrightarrow }}P{\stackrel {e}{\longrightarrow }}Q$ such that $r\circ e=1_{P}.$ In this case, we say $P$ is a retract of $Q$ .

Let $P$ be a poset. $\mathrm{Id}P$ denotes the set of ideals in $P$ ordered by set inclusion. Then, $\mathrm{Id}P$ is a dcpo in which the supremum of a directed family of ideals is given by set union. Furthermore, $\mathrm{Id}P$ is an algebraic dcpo, and the set of compact elements of $\mathrm{Id} P$ is the set of principal ideals.

3. Plotkin-Poset

In this section, we present some properties of mub-closure that will be used in the inductive construction of a Plotkin-poset.

Given a poset $P$ with property m, for any subset $A\subseteq P$ , Jung defined the $U^{\infty }$ -operator as follows Jung (Reference Jung1989):

\begin{equation*}U^{0}(A)=A,\end{equation*}

\begin{equation*}U^{n+1}(A)=\{x\in P\mid x\ \mathrm{is\ a\ minimal\ upper\ bound\ for\ some\ finite\ subset\ of\ } U^{n}(A)\},\end{equation*}

\begin{equation*}U^{\infty }(A)=\bigcup _{n=0}^{\infty }U^{n}(A).\end{equation*}

It is easy to see that $U^{\infty }(A)=\mathrm{mc}(A)$ . As a consequence, we have the following result.

Let $\mathbb{N}=\{1,2,3,\cdots \}$ be the set of natural numbers.

Lemma 3.3. Let $P$ be a poset with property m. Let $P_{1}, P_{2},\cdots , P_{n},\cdots$ be subposets of $P$ such that $P_{1}\subseteq P_{2}\subseteq \cdots \subseteq P_{n}\subseteq \cdots$ , $P=\bigcup _{n=1}^{\infty }P_{n}$ and

\begin{equation*}\mathrm{mub}_{P_{n}}(F)= \mathrm{mub}_{P}(F)\end{equation*}

for every finite subset $F\subseteq P_{n}$ , where $\mathrm{mub}_{P_{n}}(F)$ and $\mathrm{mub}_{P}(F)$ are the sets of minimal upper bounds of $F$ in $P_{n}$ and $P$ respectively. Then

\begin{equation*}\mathrm{mc}_{P_{n}}(F) = \mathrm{mc}_{P}(F)\end{equation*}

for every finite subset $F\subseteq P_{n}$ and $n\in \mathbb{N}$ , where $\mathrm{mc}_{P_{n}}(F)$ and $\mathrm{mc}_{P}(F)$ are the mub-closures of $F$ in $P_{n}$ and $P$ , respectively.

Lemma 3.4. Let $P$ be a poset. Let $P_{1}, P_{2},\cdots , P_{n},\cdots$ be subposets of $P$ such that $P_{1}\subseteq P_{2}\subseteq \cdots \subseteq P_{n}\subseteq \cdots$ and $P=\bigcup _{n=1}^{\infty }P_{n}$ . Suppose the following conditions hold for all $n\geq 1$ :

(1) for every $x\in P_{n+1}\setminus P_{n}$ , $\uparrow _{P_{n+1}} x\cap P_{n}=\emptyset$ ,

(2) for every $x\in P_{n+1}\setminus P_{n}$ , $\downarrow _{P_{n+1}} x\cap P_{n}\neq \emptyset$ implies that $\downarrow _{P_{n+1}} x\cap P_{n}$ has a greatest element.

Then

\begin{equation*}\mathrm{mub}_{P_{n}}(F) = \mathrm{mub}_{P_{n+1}}(F)= \mathrm{mub}_{P}(F)\end{equation*}

for every finite subset $F\subseteq P_{n}$ , where $\mathrm{mub}_{P_{n}}(F)$ , $\mathrm{mub}_{P_{n+1}}(F),$ and $\mathrm{mub}_{P}(F)$ are the sets of minimal upper bounds of $F$ in $P_{n}$ , $P_{n+1}$ , and $P$ , respectively.

4. The construction of a Plotkin-poset based on a finite set of circular rings

In this section, we construct a Plotkin-poset based on a finite set of circular rings. We first fix notations for discs, circular rings, and circles.

Let $\mathrm{Disc}(x, y)(r)$ be the closed disc in the $x$ - $y$ plane with center $(x, y)$ and radius $r$ . Let

\begin{equation*}\mathrm{Disc}=\{\mathrm{Disc}(x, y)(r) \mid x, y, r\in \mathbb{R},r\geq 0\}.\end{equation*}

For $D_{1}, D_{2}\in \mathrm{Disc}$ , we define $D_{1}\leq D_{2}$ if and only if $D_{1}\supseteq D_{2}.$ For $D\in \mathrm{Disc}$ , let $r_{D}$ denote the radius of $D.$ For a disc $D=\mathrm{Disc}(x, y)(r)$ and a real number $\mu$ with $\mu \gt -r$ , let $D^{+\mu }=\mathrm{Disc}(x, y)(r+\mu )$ . It is well known that $\mathrm{Disc}$ with a bottom element adjoined is a countably based $FS$ -domain Jung (Reference Jung1990b).

Let $\mathrm{Cir}(x, y)(r)$ be the circle with center $(x, y)$ and radius $r$ . For a circle $C$ , let $r_{C}$ denote the radius of $C.$

Let $\mathrm{Ring}(x, y)(r_{1},r_{2})$ be the closed circular ring in the $x$ - $y$ plane with center $(x, y)$ , where $r_{1}$ and $r_{2}$ are the radii of the inner and outer circles, respectively. For convenience, a closed circular ring is simply called a circular ring or a ring. Let

\begin{equation*}\mathrm{Ring}=\{\mathrm{Ring}(x, y)(r_{1},r_{2}) \mid x, y, r_{1}, r_{2}\in \mathbb{R}, r_{2}\gt r_{1}\gt 0\}.\end{equation*}

For a closed circular ring $R=\mathrm{Ring}(x, y)(r_{1},r_{2})$ , let

1. a. $R^{intD}$ be the closed disc $\mathrm{Disc}(x, y)(r_{1})$ ;

2. b. $R^{outD}$ be the closed disc $\mathrm{Disc}(x, y)(r_{2})$ ;

3. c. $w_{R}=r_{2}-r_{1}$ be the width of $R$ ;

4. d. $R^{int+\mu }=\mathrm{Ring}(x, y)(r_{1}+\mu ,r_{2})$ where $-r_{1}\lt \mu \lt r_{2}-r_{1}$ .

We note that the set $\mathrm{Ring}$ of closed circular rings is not currently equipped with an order.

For a family of closed discs $\{D_{i}\}_{i\in I}$ , if $\bigcap _{i\in I}D_{i}$ is non-empty, then $\bigcap _{i\in I}D_{i}$ is closed and bounded, hence compact and convex. For a set $S\subseteq \mathbb{R}^{2}$ and a disc $D\subseteq S$ , $D$ is called a maximal internal disc in $S$ if for every disc $D^{\prime }\subseteq S$ , $D\subseteq D^{\prime }$ implies $D=D^{\prime }$ .

Next, we introduce the concept of an intermediate cover for a finite set of circular rings. This is one of the basic concepts in this paper.

Definition 4.1. Let $\{R_{i}\}_{i\in I}$ be a finite set of circular rings such that $\bigcap _{i\in I}R_{i}^{intD}\neq \emptyset$ . We say a set of circular rings $\{K_{j}\}_{j\in J}$ is an intermediate cover of $\{R_{i}\}_{i\in I}$ , denoted by

\begin{equation*}\mathrm{IntCov}(\{ R_{i}\}_{i\in I}),\end{equation*}

if the following conditions hold:

(a) $K_{j}^{outD}\subseteq R_{i}^{outD}$ for every $i\in I, j\in J,$

(b) for every maximal internal disc $C$ in $\bigcap _{i\in I}R_{i}^{intD}$ , there exists a $j\in J$ such that $C\subseteq (K_{j})^{intD},$

(c) for every $j\in J$ , there exists a maximal internal disc $C$ in $\bigcap _{i\in I}R_{i}^{intD}$ such that $K_{j}^{intD}\supseteq C.$

For $K\in \mathrm{IntCov}(\{ R_{i}\}_{i\in I})$ , every element in the set $\{ R_{i} \mid i\in I\}$ is called a one-step underlying element of $K$ . The set of one-step underlying elements of $K\in \mathrm{IntCov}(\{ R_{i}\}_{i\in I})$ is denoted by

\begin{equation*}\mathrm{OSU}(K).\end{equation*}

For a subset $\mathcal{R}\subseteq \mathrm{IntCov}(\{ R_{i}\}_{i\in I})$ , let

\begin{equation*}\mathrm{OSU}(\mathcal{R})=\bigcup _{K\in \mathcal{R}}\mathrm{OSU}(K).\end{equation*}

An intermediate cover is said to have width $\omega$ if the width of every $K_{j}$ is $\omega .$ An intermediate cover for $\{R_{i}\}_{i\in I}$ with width $\omega$ is denoted by

\begin{equation*}\mathrm{IntCov}^{\omega }(\{ R_{i}\}_{i\in I}).\end{equation*}

The motivation for introducing the notion of an intermediate cover is to construct a finite set of minimal upper bounds for a family of circular rings with a non-empty intersection of inner closed discs.

Condition (a) means that each $K_{j}^{outD}$ is contained in every $R_{i}^{outD}$ . Condition (b) is equivalent to the statement that for every disc $D$ contained in $\bigcap _{i\in I}R_{i}^{intD}$ , there exists an $j\in J$ such that $D\subseteq (K_{j})^{intD}.$ This is not equivalent to $\bigcap _{i\in I}R_{i}^{intD}\subseteq \bigcup _{i\in I}(K_{i})^{intD}.$ Condition (c) ensures that every element in an intermediate cover covers at least one maximal internal disc.

The width of an intermediate cover is less than or equal to the width of every $R_{i}$ . If $K$ belongs to an intermediate cover, then the radius of the outer circle of $K$ is less than or equal to the radius of the outer circle of every $R_{i}$ , that is, $r_{K^{outD}}\leq r_{R_{i}^{outD}}$ for every $i\in I.$

To meet our requirements, we focus on the existence and construction methods of intermediate covers. Next, we prove that an intermediate cover exists for any finite set of circular rings with a non-empty intersection of inner closed discs. More importantly: (1) under certain conditions, an intermediate cover contains exactly one circular ring; (2) the decrease in the width of circular rings in a cover can be controlled by a lower bound; (3) the radius of the outer circles decreases faster than the width of the rings in a cover. These properties will be used to guarantee that we can construct a finite set of upper bounds for a finite set of rings.

The Vietoris topology $\tau _{V}$ on $\mathrm{Disc}$ is the subspace topology induced by the Vietoris topology Michael (Reference Michael1951); Nadler (Reference Wanes and Nadler1999) on the hyperspace $CL(\mathbb{R}^{2})$ of all non-empty closed subsets of $\mathbb{R}^{2}$ . The basic open sets in $(\mathrm{Disc}, \tau _{V})$ are of the form:

\begin{equation*}\lt U_{1}, U_{2}, \cdots , U_{k}\gt =\Bigg\{D\in \mathrm{Disc} \mid D\subseteq \bigcup _{i=1}^{k}U_{i}\ \mathrm{and}\ D\cap U_{i}\neq \emptyset \ \mathrm{for\ each}\ i\Bigg\},\end{equation*}

where $U_{1}, U_{2}, \cdots , U_{k}$ are open sets in $\mathbb{R}^{2}$ and $k\in \mathbb{N}.$ Let $\mathbb{R}^{+}=[0, \infty )$ . Let $\mathbb{R}^{2}\times \mathbb{R}^{+}$ be the parametrized space equipped with the product topology. The bijection $\phi : \mathbb{R}^{2}\times \mathbb{R}^{+} \rightarrow \mathrm{Disc}$ , defined by $((x, y), r)\mapsto \mathrm{Disc}(x, y)(r)$ , is a homeomorphism.

Proof. An intermediate cover for $\{\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon )\}_{i\in I}$ is also an intermediate cover for $\{R_{i}\}_{i\in I}$ . Thus, we work with $\{\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon )\}_{i\in I}.$ For $i_{1}\in I$ , if there exists an $i_{2}\in I$ such that $\mathrm{Disc}(x_{i_{1}}, y_{i_{1}})(r_{i_{1}})\subseteq \mathrm{Disc}(x_{i_{2}}, y_{i_{2}})(r_{i_{2}})$ , then an intermediate cover for $\{\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon )\}_{i\in I\setminus \{i_{2}\}}$ is also an intermediate cover for $\{\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon )\}_{i\in I}$ . Thus, we may further assume that no disc in $\{\mathrm{Disc}(x_{i}, y_{i})(r_{i})\}_{i\in I}$ contains another.

We identify the set of closed discs in the plane with $\mathbb{R}^{2}\times \mathbb{R}^{+}$ via the homeomorphism $\phi$ . Let $C=\bigcap _{i\in I}R_{i}^{intD}$ . Then, $C$ is a non-empty closed set. Let $D\subseteq C$ be a closed disc with center $(x_{0}, y_{0})$ and radius $r_{0}$ . Let

\begin{equation*}\widehat {D}=\{((x, y), r) \mid D\subseteq \mathrm{Disc}(x, y)(r)\subseteq C\}.\end{equation*}

Then, $\widehat {D}$ is sequentially closed in $\mathbb{R}^{2}\times \mathbb{R}^{+}$ . Thus, $\widehat {D}$ is closed and bounded, hence compact, in $\mathbb{R}^{2}\times \mathbb{R}^{+}$ . Since the projection onto the $r$ -coordinate is continuous, there exists a closed disc $D^{*}\in \widehat {D}$ with center $(x^{*}, y^{*})$ and radius $r^{*}$ such that $r^{*}$ is maximal among such discs. Thus, $D^{*}$ is a maximal closed disc containing $D$ and contained in $C$ . Hence, every closed disc contained in $C$ is contained in a maximal closed disc contained in $C$ . By the homeomorphism $\phi : \mathbb{R}^{2}\times \mathbb{R}^{+} \rightarrow \mathrm{Disc}$ , the set of closed discs contained in $C$ is compact in $(\mathrm{Disc}, \tau _{V})$ .

For every maximal closed disc $D^{*}$ in $C$ with center $(x^{*}, y^{*})$ and radius $r^{*}$ , define the ring $R_{D^{*}}=\mathrm{Ring}(x^{*}, y^{*})(r^{*}+\delta , r^{*}+\varepsilon )$ , where $0\lt \delta \lt \varepsilon$ . Then, the interior $\mathrm{int}((R_{D^{*}})^{intD})$ is an open set containing $D^{*}$ . In the Vietoris topological space $(\mathrm{Disc}, \tau _{V})$ ,

\begin{equation*}\big\{\!\lt \mathrm{int}((R_{D^{*}})^{intD})\gt \mid D^{*}\ \mathrm{is\ a\ maximal\ closed\ disc\ in\ }C\big\}\end{equation*}

is an open cover of the compact set of closed discs contained in $C$ . Thus, there exists a finite subcover

\begin{equation*}\Big\{\lt \mathrm{int}\Big(\Big(R_{D_{j}^{*}}\Big)^{intD}\Big)\gt \Big\}_{j\in J}.\end{equation*}

Then, we obtain a finite set of rings

\begin{equation*}\{R_{D^{*}}\}_{j\in J}\end{equation*}

that satisfies the properties of a finite intermediate cover with width $\varepsilon -\delta$ .

For two points $O_{1}$ and $O_{2}$ , let $d(O_{1}, O_{2})$ denote the distance between $O_{1}$ and $O_{2}$ .

Proof. An intermediate cover for $\{\mathrm{Ring}(x_{1}, y_{1})(r_{1},r_{1}+\varepsilon ), \mathrm{Ring}(x_{2}, y_{2})(r_{2},r_{2}+\varepsilon )\}$ with width $\varepsilon -\delta$ is also an intermediate cover for $\{R_{1}, R_{2}\}$ with width $\varepsilon -\delta$ . Thus, we work with $\mathrm{Ring}(x_{1}, y_{1})(r_{1},r_{1}+\varepsilon )$ and $\mathrm{Ring}(x_{2}, y_{2})(r_{2},r_{2}+\varepsilon )$ . Furthermore, we may assume that $r_{1}\geq r_{2}$ , $x_{1}\leq 0\leq x_{2}$ , $x_{1}=-x_{2}$ , and $y_{1}=y_{2}=0$ . If $R_{1}^{intD}\cap R_{2}^{intD}$ is a singleton, then $R_{1}^{intD}\cap R_{2}^{intD}=\{(0, 0)\}$ and $r_{1}+r_{2}-d(O_{1}, O_{2})=0$ . In case (1), let $K=\mathrm{Ring}(0, 0)(\frac {1}{4}\varepsilon , \frac {3}{4}\varepsilon )$ . Then, $\{K\}$ is an intermediate cover of $\{R_{1}, R_{2}\}$ with width $\frac {1}{2}\varepsilon$ . In case (2), for every $\delta$ with $\varepsilon \gt \delta \gt 0,$ let $K=\mathrm{Ring}(0, 0)(\delta , \varepsilon )$ . Then, $\{K\}$ is an intermediate cover of $\{R_{1}, R_{2}\}$ with width $\varepsilon -\delta$ and $r_{K^{intD}}\leq r+\delta -\frac {1}{4}\varepsilon$ .

Figure 1.

Relative order of intersection points (Case 1(i)).

Table 1. Long description

A table listing values for integers with specific conditions. The table has four columns and five rows, including the header row. The columns are labeled k, 1, 2, and 3. The rows are labeled 1, 2, 3, and an ellipsis indicating continuation. The values in the table are as follows: Row 1: k, 1, 2, 4. Row 2: 1, 1, 2, 4. Row 3: 2, 3, 5. Row 4: 3, 6. The table defines the values of k for integers with specific conditions and lists the values accordingly.

Now assume that $R_{1}^{intD}\cap R_{2}^{intD}$ contains more than two points. Then the circles $\mathrm{Cir}(x_{1}, y_{1})(r_{1})$ and $\mathrm{Cir}(x_{2}, y_{2})(r_{2})$ intersect at two points. Thus, the circles $\mathrm{Cir}(x_{1}, y_{1})(r_{1}+\varepsilon )$ and $\mathrm{Cir}(x_{2}, y_{2})(r_{2}+\varepsilon )$ also intersect at two points. Suppose the intersection points of the line $O_{1}O_{2}$ with $\mathrm{Cir}(x_{1}, 0)(r_{1}+\varepsilon )$ , $\mathrm{Cir}(x_{1}, 0)(r_{1})$ , $\mathrm{Cir}(x_{2}, 0)(r_{2}+\varepsilon )$ , $\mathrm{Cir}(x_{2}, 0)(r_{2})$ are $A, D$ ; $B, C$ ; $E, H$ ; $F, G$ respectively. The relative order of $A, B, O_{1}, C, D$ is fixed. The relative position of $E, F, O_{2}, G, H$ is also fixed. Since $\mathrm{Disc}(x_{1}, y_{1})(r_{1})\cap \mathrm{Disc}(x_{2}, y_{2})(r_{2})$ contains more than two points, the position of $B, F, C, G$ is fixed. Since the circles $\mathrm{Cir}(x_{1}, y_{1})(r_{1}+\varepsilon )$ and $\mathrm{Cir}(x_{2}, y_{2})(r_{2}+\varepsilon )$ intersect at two points, the position of $A, E, D, H$ is fixed. The length of $FC$ is $r_{1}+r_{2}-d(O_{1}, O_{2}).$

(1) (i) Suppose $r\leq \frac {1}{4}\varepsilon$ . Then, $|FG|=2r\leq \frac {1}{2}\varepsilon \lt \varepsilon \lt |FC|+|CD|=|FD|$ . Thus, the point $G$ lies between $C$ and $D$ . Hence, $|CG|\lt |FG|=2r\leq \frac {1}{2}\varepsilon$ and $|GD|=|CD|-|CG|=\varepsilon -|CG|\gt \frac {1}{2}\varepsilon$ . The configuration for this case is shown in Figure 1.

(ii) Suppose $|FC|\geq 2r-\frac {1}{2}\varepsilon$ . Since $|FC|=|FG|-|CG|=2r-|CG|$ , we have $|CG|=$ $2r-|FC|\leq \frac {1}{2}\varepsilon$ . Thus, $G$ lies between $C$ and $D$ and $|GD|\geq \frac {1}{2}\varepsilon$ . The configuration for this case is shown in Figure 2.

Let $K=\mathrm{Ring}(x_{2}, 0)(r, r+\frac {1}{2}\varepsilon )$ . Then, $K$ is the desired ring.

(2) By Lemma 4.2, there exists a finite intermediate cover $\{K_{j}\}_{j\in J}$ of $\{R_{1}, R_{2}\}$ with width $\varepsilon -\delta$ . We only need to show that $r_{K_{j}^{intD}}\leq r+\delta -\frac {1}{4}\varepsilon$ for all $j\in J$ .

The configuration for this case is shown in Figure 3, where the relative order of $E, F, C, D$ is fixed. This fact will be used below.

Suppose $r\gt \frac {1}{4}\varepsilon$ and $|FC|\lt 2r-\frac {1}{2}\varepsilon$ . Then, $\frac {1}{2}|ED|=\frac {1}{2}(2\varepsilon +|FC|)\lt r+\frac {3}{4}\varepsilon .$ Let $x_{0}=$ $\frac {1}{2}(x_{F}+x_{C})$ , where $x_{F}$ and $x_{C}$ are the $x$ -coordinates of $F$ and $C$ , respectively. For a $0\lt \delta \lt \varepsilon$ , let $K_{1}=\mathrm{Ring}(x_{0}, 0)(\frac {1}{2}|FC|+\delta , \frac {1}{2}|ED|).$ Then, $w_{K_{1}}=\varepsilon -\delta ,$ $r_{K_{1}^{outD}}=\frac {1}{2}|ED|\leq (r+\varepsilon )-\frac {1}{4}\varepsilon$ , $r_{K_{1}^{intD}}\leq r+\delta -\frac {1}{4}\varepsilon$ . Furthermore, $r_{K_{j}^{intD}}\leq r_{K_{1}^{intD}}\leq r+\delta -\frac {1}{4}\varepsilon$ for every $j\in J$ .

Figure 2.

Relative order of intersection points (Case 1(ii)).

Table 1. Long description

A table listing values for integers with specific conditions. The table has four columns and five rows, including the header row. The columns are labeled k, 1, 2, and 3. The rows are labeled 1, 2, 3, and an ellipsis indicating continuation. The values in the table are as follows: Row 1: k, 1, 2, 4. Row 2: 1, 1, 2, 4. Row 3: 2, 3, 5. Row 4: 3, 6. The table defines the values of k for integers with specific conditions and lists the values accordingly.

Figure 3.

Position relation of intersection points (Case 2).

Table 1. Long description

A table listing values for integers with specific conditions. The table has four columns and five rows, including the header row. The columns are labeled k, 1, 2, and 3. The rows are labeled 1, 2, 3, and an ellipsis indicating continuation. The values in the table are as follows: Row 1: k, 1, 2, 4. Row 2: 1, 1, 2, 4. Row 3: 2, 3, 5. Row 4: 3, 6. The table defines the values of k for integers with specific conditions and lists the values accordingly.

The notion of an extended intermediate cover can be regarded as a strengthened version of an intermediate cover. An extended intermediate cover for $\{R_{i}\}_{i\in I}$ not only covers discs in $\bigcap _{i\in I} R_{i}^{intD}$ but also covers discs $D^{+\delta }$ where $D$ is contained in $\bigcap _{i\in I} R_{i}^{intD}$ . This is not equivalent to (precisely, weaker than) covering all discs in $\bigcap _{i\in I} (R_{i}^{intD})^{+\delta }$ .

A finite extended intermediate cover for $\{ R_{i} \mid i\in I\}$ with width $w$ , greatest inner radius $r$ (i.e., $\mathrm{max}\{r_{(K_{j})^{intD}} \mid j\in J\}=r$ ) and $\delta$ extension is denoted by

\begin{equation*}\mathrm{ExtIntCov}_{r}^{w}(\{ R_{i}\}_{i\in I})_{\delta }.\end{equation*}

By this lemma, we can construct an extended intermediate cover from an intermediate cover. Thus, we have the following proposition.

For convenience in describing how to construct a Plotkin poset from an arbitrary finite family of circular rings, we adopt the following convention.

Now we can construct a finite intermediate cover for a finite set of rings. To construct an infinite poset with the finite mub property based on rings, we start with a single ring and inductively add new rings one by one. When a new ring is added to a finite set, new subsets with non-empty intersections of inner discs may arise. Next, we describe the process of constructing a poset of rings with the finite mub property when a new ring is added.

Let $R_{i}=\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon _{i})\in \mathrm{Ring},$ where $x_{i}, y_{i}, r_{i}, \varepsilon _{i} \in \mathbb{R}$ , $r_{i}\gt 0, \varepsilon _{i}\gt 0,$ $i\in I$ , and let $I$ be a finite index set. Given a circular ring $R^{\ast }=\mathrm{Ring}(x^{\ast }, y^{\ast })(r^{\ast },r^{\ast }+\varepsilon ^{\ast })\in \mathrm{Ring}$ , let $r=\mathrm{min}\{r_{i} \mid$ $i\in I\}\wedge r^{\ast }$ and $\varepsilon =\mathrm{min}\{\varepsilon _{i} \mid i\in I\}\wedge \varepsilon ^{\ast }.$ We construct a sequence of sets of circular rings as follows.

Step 0: Let

\begin{equation*}L_{0}:=\{R_{i} \mid i\in I\}\cup \{R^{\ast }\}.\end{equation*}

Step 1: Set $\delta _{1}=\frac {1}{2^{3}}\varepsilon ,$ and define

\begin{equation*}L_{1}:=\bigcup _{R^{\ast }\in \mathcal{R}\subseteq L_{0}}\mathrm{ExtIntCov}_{r_{1}}^{\omega_{1}}(\mathcal{R})_{\delta _{1}},\end{equation*}

where $r_{1}\leq r+2\delta _{1}-\frac {1}{4}\varepsilon$ and $\omega_{1}=\varepsilon -2\delta _{1}.$

(Note: (1). The condition $R^{\ast }\in \mathcal{R}$ ensures that we only construct intermediate covers for new subsets containing $R^{\ast }$ ; the rest remain unchanged. (2). By Convention 4.9, $\mathrm{ExtIntCov}_{r_{1}}^{\omega_{1}}(\mathcal{R})_{\delta _{1}}$ is empty unless $\mathcal{R}$ has a non-empty intersection of inner discs).

Step 2: Set $\delta _{2}=\frac {1}{2^{4}}\varepsilon ,$ and define

\begin{equation*}L_{2}:=\bigcup _{\mathcal{R}\subseteq L_{1}}\mathrm{ExtIntCov}_{r_{2}}^{\omega_{2}}(\mathcal{R}\cup \mathrm{OSU}(\mathcal{R}))_{\delta _{2}},\end{equation*}

where $r_{2}\leq r_{1}+2\delta _{2}-\frac {1}{4}\omega_{1}$ and $\omega_{2}=\omega_{1}-2\delta _{2}.$

(Note: A new condition is introduced here. This ensures that we construct intermediate covers for families of rings whose inner disc intersections are contained in the inner discs of the underlying elements. In turn, this guarantees that the centers of newly generated rings lie within the inner discs of the underlying rings.)

Step 3: Set $\delta _{3}=\frac {1}{2^{5}}\varepsilon ,$ and define

\begin{equation*}L_{3}:=\bigcup _{\mathcal{R}\subseteq L_{2}}\mathrm{ExtIntCov}_{r_{3}}^{\omega_{3}}\left (\mathcal{R}\cup \mathrm{OSU}(\mathcal{R})\cup \mathrm{OSU}^{2}(\mathcal{R})\right )_{\delta _{3}}\!,\end{equation*}

where $r_{3}\leq r_{2}+2\delta _{3}-\frac {1}{4}\omega_{2}$ and $\omega_{3}=\omega_{2}-2\delta _{3}.$

(Note: We define $\mathrm{OSU}^{2}(\mathcal{R})=\mathrm{OSU}(\mathrm{OSU}(\mathcal{R}))$ . For $i\geq 3$ , $\mathrm{OSU}^{i}(\mathcal{R})$ is defined inductively.)

\begin{equation*}\vdots \end{equation*}

Step n: Set $\delta _{n}=\frac {1}{2^{n+2}}\varepsilon ,$ and define

\begin{equation*}L_{n}:=\bigcup _{\mathcal{R}\subseteq L_{n-1}}\mathrm{ExtIntCov}_{r_{n}}^{\omega_{n}}\left (\mathcal{R}\cup \bigcup _{i=1}^{n-1}\mathrm{OSU}^{i}(\mathcal{R})\right )_{\delta _{n}},\end{equation*}

where $r_{n}\leq r_{n-1}+2\delta _{n}-\frac {1}{4}\omega_{n-1}$ and $\omega_{n}=\omega_{n-1}-2\delta _{n}.$

(Note: Intermediate covers for subsets of rings are added in each step. Subsets are naturally ordered by inclusion, and this ordering extends to their intersections. However, the structure of an intermediate cover does not depend on the order of the sets involved. An intermediate cover is uniquely determined by the collection of subsets, not by their inclusion order. For this reason, we may safely disregard the inclusion order when introducing intermediate covers, as it has no impact on the final construction.)

Finally, define

\begin{equation*}L:=\bigcup _{i=0}^{n}L_{i}.\end{equation*}

Since $\sum _{i=1}^{n}\delta _{i}=\frac {1}{2^{2}}(1-\frac {1}{2^{n}})\varepsilon ,$ $\omega _{k}=\frac {1}{2}\varepsilon +\frac {1}{2^{k+1}}\varepsilon ,$ and $\sum _{k=1}^{n-1}\omega _{k}=\frac {n}{2}\varepsilon -\frac {1}{2^{n}}\varepsilon$ , we have

\begin{equation*}r_{n}\leq r+2\sum _{i=1}^{n}\delta _{i}-\frac {1}{4}\left (\varepsilon +\sum _{k=1}^{n-1}\omega _{k}\right )\leq r+\left (\frac {1}{4}-\frac {n}{8}-\frac {1}{2^{n+2}}\right )\varepsilon .\end{equation*}

The right-hand side of this inequality becomes negative for sufficiently large $n$ , so there exists a smallest $N\in \mathbb{N}$ such that $r_{N-1}\leq \frac {1}{4}\varepsilon .$ Thus,

\begin{align*} &\bigcup _{\mathcal{R}\subseteq L_{N+1}}\mathrm{IntCov}(\mathcal{R})=\emptyset . & &\mathrm{(Terminal\ Condition\ T)} \end{align*}

This implies that the intermediate cover construction process terminates after the $N$ -th step. Since $\delta _{N}=\frac {1}{2^{N+2}}\varepsilon$ , the extension in the final step is $\frac {1}{2^{N+2}}\varepsilon$ . In this case, we denote $L$ by

\begin{equation*}\mathrm{PLP}_{\delta }(\{R_{i}\}_{i\in I},R^{\ast })\ (\mathrm{or}\ \mathrm{PLP}(\{R_{i}\}_{i\in I},R^{\ast })\ \mathrm{for\ short}),\end{equation*}

where $\delta =\frac {1}{2^{N+2}}\varepsilon.$

We now define a suitable partial order on $\mathrm{PLP}(\{R_{i} \mid i\in I\},R^{\ast })$ such that elements of an intermediate cover can act as minimal upper bounds. Suppose a partial order $\leq$ is defined on $L_{0}$ . For $R, R^{\prime }\in \mathrm{PLP}(\{R_{i} \mid i\in I\},R^{\ast })$ , we extend the partial order by defining $R\leq R^{\prime }$ if and only if one of the following conditions holds:

1. 1. $R\leq R^{\prime }$ in $(L_{0}, \leq )$ ;

2. 2. $R$ is an underlying element of $R^{\prime }$ ;

3. 3. $R= R^{\prime }$ .

Proposition 4.12. Let $I$ be a finite index set, and for each $i\in I$ , let $R_{i}=\mathrm{Ring}(x_{i}, y_{i})(r_{i},r_{i}+\varepsilon _{i})\in \mathrm{Ring}$ where $x_{i}, y_{i}, r_{i}, \varepsilon _{i} \in \mathbb{R}$ , $r_{i}\gt 0,$ and $\varepsilon _{i}\gt 0$ . Let $R^{\ast }=\mathrm{Ring}(x^{\ast }, y^{\ast })(r^{\ast },r^{\ast }+\varepsilon ^{\ast })\in \mathrm{Ring}$ , and set $\varepsilon =\mathrm{min}\{\varepsilon _{i} \mid i\in I\}\wedge \varepsilon ^{\ast }.$ Let $D$ be a disc.

(1) If there exists a non-empty subset $I_{0}\subseteq I$ such that $D\subseteq (\bigcap _{i\in I_{0}}R_{i}^{intD})\cap (R^{\ast })^{intD}$ , then the set

\begin{equation*}\mathrm{Under}(D):=\left \{Q\in \mathrm{PLP}(\{R_{i} \mid i\in I\},R^{\ast }) \mid Q^{intD}\supseteq D\right \}\end{equation*}

has a greatest element.

(2) If there exists a non-empty subset $I_{0}\subseteq I$ such that the center of $D$ lies in $(\bigcap _{i\in I_{0}}R_{i}^{intD})\cap (R^{\ast })^{intD}$ , and there exists $\mu$ with $0\lt \mu \lt \frac {1}{2^{3}}\varepsilon$ such that $D^{-\mu }\subseteq (\bigcap _{i\in I_{0}}R_{i}^{intD})\cap (R^{\ast })^{intD}$ , then the set

\begin{equation*}\mathrm{Under}(D):=\left \{Q\in \mathrm{PLP}_{\frac {1}{2^{N+2}}\omega }(\{R_{i} \mid i\in I\},R^{\ast }) \mid Q^{intD}\supseteq D\right \}\end{equation*}

has a greatest element.

Proof. Let $L_{j}$ denote the sets defined in the construction of $\mathrm{PLP}_{\frac {1}{2^{N+2}}\omega }(\{R_{i}\}_{i\in I},R^{\ast })$ for $j=0,1,\cdots , n.$ Define

\begin{equation*}\mathrm{Under}(D)_{1}:=\left \{Q\in \bigcup _{R^{*}\in \mathcal{R}\subseteq L_{0}}\mathrm{ExtIntCov}_{r_{1}}^{w_{1}}(\mathcal{R}) \mid Q^{intD}\supseteq D\right \},\end{equation*}

and for $k=2, 3, \cdots , n$ ,

\begin{equation*}\mathrm{Under}(D)_{k}:=\left \{Q\in \bigcup _{\mathcal{R}\subseteq L_{k-1}}\mathrm{ExtIntCov}_{r_{k}}^{w_{k}}\left (\mathcal{R}\cup \bigcup _{t=1}^{k-1}\mathrm{OSU}^{t}(\mathcal{R})\right ) \mid Q^{intD}\supseteq D\right \}.\end{equation*}

(1) Since $D\subseteq (\bigcap _{i\in I_{0}}R_{i}^{intD})\cap (R^{\ast })^{intD}$ , we have $\mathrm{Under}(D)_{1}\neq \emptyset .$ If $\mathrm{Under}(D)_{1}$ is a singleton, then this element is the desired greatest element. If $|\mathrm{Under}(D)_{1}|\gt 1$ , then $\mathrm{Under}(D)_{2}\neq \emptyset .$ If $|\mathrm{Under}(D)_{2}|\gt 1$ , we repeat the process. By the terminal condition, there exists $n_{D}\in \mathbb{N}$ such that $\mathrm{Under}(D)_{n_{D}}$ is a singleton; this is the desired greatest element.

(2) By the definition of an extended intermediate cover, $\mathrm{Under}(D)_{1}\neq \emptyset .$ The result then follows by arguments analogous to those in (1).

5. The construction of a bifinite domain based on a denumerable family of circular rings

In this section, we construct a Plotkin poset from a countable family of circular rings, whose ideal completion is a bifinite domain.

To index the elements of a countable family of rings properly, we first establish some preliminary notation.

Let $\mathbb{Q}$ denote the set of rational numbers, and let $\mathbb{Q}^{+}=\{q\in \mathbb{Q} \mid q\gt 0\}.$

For integers $k, l\in \mathbb{N}$ with $1\leq k\leq l$ , define $a_{k,l}=\frac {(l-1)l}{2}+k.$ The values of $a_{k,l}$ are listed in Table 1. For any two such indices $a_{k_{1},l_{1}}$ and $a_{k_{2},l_{2}}$ , the following hold:

1. 1. $a_{k_{1},l_{1}}\lt a_{k_{2},l_{2}}$ if and only if either $l_{1}=l_{2}$ and $k_{1}\lt k_{2}$ , or $l_{1}\lt l_{2}$ ;

2. 2. $a_{k_{1},l_{1}}= a_{k_{2},l_{2}}$ if and only if $l_{1}=l_{2}$ and $k_{1}=k_{2}$ .

We first note that $\mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ is a countable set, and thus there exists a bijection between $\mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ and $\mathbb{N}$ . For any element $(q_{1},q_{2},q_{3})\in \mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ , we denote its image under this bijection by $N_{(q_{1},q_{2},q_{3})}$ .

Table 1.

The value of $a_{k,l}$

Table 1. Long description

A table listing values for integers with specific conditions. The table has four columns and five rows, including the header row. The columns are labeled k, 1, 2, and 3. The rows are labeled 1, 2, 3, and an ellipsis indicating continuation. The values in the table are as follows: Row 1: k, 1, 2, 4. Row 2: 1, 1, 2, 4. Row 3: 2, 3, 5. Row 4: 3, 6. The table defines the values of k for integers with specific conditions and lists the values accordingly.

Next, we introduce a set of closed circular rings with rational centers and rational radii. Let

\begin{equation*}\mathrm{QRing}\end{equation*}

be the following set of closed circular rings in the plane:

\begin{align*} \left \{\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {2}{10^{a_{k,l}}}\Big) \mid q_{1},q_{2}\in \mathbb{Q}, q_{3}\in \mathbb{Q}^{+}, k=N_{(q_{1},q_{2},q_{3})}, l\in \mathbb{N}, k\leq l\right \}. \end{align*}

The centers and radii of rings in $\mathrm{QRing}$ are rational numbers. The widths of rings in $\mathrm{QRing}$ are pairwise distinct. For

\begin{equation*}R=\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {2}{10^{a_{k,l}}}\Big), R^{\prime }=\mathrm{Ring}(q_{1}^{\prime },q_{2}^{\prime })\Big(q_{3}^{\prime }+\frac {1}{10^{a_{k^{\prime },l^{\prime }}}}, q_{3}^{\prime }+\frac {2}{10^{a_{k^{\prime },l^{\prime }}}}\Big)\in \mathrm{QRing},\end{equation*}

we define $R\leq R^{\prime }$ if and only if one of the following conditions holds:

1. 1. $a_{k,l}\leq a_{k^{\prime }, l^{\prime }}$ and $R^{intD}\supseteq (R^{\prime })^{outD},$

2. 2. $R= R^{\prime }.$

Then, $(\mathrm{QRing},\leq )$ is a poset. For $R, R^{\prime }\in \mathrm{QRing}$ , if $R\lt R^{\prime }$ , then $w_{R^{\prime }}\leq \frac {1}{10}w_{R}.$ This means that elements in $\mathrm{QRing}$ are ordered not only by inclusion but also such that greater elements have narrower widths. For every $R\in \mathrm{QRing}$ , since there are finitely many elements below $R$ , $R$ is compact. Furthermore, $\mathrm{QRing}$ is an algebraic poset.

It is noteworthy that for every triple of rationals $(q_{1},q_{2},q_{3})\in \mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ ,

\begin{equation*}\left \{\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {2}{10^{a_{k,l}}}\Big)\in \mathrm{QRing} \mid k=N_{(q_{1},q_{2},q_{3})}, l\in \mathbb{N}, k\leq l\right \}\end{equation*}

is totally ordered, and

\begin{equation*}\mathrm{Disc}(q_{1},q_{2})(q_{3})=\bigcap \left \{\mathrm{Disc}(q_{1},q_{2})\Big(q_{3}+\frac {2}{10^{a_{k,l}}}\Big) \mid k=N_{(q_{1},q_{2},q_{3})}, l\in \mathbb{N}, k\leq l\right \}.\end{equation*}

For a disc $D\in \mathrm{Disc}$ , let $D^{o}$ denote the interior of $D$ . Furthermore, for every $D\in \mathrm{Disc}$ , the set $\{R\in \mathrm{QRing} \mid D\subseteq (R^{intD})^{o}\}$ is a directed subset of $\mathrm{QRing}$ , and we have

\begin{equation*}D=\bigcap \left \{R^{outD} \mid R\in \mathrm{QRing},D\subseteq (R^{intD})^{o}\right \}.\end{equation*}

These properties are useful for the construction of a retraction map later.

For $n\in \mathbb{N}$ , there is a unique pair of natural numbers $k, l$ such that $a_{k,l}=n$ . Furthermore, there is a unique triple of rational numbers $(q_{1},q_{2},q_{3})$ such that $N_{(q_{1},q_{2},q_{3})}=k.$ Let

\begin{equation*}\mathrm{QR}_{n}=\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {2}{10^{a_{k,l}}}\Big)\end{equation*}

such that $a_{k,l}=n$ and $N_{(q_{1},q_{2},q_{3})}=k.$ . We have

1. 1. $w_{\mathrm{QR}_{n}}=\frac {1}{10^{n}}$ ;

2. 2. $\{\mathrm{QR}_{n} \mid n\in \mathbb{N}\}=\mathrm{QRing}.$

Thus, elements in $\mathrm{QRing}$ are indexed such that a larger index corresponds to a smaller width. For $n_{1}, n_{2}\in \mathbb{N}$ with $n_{1}\lt n_{2}$ , if there exist natural numbers $k, l_{1}, l_{2}$ with $l_{1}\lt l_{2}$ such that $a_{k,l_{1}}=n_{1}$ and $a_{k,l_{2}}=n_{2}$ , then $(\mathrm{QR}_{n_{1}})^{intD}\supseteq (\mathrm{QR}_{n_{2}})^{outD}.$

Next, we inductively construct a Plotkin-poset based on $\mathrm{QRing}$ . The main idea is as follows. We start with the first element of $\mathrm{QRing}$ . Then, we use the next element from $\mathrm{QRing}$ to construct the Plotkin-poset in the next step. When constructing the next stage, we work “disjointly” from the existing structure so that minimal upper bounds remain unchanged. To achieve this separation, we use primed elements. The pattern will become clearer from Step 3 onward.

Step 1: For $n=1,$ let

\begin{equation*}\mathrm{QR}_{1}^{\prime }=\mathrm{QR}_{1},\end{equation*}

\begin{equation*}L_{1}=\{\mathrm{QR}_{1}^{\prime }\}.\end{equation*}

Step 2: For $n=2,$ let

\begin{equation*}\mathrm{QR}_{2}^{\prime }=\mathrm{QR}_{2},\end{equation*}

\begin{equation*}L_{2}^{\prime }=\{\mathrm{Ring}(x, y)\Big(r, r+\frac {1}{10^{2}}\Big) \mid \mathrm{Ring}(x, y)(r, r^{\prime })\in L_{1}\},\end{equation*}

\begin{equation*}L_{2}=L_{1}\cup \mathrm{PLP}_{\delta _{2}}(L_{2}^{\prime },\mathrm{QR}_{2}^{\prime })\setminus L_{2}^{\prime }, \omega _{2}=\frac {1}{10^{2}}, \delta _{2}=\frac {1}{2^{N_{2}+2}}\omega _{2},\end{equation*}

where $N_{2}$ is the natural number representing the number of steps in the construction of $\mathrm{PLP}_{\delta _{2}}(L_{2}^{\prime },\mathrm{QR}_{2}^{\prime })$ . The existence of $N_{2}$ is guaranteed by the termination condition $\mathrm{T}$ . The meanings of $N_{3}, \cdots$ in subsequent steps are analogous to that of $N_{2}$ .

Step 3: For $n=3,$ let

\begin{equation*}L_{3}^{\prime }=\{\mathrm{Ring}(x, y)\Big(r, r+\frac {1}{10^{3}}*\delta _{2}\Big) \mid \mathrm{Ring}(x, y)(r, r^{\prime })\in L_{2}\},\end{equation*}

\begin{equation*}\mathrm{QR}_{3}^{\prime }=\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {1}{10^{a_{k,l}}}+\frac {1}{10^{a_{k,l}}}*\delta _{2}\Big)\end{equation*}

with $a_{k,l}=3$ and $N_{(q_{1},q_{2},q_{3})}=k.$ Also let

\begin{equation*}L_{3}=L_{2}\cup \mathrm{PLP}_{\delta _{3}}(L_{3}^{\prime }, \mathrm{QR}_{3}^{\prime })\setminus L_{3}^{\prime }, \omega _{3}=\frac {1}{10^{3}}, \delta _{3}=\frac {1}{2^{N_{3}+2}}\omega _{3}.\end{equation*}

\begin{equation*}\vdots \end{equation*}

Step n: Let

\begin{equation*}L_{n}^{\prime }=\{\mathrm{Ring}(x, y)\Big(r, r+\frac {1}{10^{n}}*\delta _{n-1}\Big) \mid \mathrm{Ring}(x, y)(r, r^{\prime })\in L_{n-1}\},\end{equation*}

\begin{equation*}\mathrm{QR}_{n}^{\prime }=\mathrm{Ring}(q_{1},q_{2})\Big(q_{3}+\frac {1}{10^{a_{k,l}}}, q_{3}+\frac {1}{10^{a_{k,l}}}+\frac {1}{10^{a_{k,l}}}*\delta _{n-1}\Big)\end{equation*}

with $a_{k,l}=n$ and $N_{(q_{1},q_{2},q_{3})}=k.$ Also let

\begin{equation*}L_{n}=L_{n-1}\cup \mathrm{PLP}_{\delta _{n}}(L_{n}^{\prime },\mathrm{QR}_{n}^{\prime })\setminus L_{n}^{\prime }, \omega _{n}=\frac {1}{10^{n}}, \delta _{n}=\frac {1}{2^{N_{n}+2}}\omega _{n}.\end{equation*}

(Note: The widths of elements in $L_{n}^{\prime }$ and $\mathrm{QR}_{n}^{\prime }$ are equal to $\frac {1}{10^{n}}*\delta _{n-1}$ , which is less than the least extension $\delta _{n-1}$ in $L_{n-1}$ .)

\begin{equation*}\vdots \end{equation*}

Finally, we define

\begin{equation*}\mathrm{PQRing}=\bigcup _{n=1}^{\infty }L_{n}.\end{equation*}

Let $\mathrm{QRing}^{\prime }=\{\mathrm{QR}_{n}^{\prime } \mid n\in \mathbb{N}\}$ . Elements in $\mathrm{QRing}^{\prime }$ are in one-to-one correspondence with those in $\mathrm{QRing}$ , so we can define an order on $\mathrm{QRing}^{\prime }$ using the order on $\mathrm{QRing}$ . Then, $\mathrm{QRing}^{\prime }$ shares many properties with $\mathrm{QRing}$ , which are listed in the following proposition.

Proposition 5.1. (1) $\mathrm{QRing}^{\prime }$ is an algebraic poset.

(2) For every triple of rational numbers $(q_{1},q_{2},q_{3})\in \mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ ,

\begin{equation*}\{\mathrm{QR}_{n}^{\prime }\in \mathrm{QRing}^{\prime } \mid k=N_{(q_{1},q_{2},q_{3})}, l\in \mathbb{N}, k\leq l, n=a_{k,l}\}\end{equation*}

is a directed subset of $\mathrm{QRing}^{\prime }$ .

(3) For every triple of rational numbers $(q_{1},q_{2},q_{3})\in \mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}^{+}$ ,

\begin{equation*}\mathrm{Disc}(q_{1},q_{2})(q_{3})=\bigcap \{(\mathrm{QR}_{n}^{\prime })^{outD} \mid k=N_{(q_{1},q_{2},q_{3})}, l\in \mathbb{N}, k\leq l, n=a_{k,l}\}.\end{equation*}

(4) For every $D\in \mathrm{Disc}$ , $\{R\in \mathrm{QRing}^{\prime } \mid D\subseteq (R^{intD})^{o}\}$ is a directed subset of $\mathrm{QRing}^{\prime }$ and

\begin{equation*}D=\bigcap \{R^{outD} \mid R\in \mathrm{QRing}^{\prime }, D\subseteq (R^{intD})^{o}\}.\end{equation*}

Since $\mathrm{QR}_{i}^{\prime }\in \mathrm{PLP}(L_{i}^{\prime },\mathrm{QR}_{i}^{\prime })$ for $i=2,3,\cdots ,n$ , we have $\mathrm{QRing}^{\prime }\subseteq \mathrm{PQRing}.$ The set $\mathrm{PQRing}$ consists of two parts. One part is $\mathrm{QRing}^{\prime }$ . The other part comprises elements generated by the construction of extended intermediate covers. Every element in $\mathrm{PQRing}\setminus \mathrm{QRing}^{\prime }$ has a set of underlying elements. These underlying elements are constructed from $\mathrm{QRing}^{\prime }$ . By construction, we have $L_{1}\subseteq L_{2}\subseteq \cdots \subseteq L_{n}\subseteq \cdots .$

In the construction of $\mathrm{PQRing}$ , a new element from $\mathrm{QRing}^{\prime }$ is added at each step. In the construction of $L_{n}$ for $n\geq 2$ , we work with $L_{n}^{\prime }$ . This is to keep the minimal upper bounds for every subset of $L_{n-1}$ unchanged in subsequent steps. We will explain this in detail after an order is defined.

Elements in $L_{n}^{\prime }$ form bridges between elements in $L_{n-1}$ and elements in $L_{n}\setminus L_{n-1}$ . An element $R$ in $L_{n-1}$ is called a one-step underlying element of the corresponding element $R^{\prime }$ in $L_{n}^{\prime }$ . If $R^{\prime }\in L_{n}^{\prime }$ is an underlying element of an element $T\in L_{n}$ , then $R$ is called an underlying element of $T$ by transitivity.

For convenience in describing the order on $\mathrm{PQRing}$ , every element is regarded as an underlying element of itself.

We now define an order on $\mathrm{PQRing}$ .

According to Lemmas 3.3 and 5.6, we obtain the following result.

Define maps $e: \mathrm{Disc}\longrightarrow \mathrm{Id}(\mathrm{PQRing}_{\bot })$ and $r: \mathrm{Id}(\mathrm{PQRing}_{\bot })\longrightarrow \mathrm{Disc}$ as follows:

\begin{equation*}e(D)=\downarrow _{\mathrm{PQRing}_{\bot }}\{R\in \mathrm{QRing}^{\prime } \mid D\subseteq (R^{intD})^{o}\}\ \mathrm{for\ every}\ D\in \mathrm{Disc},\end{equation*}

\begin{equation*}r(I)=\bigcap _{R\in I} R^{outD}\ \mathrm{for\ every}\ I\in \mathrm{Id}(\mathrm{PQRing}_{\bot }).\end{equation*}

Proof. Clearly, $r$ and $e$ are well-defined and order-preserving.

Suppose $\{I_{j}\}_{j\in J}$ is a directed subset of $\mathrm{Id}(\mathrm{PQRing}_{\bot })$ . Then,

\begin{equation*}r\left (\bigvee _{j\in J}I_{j}\right )=r\left (\bigcup _{j\in J}I_{j}\right )=\bigcap _{R\in \bigcup _{j\in J}I_{j}}R^{outD}=\bigcap _{j\in J}\bigcap _{R\in I_{j}}R^{outD}=\bigvee _{j\in J}\bigcap _{R\in I_{j}}R^{outD}=\bigvee _{j\in J}r\left (I_{j}\right ).\end{equation*}

Thus, $r$ is Scott-continuous.

Suppose $\{D_{j}\}_{j\in J}$ is a directed subset of $\mathrm{Disc}$ . Then, $e\left (\bigvee _{j\in J}D_{j}\right )=e\left (\bigcap _{j\in J}D_{j}\right )=\downarrow _{\mathrm{PQRing}_{\bot }}\{R\in \mathrm{QRing}^{\prime } \mid \bigcap _{j\in J}D_{j}\subseteq (R^{intD})^{o}\}=\bigcup _{j\in J}\downarrow _{\mathrm{PQRing}_{\bot }}\{R\in \mathrm{QRing}^{\prime } \mid D_{j}\subseteq (R^{intD})^{o}\}=\bigvee _{j\in J}e(D_{j}).$ Thus, $e$ is Scott-continuous.

For $D\in \mathrm{Disc}$ ,

\begin{equation*}re(D)=\bigcap _{R\in e(D)} R^{outD}=\bigcap \{R^{outD} \mid R\in \mathrm{QRing}^{\prime },D\subseteq (R^{intD})^{o}\}=D.\end{equation*}