2. Computability via Ordered Sets

In this section, we briefly recall the fundamental notions of an order-theoretic approach to computability on uncountable sets, which was recently introduced by Hack et al. Reference Hack, Braun and Gottwald2022a). We will define a structure that carries computability from Turing machines, namely directed complete partial orders with an effective weak basis. We do not address how computability can be translated from representatives of this structure to other spaces of interest (see Hack et al. Reference Hack, Braun and Gottwald2022a and the references therein).

Before introducing directed complete partial orders, we include some definitions of the formal approach to computability on $\mathbb{N}$ based on Turing machines.

Recursively enumerable sets are, thus, the subsets of $\mathbb{N}$ whose elements can be produced in finite time, as we can introduce the natural numbers one by one in increasing order in a Turing machine and it will output one by one, each in finite time, all the elements in $A$ (possibly with repetitions). Note that there exist subsets of $\mathbb{N}$ which are not recursively enumerable (Rogers Reference Rogers1987). As we are also interested in computability on the subsets of $\mathbb{N}^2$ , we translate the notion of recursively enumerable sets from $\mathbb{N}$ to $\mathbb{N}^2$ using pairing functions. A pairing function $\langle \cdot,\cdot \rangle$ is a computable bijective function $\langle \cdot,\cdot \rangle : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ .Footnote 1 Since it is a common practice (Rogers Reference Rogers1987), we fix in the following $\langle n,m \rangle = \frac{1}{2} ( n^2+ 2nm + m^2 + 3n + m)$ , the Cantor pairing function.

Before continuing, we introduce an important concept for the following, finite maps.

Finite maps aim to translate computability from the natural numbers to another countable set $A$ . Note the definition of finite maps relies on the informal notion of effective calculability. This is the case since no general formal definition for computable maps $\alpha : \mathbb{N} \to A$ is known (Cutland Reference Cutland1980). In fact, the struggle between formal and informal notions of computability, best exemplified by Church’s thesis (Rogers Reference Rogers1987), lies at the core of computability theory and is responsible for the introduction of different formal notions of computability (Cutland Reference Cutland1980, Chapter 3). Finite maps are also known as effective denumerations (Cutland Reference Cutland1980) or effective enumerations (Scott Reference Scott1970).

We define now the order structure on which we rely to introduce computability in some (potentially uncountable) set $P$ and connect it, right after, with Turing machines.

We may think of $P$ as a set of data and of $\preceq$ as a representation of the precision (or information) relation between different elements in the set. Given $x,y \in P$ , we may read $x \preceq y$ like $y$ is at least as informative as $x$ or like $y$ is at least as precise as $x$ .

We intend now to introduce the idea of some $x \in P$ being the limit of other elements in $P$ , that is, the idea that one can generate some element $y \in P$ via a process that outputs other elements of $P$ (which approximate $y$ to arbitrary precision). This notion is formalized by the least upper bounds of directed sets.

Hence, we can generate some $x \in P$ by generating a directed set $A$ whose upper bound is $x$ , $x = \sqcup A$ . We have restricted ourselves to directed sets since we can think of them as the output of some computational process augmenting the precision or information given that, for any pair of outputs, there is a third that contains their information and, potentially, more. Directed sets are, thus, a formalization of a computational process having a direction, that is, processes gathering information in a consistent way. Of particular importance are increasing sequences or increasing chains, subsets $A \subseteq P$ where $A=(a_n)_{n\geq 0}$ , and $a_n \preceq a_{n+1}$ $\text{for all } n \geq 0$ . We can interpret increasing sequences as the output of some process where information increases every step.

Any process whose outputs increase information should tend toward some element in $P$ , that is, any directed set $A \subseteq P$ should have a supremum $\sqcup A \in P$ . A partial order with such a property is called directed complete or a dcpo. Note, for example, the partial order $P_0=\big ((0,1),\leq \big)$ is not directed complete.

Some subsets $B \subseteq P$ are able to generate all the elements in $P$ via the supremum of directed sets contained in $B$ . We refer to them as weak bases.

We are particularly interested in dcpos where countable weak bases exist, since we intend to inherit computability from Turing machines. In case we have some computational process whose outputs are in $B$ and which is approaching some $x \in P\setminus B$ , we would like to be able to provide, after a finite amount of time, the best approximation of $x$ so far. In order to do so, we need to distinguish the outputs we already have in terms of precision. This is possible if the weak basis is effective.

The intuition behind the effectivity is that we can, by finite means, get progressively more informative elements from some directed set. Note we may show a countable weak basis $B=(b_n)_{n\geq 0}$ is effective by proving the stronger property that

\begin{equation*} \{\langle n,m \rangle | b_n \preceq b_m\} \end{equation*}

is recursively enumerable. If this stronger condition is satisfied, then, for any finite subset $(b_n)_{n=1}^N \subseteq B$ where all elements are related, we can find some $n_0 \leq N$ such that $b_n \preceq b_{n_0}$ $\text{for all } n \leq N$ and, since $\alpha : \mathbb{N} \to B$ is a finite map, we can determine the best approximation so far, $\alpha (n_0)$ . We define now computable elements for dcpos with an effective weak basis.

We have achieved the goal of deriving computability (for potentially uncountable sets) from Turing machines via dcpos. Note that computable elements generalize the approach by Turing to computability on $P_{inf}(\mathbb{N})$ , the family of infinite subsets of $\mathbb{N}$ (see Hack et al. Reference Hack, Braun and Gottwald2022a). The dependence of computability on the order-theoretic model is also addressed by Hack et al. Reference Hack, Braun and Gottwald2022a).

To recapitulate, the main features of our picture are $(1)$ a map from the natural numbers to some countable set of finite labels $B=(b_n)_{n\geq 0}$ and $(2)$ a partial order $\preceq$ which can be somewhat encoded via a Turing machine and which allows us to both associate to some infinite element of interest $x \in P$ a subset of our labels $B_x \subseteq B$ which converges to it and, in some sense, to provide approximations of $x$ to arbitrary precision. As a result, the computability of $x$ reduces to whether $B_x$ can be finitely described or not.

Note that the structure of the partial order $P$ is fundamental to address higher type computability. In case $P$ is trivial (also known as discrete), that is, $x \preceq y$ if and only if $x=y$ $\text{for all } x,y \in P$ (Abramsky and Jung Reference Abramsky and Jung1994), we cannot extend computability beyond countable sets and we end up considering countable sets with finite maps toward the natural numbers. This situation, thus, reduces our approach to the theory of numberings (Badaev and Goncharov, Reference Badaev and Goncharov2000, Reference Badaev and Goncharov2008; Ershov Reference Ershov1999).

While the set of computable elements in a dcpo is countable, since the set of recursively enumerable subsets of $\mathbb{N}$ is countable (Rogers Reference Rogers1987), the cardinality of a dcpo with an effective weak basis is bounded by the cardinality of the continuum $\mathfrak{c}$ (see Hack et al. Reference Hack, Braun and Gottwald2022a). In fact, it is in the uncountable case where the order structure is of interest since the theory of numbering is insufficient.

2.1 Examples

To conclude this section, we list three examples of dcpos with effective weak bases, which will be relevant in the following.

2.1.1 The Cantor domain

If $\Sigma$ is any finite set of symbols, an alphabet, we denote by $\Sigma ^*$ the set of finite strings of symbols in $\Sigma$ and by $\Sigma ^\omega$ the set of countably infinite sequences of symbols. The union of these last two sets is called the Cantor domain or the Cantor set model (Blanck Reference Blanck2008; Martin and Mislove Reference Martin and Mislove2000) when we equip it with the prefix order. That is, the Cantor domain is the pair $(\Sigma ^{\infty },\preceq _C)$ , where

(1) \begin{equation} \begin{split} \Sigma ^\infty &:= \Big \{x\, \Big |\, x:\{1,\ldots,n\} \to \Sigma,\text{ }0\leq n\leq \infty \Big \}, \\ x \preceq _C y &\iff |x| \leq |y| \text{ and } x(i)=y(i) \text{ } \text{for all } i \leq |x|, \end{split} \end{equation}

$|s|$ is the cardinality of the domain of $s \in \Sigma ^\infty$ , and $|\Sigma |\lt \infty$ . One can see $\Sigma ^*$ is an effective weak basis for $\Sigma ^{\infty }$ (Hack et al. Reference Hack, Braun and Gottwald2022a).

2.1.2 The interval domain

The interval domain (Di Gianantonio Reference Di Gianantonio1996; Edalat and Sünderhauf Reference Edalat and Sünderhauf1999; Scott Reference Scott1970) consists of the pair $(\mathscr{I}, \sqsubseteq)$ , where

(2) \begin{equation} \begin{split} \mathscr{I}:=& \Big \{[a,b] \subseteq \mathbb{R}\, \Big |\, a,b \in \mathbb{R},\text{ }a \leq b\Big \}\cup \Big \{\perp \Big \}, \text{ and } \\ x \sqsubseteq y \iff & x= \perp \text{or } x=[a,b], y=[c,d], a \leq c \text{ and } d \leq b. \end{split} \end{equation}

Note that one can see $B_{\mathscr{I}} := \{[p,q] \subseteq \mathbb{R}|p \leq q,\text{ }p,q \in \mathbb{Q}\} \cup \{\perp \}$ is an effective weak basis for $(\mathscr{I}, \sqsubseteq)$ . If $P$ is a partial order, we will denote by $\perp$ an element $x \in P$ , if it exists, such that $x \preceq y$ $\text{for all } y \in P$ , as we just did for the interval domain.

2.1.3 Majorization

For any $n \geq 2$ , majorization (Gottwald and Braun Reference Gottwald and Braun2019; Martin Reference Martin2006; Marshall et al. Reference Marshall, Olkin and Arnold1979) consists of the pair $(\Lambda ^n,\preceq _M)$ , where

(3) \begin{equation} \begin{split} \Lambda ^n := \bigg \{ x\in [0,1]^n \,\bigg |\, &\sum _{i=1}^n x_i = 1 \text{ and } (\text{for all } i\lt n)\text{ } x_i \geq x_{i+1} \bigg \}, \\ x \preceq _M y &\iff (\text{for all } i\lt n)\text{ } s_i(x) \leq s_i(y), \end{split} \end{equation}

and $s_i(x) := \sum _{j=1}^i x_j$ $\text{for all } i\lt n$ . Note that majorization has an effective weak basis, as the following proposition (which we prove in the Appendix A.2.1) states.

Proposition 1. If $n\geq 2$ , then $\mathbb{Q}^n \cap \Lambda ^n$ is an effective weak basis for majorization.

Proposition 1 relies on a stronger property fulfilled by majorization, which we state in Lemma 1 and prove in the Appendix A.2.4.

Lemma 1. If $x \in \Lambda ^n\setminus \{\perp \}$ , then $\text{for all } \varepsilon \gt 0$ there exists some $q \in \mathbb{Q}^n \cap \Lambda ^n$ such that $s_k(x)-\varepsilon \lt s_k(q) \lt s_k(x)$ $\text{for all } k\lt n$ .

3. Order Density and Weak Bases

In the approach to computability from Section 2, dcpos with countable weak bases are the fundamental structure allowing to define computability. In this section, we relate order density properties to countable weak bases. In order to do so, we need some extra terminology. If $P$ is a partial order and $x,y \in P$ , we say $y$ is strictly preferred to $x$ or $x$ is strictly below $y$ for $x,y \in P$ and denote it by $x \prec y$ if $x \preceq y$ and $\neg (y \preceq x)$ hold. In case we have $\neg (x \preceq y)$ and $\neg (y \preceq x)$ , we say $x$ and $y$ are incomparable and denote it by $x \bowtie y$ . We introduce now two order density properties, which will play a major role in the following.

To exemplify the previous properties, note that $\Sigma ^*$ is a countable Debreu dense subset of the Cantor domain $\Sigma ^\infty$ since, given $a,b \in \Sigma ^\infty$ such that $a \prec _C b$ , we have $a \in \Sigma ^*$ and, taking $d=a$ , we get $a \preceq _C d \preceq _C b$ . Moreover, note that $\Sigma ^*$ is a countable Debreu upper dense subset of the Cantor domain since, given $a,b \in \Sigma ^\infty$ such that $a \bowtie b$ , we can either pick $d=b$ if $b \in \Sigma ^*$ or some $d \in \Sigma ^*$ such that $a \bowtie d \prec b$ if $b \in \Sigma ^\omega$ . Lastly, in particular, the Cantor domain is Debreu upper separable.

As a first relation between order density and weak bases, we show, in Proposition 2, that having a countable weak basis implies there is a countable Debreu upper dense subset and also a countable set that fulfills a weak form of Debreu density. In fact, although we are primarily interested in the countable case, we show a more general relation, where countability plays no role.

Because of the proof of Proposition 2, it seems having a countable weak basis is insufficient for a countable Debreu dense subset to exist. This is indeed the case, as we show in Proposition 3 via a counterexample.

Proof. Consider $P := ([0,1]\cup [2,3],\preceq)$ where

\begin{equation*} x \preceq y \iff \begin {cases} x\leq y \text { and } x,y \in [0,1], \text { or}\\ x \leq y \text { and } x,y \in [2,3], \text { or}\\ x+2 \leq y. \end {cases} \end{equation*}

(See Fig. 1 for a representation of $P$ .) Since $P$ is clearly a partial order, we show now it is directed completely. Take $A \subseteq P$ directed. If we have either $A \subseteq [0,1]$ or $A \subseteq [2,3]$ , then $\sqcup A$ exists and is the supremum of $A$ in the usual $\mathbb{R},\leq)$ sense. If $A$ has elements in both, then it is easy to see that the supremum of $A \cap [2,3]$ with respect to $\leq$ is the supremum of $A$ with respect to $\preceq$ . Analogously, $B := \mathbb{Q} \cap P$ is a countable weak basis. To conclude, assume $D \subseteq P$ is Debreu dense set. There exist, then, $d_x \in D$ such that $x \preceq d_x \preceq x+2$ $\text{for all } x \in [0,1]$ . By definition, we have $d_x \in \{x,x+2\}$ . In particular, $d_x \neq d_y$ $\text{for all } x,y \in [0,1]$ $x \neq y$ and $D$ is uncountable.

Figure 1.

Representation of a dcpo, defined in Proposition 3, with a countable weak basis and no countable Debreu dense subset. In particular, we show $A := [0,1]$ , $B := [2,3]$ , and how $x,y,z \in A$ , $x\lt y\lt z$ , are related to $x+2,y+2,z+2 \in B$ . Notice an arrow from an element $w$ to an element $t$ represents $w \prec t$ .

We consider now the converse of Proposition 2, that is, whether countable weak bases exist for any Debreu separable dcpo. Some extra terminology is needed for this purpose.

Intuitively, non-trivial (trivial) elements are those for which there is (there is not) a computational process that converges to them without ever actually outputting them. This is important since we may have some element that requires infinite precision, like say $\pi$ in decimal representation, which may be achievable to arbitrary precision via some algorithm that only outputs elements with a finite representation.

We prove, in Theorem 1, that if $P$ is a dcpo with a weak basis $B\subseteq P$ and a countable Debreu dense subset, then there exists a dcpo $Q$ with a countable weak basis such that for the non-trivial set of $B$ is included in $Q$ , $\mathscr{N}_B \subseteq Q$ . In order to do so, we first need two lemmas. We start, in Lemma 2, recalling the straightforward fact that, whenever the supremum of a directed set $A$ is not contained in the set, we can find $\text{for each } a \in A$ some $b \in A$ that is strictly preferred to $a$ , $a \prec b$ (see Hack et al. Reference Hack, Braun and Gottwald2022a, Lemma 1). Right after, in Lemma 3, we show that, if $x \in P$ has a non-trivial directed set $A \subseteq P$ and there exists a countable Debreu dense subset $D\subseteq P$ , then $x$ has a non-trivial increasing sequence contained in $D$ . In order to do so, we first consider the elements from $D$ , which are between (in $\preceq$ ) those in $A$ , and then profit from the countability of $D$ to build an increasing sequence converging to the supremum of $A$ .

Lemma 3. If $P$ is a dcpo with a countable Debreu dense subset $D\subseteq P$ , $x \in P$ is an element with a non-trivial directed set $A \subseteq P$ , and

(4) \begin{equation} D_A:=\{d \in D\, | \, \exists a,b \in A \text{ s.t. } a \preceq d \preceq b \} \end{equation}

is a subset of $D$ with a numeration $D_A=(d_n)_{n\geq 0}$ , then $(d^{\prime}_n)_{n\geq 0}$ is a non-trivial increasing sequence for $x$ , where

(5) \begin{equation} \begin{split} d_0^{\prime} &:= d_0,\\ d_n^{\prime} &:= d_{m_n} \text{ for all } n \geq 1, \end{split} \end{equation}

and $m_n \geq 0$ fulfills $d_{n-1}^{\prime} \preceq d_{m_n}$ and $d_n \preceq d_{m_n}$ for all $n \geq 1$ .

Note, as a consequence of Lemma 3, whenever a dcpo $P$ is Debreu separable, any directed set $A \subseteq P$ contains an increasing sequence $(a_n)_{n\geq 0} \subseteq A$ such that $\sqcup (a_n)_{n\geq 0}=\sqcup A$ (see Proposition 24 in the Appendix 1).

Using Lemmas 2 and 3, we can now prove Theorem 1, our first construction of countable weak bases using order density properties. Intuitively, Theorem 1 considers $Q$ , a subset of some dcpo $P$ , and takes all the elements in some basis that can be achieved non-trivially and profits from the sequences in Lemma 3 to achieve them via a countable Debreu dense subset. Lastly, it adds elements to $Q$ in order to assure it is a dcpo.

Theorem 1. If $P$ is a dcpo with a countable Debreu dense set $D\subseteq P$ and a weak basis $B\subseteq P$ , then the dcpo $(Q,\preceq)$ fulfills $D \cup \mathscr{N}_B \subseteq Q$ and has $D$ as countable weak basis, where

\begin{equation*} Q := D \cup \{x \in P\setminus D\,|\, \exists \text { a directed set } A \subseteq D \text { s.t. } \sqcup A=x\} \end{equation*}

and $\preceq$ is the restriction to $Q$ of the partial order in $P$ .

Proof. We will show $(Q,\preceq)$ is a dcpo with a countable weak basis such that $D \cup \mathscr{N}_B \subseteq Q$ . Note $\mathscr{N}_B \subseteq Q\setminus D$ by Lemma 3. To conclude the proof, we only need to show $Q$ is directed completely as, after this is established, it is clear by definition that $D \subseteq Q$ is a countable weak basis of $Q$ . Take, thus, a directed set $A \subseteq Q$ with $\sqcup A \not \in A$ , as the opposite is straightforward. Note $\sqcup A \in P$ , since $P$ is a dcpo, and we intend to show $\sqcup A \in Q$ . To begin with, consider

\begin{equation*} A^{\prime} := A \cup \{d \in D\,|\, \exists a,b \in A \text { s.t. } a \preceq d \preceq b\} \subseteq Q, \end{equation*}

which is straightforwardly a directed set. Note $\sqcup A^{\prime}$ exists in $P$ and, actually, $\sqcup A^{\prime}=\sqcup A$ , as we have $(1)$ $\text{for all } a^{\prime} \in A^{\prime}$ there exists some $a \in A$ such that $a^{\prime}\preceq a$ , which leads to $a^{\prime} \preceq \sqcup A$ $\text{for all } a^{\prime} \in A$ , and $(2)$ if $a^{\prime}\preceq y$ $\text{for all } a^{\prime}\in A^{\prime}$ then $a \preceq y$ $\text{for all } a \in A$ and we get $\sqcup A \preceq y$ . Consider now

\begin{equation*} A^{\prime\prime} := \Big (A^{\prime} \cap D\Big) \bigcup _{x \in A^{\prime}\setminus D} B_x, \end{equation*}

where, $\text{for all } x \in A^{\prime}\setminus D$ , $B_x \subseteq D$ is a directed set, which exists by definition of $Q$ , such that ${\sqcup}B_x=x$ . We conclude by showing that $(1)$ $A^{\prime\prime}$ is directed and that $(2)$ $\sqcup A^{\prime\prime}=\sqcup A$ , which imply, since $A^{\prime\prime} \subseteq D$ , that $\sqcup A = \sqcup A^{\prime\prime} \in Q$ . As a result, $Q$ is directed complete, as we intended to prove. To show $(1)$ , we take $a,b \in A^{\prime\prime}$ and consider four cases: (a) $a,b \in B_x$ for some $x \in A^{\prime}\setminus D$ , (b) $a,b \in A^{\prime} \cap D$ , (c) $a \in B_x, b \in B_{x^{\prime}}$ with $x,x^{\prime} \in A^{\prime}\setminus D$ $x \neq x^{\prime}$ and (d) $a \in A^{\prime} \cap D$ and $b \in B_x$ for some $x \in A^{\prime}\setminus D$ . In (a) there exists some $c \in B_x$ such that $a,b \preceq c$ since $B_x$ is a directed set. (b) holds as there exists some $y \in A$ such that $a,b \preceq y$ and some $y^{\prime} \in A$ such that $y \prec y^{\prime}$ by Lemma 2. We obtain there exists some $c \in A^{\prime} \cap D$ such that $a,b\preceq y \preceq c \preceq y^{\prime}$ . (c) holds as there exists some $y \in A$ such that $x,x^{\prime}\preceq y$ and we can follow (b) to get some $c \in A^{\prime} \cap D$ such that $a \preceq x \preceq c$ and $b \preceq x^{\prime} \preceq c$ . (d) holds similarly to (c). In order to finish, we only need to show $(2)$ holds. Since $\text{for all } a^{\prime\prime} \in A^{\prime\prime}$ there exists some $a^{\prime} \in A^{\prime}$ such that $a ^{\prime\prime}\preceq a^{\prime}$ , we have $a ^{\prime\prime}\preceq \sqcup A^{\prime}=\sqcup A$ $\text{for all } a^{\prime\prime} \in A^{\prime\prime}$ . Moreover, if we have $a^{\prime\prime}\preceq z$ $\text{for all } a^{\prime\prime}\in A^{\prime\prime}$ , then we have, by definition of $A^{\prime\prime}$ , $a^{\prime} \preceq z$ $\text{for all } a^{\prime}\in A^{\prime} \cap D$ and, by definition of $B_{a^{\prime}}$ , $a^{\prime}\preceq z$ $\text{for all } a^{\prime}\in A^{\prime}\setminus D$ . Thus, $\sqcup A=\sqcup A^{\prime} \preceq z$ and, hence, $\sqcup A^{\prime\prime}=\sqcup A^{\prime}=\sqcup A$ .

Theorem 1 has the following straightforward strengthening.

From the proof of Corollary 1, which relies on Lemma 3, Debreu separability alone seems to be insufficient to build a countable weak basis. This is precisely the case, as we show in Proposition 4 via a counterexample. In fact, our counterexample shows that even strengthening the hypothesis to Debreu upper separability is insufficient.

Proof. Take $P := (\Sigma ^* \cup \Sigma ^\omega,\preceq)$ , where $\Sigma$ is finite and

\begin{equation*} x \preceq y \iff \begin {cases} x=y, \text { or}\\ x \in \Sigma ^*,\text { } y \in \Sigma ^\omega \text { and } x \preceq _C y, \end {cases} \end{equation*}

with $\preceq _C$ the partial order from the Cantor domain. (See Fig. 2 for a representation with $\Sigma =\{0,1\}$ .) As we directly see $P$ is a partial order, we begin by showing $P$ is directed complete. Consider a directed set $A \subseteq P$ with $|A|\geq 2$ . (If $A=\{a\}$ for some $a \in P$ , then $\sqcup A= a$ and we have finished.) If $A \cap \Sigma ^\omega = \emptyset$ , notice $A$ is not directed, since given $x,y \in A \cap \Sigma ^*$ $x \neq y$ , there is no $z \in \Sigma ^*$ such that $x,y \preceq z$ . Thus, there exists some $x_A \in A \cap \Sigma ^\omega$ . Notice we actually have $A \cap \Sigma ^\omega =\{x_A\}$ , as given $y \in \Sigma ^\omega$ $y \neq x_A$ there is no $z \in P$ such that $x_A,y\preceq z$ . Analogously, we obtain $y \preceq x_A$ $\text{for all } y \in A$ and, thus, $\sqcup A=x_A$ . We conclude $P$ is directed complete. We notice now $\Sigma ^*$ is a countable Debreu dense and Debreu upper dense subset of $P$ . If $x \prec y$ for some $x,y \in P$ , then $x \in \Sigma ^*$ by definition and we conclude $\Sigma ^*$ is Debreu dense. If $x \bowtie y$ with $x,y \in \Sigma ^\omega$ , then there exists some $d \in \Sigma ^*$ such that $x \bowtie _c d$ and $d \preceq _c y$ . If $y \in \Sigma ^\omega$ and $x \in \Sigma ^*$ , we consider some $d \in \Sigma ^*$ such that $d \preceq _C y$ and get $x \bowtie d \preceq y$ . If either $x \in \Sigma ^\omega$ and $y \in \Sigma ^*$ or $x,y \in \Sigma ^*$ , then we take $d=y$ . We obtain $P$ is Debreu upper separable. To finish, we will show any weak basis $B \subseteq P$ is uncountable. By definition of weak basis, there exists some directed set $A_x \subseteq B$ such that $x = \sqcup A_x$ $\text{for all } x \in \Sigma ^\omega$ . As discussed above, this implies $x \in A_x$ . Hence, $\Sigma ^\omega \subseteq B$ and $B$ is uncountable.

Note that, in the proof of Hack et al. Reference Hack, Braun and Gottwald2023, Proposition 4 (iii)), we introduced the counterexample we used in Proposition 4. Note, also, $\Sigma ^\omega \subseteq \mathscr{T}_B$ for any weak basis $B$ of the dcpo in the proof of Proposition 4. Before we continue relating weak bases to order density properties, we define the minimal elements of a partial order. If $P$ is a partial order, then the set of minimal elements of $P$ , denoted by $\text{min}(P)$ , is

\begin{equation*} \text {min}(P) := \{x \in P\,| \text { there is no } y \in P \text { such that } y \prec x\}. \end{equation*}

Notice, for the Cantor domain, $\text{min}(P)=\{\perp \}$ . $\text{min}(P)$ is related to order density properties since, as we note in Lemma 4, it is countable whenever countable Debreu upper dense subsets exist.

Clearly, we cannot eliminate the hypothesis in Lemma 4 as, for example, $\text{min}(P)$ is uncountable for any uncountable set with the trivial partial order.

Figure 2.

Representation of a dcpo, defined in Proposition 4, which is Debreu upper separable and has no countable weak basis. In particular, we show $\Sigma ^\omega$ and $\Sigma ^*$ for $\Sigma =\{0,1\}$ and how $010101\ldots,000000\ldots,010000\ldots \in \Sigma ^\omega$ are related to $0101, 0, 01 \in \Sigma ^*$ . Notice an arrow from an element $w$ to an element $t$ represents $w \prec t$ .

Since, by Proposition 4, Debreu upper separability is not enough for countable weak bases to exist, we consider stronger order density properties.

Notice that, although the Cantor domain $\Sigma ^\infty$ is Debreu separable, it is not order separable since, if $s \in \Sigma ^*$ and $\beta \in \Sigma$ , then we have $s \prec s\beta$ and, $\text{for all } t \in \Sigma ^*$ such that $t \prec s\beta$ , $t \preceq s$ also holds. As we show in Proposition 5, order separability is sufficient to build a countable weak basis for $P\setminus \text{min}(P)$ the basic idea is similar to the one for Lemma 3 and, taking into account Lemma 4, we can extend the result to $P$ by also assuming the existence of a countable Debreu upper dense subset.

Proof. We begin showing the first statement. Notice, if $P$ is a dcpo, then $P\setminus \text{min}(P)$ is a dcpo as well. Take $D \subseteq P$ a countable order dense subset of $P$ , which we can choose w.l.o.g. such that $D \cap \text{min}(P) = \emptyset$ . We will show $D$ is a countable weak basis for $P\setminus \text{min}(P)$ . In particular, we will show the following lemma.

Lemma 5. If $P$ is a dcpo, $D \subseteq P$ is a countable order dense subset and $x,y \in P$ such that $x \prec y$ , then $D^{\prime} =(d_n^{\prime})_{n \geq 0} \subseteq D$ is an increasing chain such that $\sqcup D^{\prime} =y$ , where

(6) \begin{equation} \begin{alignedat}{2} d_0^{\prime} &:= d_{m_0} \text{ and } &&m_0 := \text{min} \{n \geq 0\, |\, x \prec d_n \prec y\},\\ d_n^{\prime} &:= d_{m_n} \text{ and } &&m_n := \text{min}\{n \geq m_{n-1}\, |\, d_{n-1}^{\prime} \prec d_n \prec y\} \text{ for all } n \geq 1. \end{alignedat} \end{equation}

Proof. Take some $y \in P\setminus \text{min}(P)$ and $D=(d_n)_{n \geq 0}$ a numeration of $D$ . Since $y \not \in \text{min}(P)$ , there exists some $x \in P$ such that $x \prec y$ . Moreover, by definition of order separability, we have that $m_n$ exists for all $n \geq 0$ and, hence, $D^{\prime} =(d_n^{\prime})_{n \geq 0} \subseteq D$ in (6) is well-defined. Since $D^{\prime}=(d_n^{\prime})_{n\geq 0}$ is an increasing sequence by construction, it has a supremum $\sqcup D^{\prime}$ . Given that $y$ is an upper bound of $(d_n^{\prime})_{n\geq 0}$ by construction, we have $\sqcup D^{\prime} \preceq y$ . To finish the proof of the first statement, we assume $\sqcup D^{\prime} \prec y$ and get a contradiction. If that was the case, there would be some $\overline{n} \geq 0$ such that $\sqcup D^{\prime} \prec d_{\overline{n}} \prec y$ by order separability. Consider, thus, $\overline{m} := \text{max} \{ n \lt \overline{n}| d_n \in D^{\prime}\}$ . Since $d_{\overline{m}+1},\ldots,d_{\overline{n}-1} \not \in D^{\prime}$ by definition of $\overline{m}$ and $d_{\overline{m}} \prec d_{\overline{n}} \prec y$ by transitivity, we would have, assuming $d_n^{\prime}=d_{\overline{m}}$ for some $n\geq 0$ w.l.o.g., $d_{n+1}^{\prime}=d_{\overline{n}}$ , a contradiction. Thus, $D^{\prime}$ is an increasing chain such that $ \sqcup D^{\prime} = y$ .

Hence, by Lemma 5, $D$ is a countable weak basis for $P\setminus \text{min}(P)$ . Regarding the second statement, notice we can take $A_x= \{x\}$ as directed set with $\sqcup A_x=x$ $\text{for all } x \in \text{min}(P)$ , and, since $\text{min}(P)$ is countable by Lemma 4, we have $D \cup \text{min}(P)$ is a countable weak basis for $P$ .

Since order density of $P$ is enough to introduce computability on $P\setminus \text{min}(P)$ while, again by Proposition 4, Debreu separability is not, we take a stronger version of Debreu separability as hypothesis in Proposition 6. In particular, we show there are countable weak bases for $P\setminus \text{min}(P)$ whenever countable Debreu dense subset satisfying a specific property exists. To express such a property, we recall some definitions in by Bridges and Mehta (Reference Bridges and Mehta2013). If $x,y \in P$ , then we say $y$ is an immediate successor of $x$ and $x$ is an immediate predecessor of $y$ if both $x \prec y$ and $(x, y) := \{z \in P|x \prec z \prec y\} = \emptyset$ hold. In this scenario, $(x, y)$ is called a jump.

Proposition 6. If $P$ is a dcpo and $D \subseteq P$ is a countable Debreu dense subset whose elements have a finite number of immediate successors, then $(Q_1\cup D)\setminus \text{min}(P)$ is a countable weak basis for $P\setminus \text{min}(P)$ , where

(7) \begin{equation} Q_1 := \left \{ y \in P\, |\, \exists x \in P \text{ s.t. } (x,y)=\emptyset \right \}. \end{equation}

Furthermore, if $P$ also has a countable Debreu upper dense subset, then $Q_1 \cup D \cup \text{min}(P)$ is a countable weak basis for $P$ .

Proof. We begin showing the first statement. Notice, if $P$ is a dcpo, then $P\setminus \text{min}(P)$ is also a dcpo. Take $D\subseteq P$ a countable Debreu dense subset of $P$ whose elements have a finite number of immediate successors and $y \in P\setminus \text{min}(P)$ . Notice $y \in Q_0 \cup Q_1$ , where

\begin{equation*} Q_0 := \{ y \in P\,|\, \text {for all } x \in P \text { s.t. } x \prec y \text { } \exists z \in P \text { s.t. } x \prec z \prec y\} \end{equation*}

and $Q_1$ is defined as in (7). If $y \in Q_0$ , we take some $x \prec y$ and we can emulate the construction of $(d^{\prime}_n)_{n \geq 0}$ in Lemma 5, (6), to construct an increasing sequence contained in $D$ whose supremum is $y$ . We just have to notice, by Debreu separability and definition of $Q_0$ , whenever $x \prec y$ for $x,y \in P$ , there exist $z \in P$ and $m\geq 0$ such that $x \prec z \preceq d_m \prec y$ . Conversely, if $y \in Q_1$ , there exists some $x \in P$ such that $(x,y)=\emptyset$ . If $y \not \in D$ , then $x \in D$ by Debreu separability. If we denote by $(s_n)_{n\geq 0}$ a numeration of the immediate successors of the elements in $D$ , which can be constructed since each member of $D$ has a finite number of immediate successors and $D$ is countable by hypothesis, we have $Q_1 \subseteq D\cup (s_n)_{n\geq 0}$ and, thus, $Q_1$ is countable. We conclude $(Q_1\cup D)\setminus \text{min}(P)$ is a countable weak basis for $P\setminus \text{min}(P)$ . For the second statement, since $\text{min}(P)$ is countable by Lemma 4, we conclude $Q_1 \cup D \cup \text{min}(P)$ is a countable weak basis for $P$ .

Note the hypothesis in the first statement of Proposition 6 is weaker than order separability since, whenever a partial order is dense, immediate successors do not exist.Footnote 2 Notice, also, the Cantor domain satisfies the hypotheses in Proposition 6, since $\Sigma ^*$ is a countable Debreu dense subset of $\Sigma ^\infty$ and each $s \in \Sigma ^*$ has exactly $|\Sigma |$ immediate successors. The converse of Proposition 6 is, again by Proposition 3, false.

To summarize, the main results in this section are Theorem 1 and Propositions 5 and 6. In the first one, we show how one can profit from Debreu density to introduce a dcpo that includes computability on the non-trivial elements of any weak basis $B$ . In the second, we use the stronger property of order separability to extend computability to all non-minimal elements. In the last one, we show that we can achieve the same results as in Proposition 5 by asking for the existence of a countable Debreu separable subset whose elements only have a finite number of immediate successors. Lastly, we extend computability to the whole dcpo in Propositions 5 and 6 by also requiring the existence of a countable Debreu upper dense subset.

4. Uniform Computability via Ordered Sets

Following Hack et al. Reference Hack, Braun and Gottwald2022a), in Section 2, we considered an element $x$ in some dcpo $P$ with a countable weak basis $B\subseteq P$ to be computable if there exists some directed set $B_x \subseteq B$ such that ${\sqcup}B_x =x$ and whose associated subset of the natural numbers $\alpha ^{-1}(B_x)$ is recursively enumerable. We consider now a stronger approach, where we associate to each $x \in P$ a unique directed set $B_x \subseteq B$ such that $\alpha ^{-1}(B_x)$ is recursively enumerable if and only if $x$ is computable. In order to do so, $B_x$ should be fundamentally related to $x$ , that is, the information in every $b \in B_x$ should be gathered by any process which computes $x$ . The differences between the approach in Section 2 and the one here are discussed in Hack et al. Reference Hack, Braun and Gottwald2022a). Crucially, while the more general approach only enables to introduce computable elements, computable functions can also be defined in this stronger framework, which was introduced by Scott in Scott (Reference Scott1970). In fact, the uniform approach to computability we discuss in this section is known as domain theory (Cartwright et al. Reference Cartwright, Parsons and AbdelGawad2016; Edalat and Sünderhauf Reference Edalat and Sünderhauf1999; Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2012; Mislove Reference Mislove1998; Scott Reference Scott1982; Stoltenberg-Hansen Reference Stoltenberg-Hansen, Lindstrom, Lindström and Griffor1994, Reference Stoltenberg-Hansen2001). Note that, as in Section 2, we only introduce the order structure used to define computability and do not address how to translate it to other spaces of interest (see Hack et al. Reference Hack, Braun and Gottwald2022a and the references therein).

Before we continue, we introduce the Scott topology, which will play a major role in the following.

Note that the Scott topology characterizes the partial order in $P$ , that is,

(8) \begin{equation} x \preceq y \iff x \in O \text{ implies } y \in O \text{ } \text{for all } O \in \sigma (P) \end{equation}

(Abramsky and Jung Reference Abramsky and Jung1994, Proposition 2.3.2. Note that, by (8), the Scott topology satisfies the $T_0$ topological separation axiom. (We say a topological space $(X,\tau)$ is a $T_0$ space or a Kolmogorov space if, for every pair of distinct points $x,y \in X$ , there exists an open set $O \in \tau$ such that either $x \in O$ and $y \not \in O$ or $y \in O$ and $x \not \in O$ hold Kelley Reference Kelley2017.)

We can think of the Scott topology as the family of properties on the data set $P$ which allows us to distinguish the elements in $P$ (Smyth Reference Smyth1983). In particular, by definition, if some computational processes have a limit, then any property of the limit is verified in finite time, and, since $\sigma (P)$ is $T_0$ , these properties are enough to distinguish between the elements of $P$ .

In order to attach to each element in a dcpo, a unique subset of $\mathbb{N}$ , which is equivalent to it for all computability purposes, we recall the way-below relation.

We think of an element way-below another as containing essential information about the latter, since any computational process producing the latter cannot avoid gathering the information in the former. For example, if is a directed set such that $\sqcup A_1 =x$ , then, if $A_2 \subseteq P$ is a directed set where $\sqcup A_2=x$ , there exists $\text{for all } a_1 \in A_1$ some $a_2 \in A_2$ such that $a_1 \preceq a_2$ . We can regard $A_1$ , thus, as a canonical computational process that yields $x$ . (To be more precise, one can show that the computability, in the sense of Definition 15, of an element is equivalent to the computability of a specific subset of the basis. This is formalized in by Hack et al. Reference Hack, Braun and Gottwald2022a, Proposition 2.) The significance of the introduction of $\ll$ for computability is discussed by Hack et al. Reference Hack, Braun and Gottwald2022a). We simply note, for the moment, that $x \ll y$ implies $x \preceq y$ and that, if $x \preceq y$ and $y \ll z$ , then $x \ll z$ $\text{for all } x,y,z\in P$ .

We introduce now, in a canonical way, computable elements in a dcpo. To do so, we first define effective bases.

Note that bases are exactly like weak bases except for the fact that they achieve any element $x \in P$ using elements of $P$ , which contain essential information about $x$ . Recall, if $B$ is a basis, then it is a weak basis, and is a topological basis for $\sigma (P)$ (Abramsky and Jung Reference Abramsky and Jung1994; Hack et al. Reference Hack, Braun and Gottwald2022a). A dcpo is called continuous or a domain if bases exist. As in the case of weak bases, we are interested in dcpos with a countable basis or $\omega$ -continuous dcpos, since computability can be introduced via Turing machines there. Note that $\Sigma ^\infty$ is $\omega$ -continuous, as $B=\Sigma ^*$ is a countable basis.

Definition 14 (Effective basis, Edalat and Sünderhauf Reference Edalat and Sünderhauf1999; Stoltenberg-Hansen and Tucker Reference Stoltenberg-Hansen and Tucker2008). We say a basis $B \subseteq P$ is effective if there is a finite map, which enumerates $B$ , $B=(b_n)_{n\geq 0}$ , there is a bottom element $\perp \in B$ and

\begin{equation*} \{\langle n,m \rangle \, |\, b_n \ll b_m\} \end{equation*}

is recursively enumerable.

Effectivity for bases has the same purpose as for weak bases, although it poses stronger requirements. A discussion in this regard can be found in Hack et al. (Reference Hack, Braun and Gottwald2022a). We assume w.l.o.g. $b_0 = \perp$ in the following. Note that the relevance of the bottom element is discussed in Hack et al. Reference Hack, Braun and Gottwald2022a). We can now define computable elements.

Definition 15 (Computable element, Edalat and Sünderhauf Reference Edalat and Sünderhauf1999). If $P$ is a dcpo with an effective basis $B=(b_n)_{n\geq 0}$ , we say $x \in P$ is computable provided

\begin{equation*} \{n \in \mathbb {N}\, |\, b_n \ll x\} \end{equation*}

is recursively enumerable.

The dependence of the set of computable elements on the model in this approach is addressed in Hack et al. Reference Hack, Braun and Gottwald2022a).

Before introducing computable functions, we need some extra terminology. We call a map $f:P \rightarrow Q$ between dcpos $P,Q$ a monotone if $x \preceq y$ implies $f(x) \preceq f(y)$ . Furthermore, we call $f$ continuous if it is monotone and, for any directed set $A \subseteq P$ , we have $f({\sqcup} A) = \sqcup f(A)$ (Abramsky and Jung Reference Abramsky and Jung1994; Scott Reference Scott1970). Note that this definition is equivalent to the usual topological definition of continuity (see Kelley Reference Kelley2017) applied to the Scott topology. In fact, dcpos constitute the objects of the category DCPO, whose morphisms are the functions that are continuous in the Scott topology (Lawson Reference Lawson1998). Note, also, if $B \subseteq P$ is a weak basis and $f$ is continuous, then $f(P)$ is determined by $f(B)$ . This is the case since, if $x \in P$ , then there exists some directed set $B_x \subseteq B$ such that ${\sqcup}B_x=x$ . Thus, by continuity of $f$ , $f(x)=f({\sqcup}B_x)=\sqcup f(B_x)$ . We can consider the monotonicity of $f$ in the definition of continuity as a mere technical requirement to make sure $\sqcup f(A)$ exists for any directed set $A \subseteq P$ .

We introduce now computable functions.

Definition 16 (Computable function, Edalat and Sünderhauf Reference Edalat and Sünderhauf1999). We say a function $f:P \to Q$ between two dcpos $P$ and $Q$ with effective bases $B=(b_n)_{n\geq 0}$ and $B^{\prime}=(b^{\prime}_n)_{n\geq 0}$ , respectively, is computable if $f$ is continuous and the set

(9) \begin{equation} \{\langle n,m \rangle \, |\, b^{\prime}_n \ll f(b_m)\} \end{equation}

is recursively enumerable.

Intuitively, the idea that any formal definition of computable function is meant to capture is that of a map that sends computable inputs to computable outputs (Braverman Reference Braverman2005; Ko Reference Ko2012). Indeed, this is the case for this definition (see Edalat and Sünderhauf Reference Edalat and Sünderhauf1999, Theorem 9) and Hack et al. Reference Hack, Braun and Gottwald2022a). Moreover, we can think of continuity in $\sigma (P)$ as a weak form of computability. The dependence of the set of computable functions on the model in this approach is addressed in Hack et al. Reference Hack, Braun and Gottwald2022a).

To conclude, we can return to the examples from Section 2.1. In particular, one can see $\Sigma ^*$ is an effective basis for $\Sigma ^\infty$ (Hack et al. Reference Hack, Braun and Gottwald2022a) and $B_{\mathscr{I}}$ is known to be an effective basis of $(\mathscr{I},\sqsubseteq)$ (Edalat and Sünderhauf Reference Edalat and Sünderhauf1999). Moreover, $B_n := \Lambda ^n \cap \mathbb{Q}^n$ is a countable basis for majorization for any $n \geq 2$ and the following lemma, which we prove in Appendix A.2.2 (see also Appendix A.2.3), holds.

Furthermore, one can slightly modify the proof of Proposition 1 in the Appendix A.2.1 to conclude $B_n$ is an effective basis. The key property we use is Martin and Panangaden Reference Martin and Panangaden2008, Lemma 5.1), where it was shown

(10) \begin{equation} x \ll _M y \iff x = \perp \text{ or } s_k(x)\lt s_k(y) \text{ } \text{for all } k\lt n \end{equation}

$\text{for all } x,y \in \Lambda ^n$ . Notice, Lemma 6 improves upon (Martin Reference Martin2006, Theorem 1.3), establishing $\omega$ -continuity instead of just continuity for all $(\Lambda ^n,\preceq _M)$ with $n \geq 2$ .

5. Order Density and Bases

In this section, we continue relating order density properties to computability, specifically, to the uniform approach introduced in Section 4. We begin, in Section 5.1, mimicking the relations in Section 3, this time for bases. After establishing the discrepancy between bases and order density in general, we introduce, in Section 5.2, a property under which they coincide, namely, conditional connectedness. Importantly, the Cantor domain fulfills this property. Notice, for generality, we will assume bases do not necessarily have a bottom element in this section.

We can summarize the relation between Sections 3 and 5 as follows: First, we show, in Propositions 8 and 9, respectively, that the results regarding the existence of countable weak bases in Propositions 5 and 6 can be emulated for bases by restricting ourselves to continuous dcpos. Then, we show, in Theorem 2, that the positive and negative results regarding the relation between weak bases and order density properties (see Propositions 2, 3, and 4, and Theorem 1) can be improved upon to reach equivalence by additionally asking for conditional connectedness.

5.1 Continuous Debreu separable dcpos

We consider here continuous dcpos and extend the relationship between order density and computability from Section 3, focusing on connecting the former with countable bases. We begin introducing a definition and two lemmas that will be useful in the following. We say an element $x \in P$ is compact if $x \ll x$ holds and denote by $\text{K}(P)$ the set of compact elements of $P$ (Abramsky and Jung Reference Abramsky and Jung1994). Note that $\text{K}(P) \subseteq B$ for any basis $B \subseteq P$ .

Lemma 7. All compact elements in a dcpo $P$ are trivial $\text{K}(P) \subseteq \mathscr T_P$ . The converse, however, is not true.

Proof. Take $x \in \text{K}(P)$ and a directed set $A$ such that $\sqcup A=x$ . By definition of compact element, there exists some $a \in A$ such that $x \preceq a$ . As $a \preceq \sqcup A$ by definition, then, by antisymmetry, we get $x =\sqcup A = a \in A$ . To show the converse is false, take the dcpo $([0,1],\preceq)$ , where $x \preceq y$ if $x \leq y$ $\text{for all } x,y \in (0,1]$ , $0 \preceq 0$ and $0 \preceq 1$ . Note that $0 \bowtie y$ $\text{for all } y \in (0,1)$ . While the only directed set $A$ with $\sqcup A=0$ is $A=\{0\}$ , we also have $A^{\prime} = (0,1)$ is directed with $\sqcup A^{\prime}=1$ and $0 \bowtie a$ $\text{for all } a \in A^{\prime}$ . Thus, $\neg (0 \ll 1)$ and $0 \not \in \text{K}(P)$ as, otherwise, we would have $0 \ll 0 \preceq 1$ , hence $0 \ll 1$ .

Lemma 8. If $P$ is a continuous dcpo, then $\sigma (P)$ is second countable if and only if $P$ is $\omega$ -continuous. However, there exist dcpos without bases where $\sigma (P)$ is second countable.

Proof. The first statement is already known, see Lawson Reference Lawson1998, Proposition 3.4). For the second statement, take $P := \big (\big [0,1\big ],\preceq \big)$ , where, for all $x,y \in [0,1]$ ,

\begin{equation*} x \preceq y \iff \begin {cases} x \leq y &\text { if } x,y \in \big [0,\frac {1}{2}\big ],\\[4pt] y \leq x & \text { if } x,y \in \big [\frac {1}{2},1\big ]. \end {cases} \end{equation*}

It is easy to see $P$ has no basis (Hack et al. Reference Hack, Braun and Gottwald2022a). To conclude, we note that $\big \{(p,q), [0,q),(p,1]\big |0 \leq p \lt \frac{1}{2}\lt q \leq 1, p,q \in \mathbb{Q}\big \}$ is a countable basis for $\sigma (P)$ . Take, thus, $x \in O \in \sigma (P)$ and assume w.l.o.g. $x \lt \frac{1}{2}$ . If $x \gt 0$ , since $O \in \sigma (P)$ and $x \in O$ , then there exists some $y\lt x$ such that $y \in O$ , otherwise $D_x := \{y \in P|y\lt x\}$ fulfills $\sqcup D_x=x$ and $D_x \cap O = \emptyset$ , a contradiction. Arguing analogously, we have there exists some $z\gt \frac{1}{2}$ such that $z \in O$ . Since $O$ is upper closed, $x \in (p_x,q_x) \subseteq O$ , where $p_x,q_x \in \mathbb{Q}$ , $y \leq p_x \leq x$ and $\frac{1}{2} \leq q_x \leq z$ . If $x=0$ , we can follow argue analogously and conclude $x \in [0,q) \subseteq O$ for some $q \in \mathbb{Q}$ , $ \frac{1}{2}\lt q$ .

Regarding the Cantor domain, Note that is a countable topological basis for its Scott topology. Lemma 8 establishes that, in order to relate order density with countable bases, we can relate the former to countability axioms on $\sigma (P)$ . As a starting link, we relate, in Proposition 7, continuous Debreu separable dcpos with these axioms. In particular, we show Debreu separability of $P$ implies $\sigma (P)$ is first countable for continuous dcpos. Furthermore, whenever the set of compact elements is countable, we show $P$ is $\omega$ -continuous, that is, its Scott topology is second countable.

Proposition 7. Take a dcpo $P$ and a countable Debreu dense subset $D \subseteq P$ . If $P$ is continuous, then $\sigma (P)$ is first countable. Moreover, if $\text{K}(P)$ is countable, then $D \cup \text{K}(P)$ is a countable basis for $P$ .

Proof. We begin with the first statement. Take $x \in P$ , a basis $B\subseteq P$ and a Debreu dense subset $D\subseteq P$ . We will show there exists a countable family $(U_n^x)_{n\geq 0} \subseteq \sigma (P)$ such that, if $x \in O \in \sigma (P)$ , then there exists some $n_0 \geq 0$ such that $x \in U_{n_0}^x \subseteq O$ . If $x \in \text{K}(P)$ , we take and notice, if $O \in \sigma (P)$ , then we have whenever $x \in O$ , as by definition. If $x \not \in \text{K}(P)$ , then there exists some directed set such that ${\sqcup}B_x=x \not \in B_x$ by definition of basis. By Lemma 3, there exists an increasing sequence $(d^{\prime}_n)_{n\geq 0} \subseteq D$ (defined by (5) and (4)) such that $\sqcup (d^{\prime}_n)_{n\geq 0}=x \not \in (d^{\prime}_n)_{n\geq 0}$ . Since, by construction, there exists some $b_n \in B_x$ such that $d_n \preceq b_n \ll x$ , we have $d^{\prime}_n \ll x$ $\text{for all } n \geq 0$ . Thus, if $x \not \in \text{K}(P)$ , we take $\text{for all } n\geq 0$ since, given some $O \in \sigma (P)$ such that $x \in O$ , then, by definition, there exists some $n_0 \geq 0$ such that $d^{\prime}_{n_0} \in O$ because $\sqcup (d^{\prime}_n)_{n\geq 0} =x$ . In particular, . Regarding the second statement, we notice, following the first part, $\text{for all } x \in P$ , $O \in \sigma (P)$ such that $x \in O$ there exists some $n_0\geq 0$ such that $x \in U_{n_0} \subseteq O$ , where

(11)

since $\text{K}(P)$ is countable by assumption. We conclude $\sigma (P)$ is second countable and, by Lemma 8, $P$ is $\omega$ -continuous. In particular, $D \cup \text{K}(P)$ is a countable basis.

Remark 4 (Implication for computability). By Proposition 7, we can define computable elements (in the sense of Definition 15) on a continuous dcpo $P$ with a countable Debreu dense subset $D$ and countable compact elements $\text{K}(P)$ whenever $D \cup \text{K}(P)$ is effective. Moreover, if that is the case for two dcpos $P_0,P_1$ , then we can define computable functions (in the sense of Definition 16) between them $f:P_0 \to P_1$ .

Note that there exist Debreu separable dcpos where $\sigma (P)$ is not first countable (take, e.g., the dcpo in Hack et al. Reference Hack, Braun and Gottwald2022a, Proposition 5) and continuous Debreu separable dcpos where $\text{K}(P)$ is uncountable (like $P := (A \cup B,\preceq)$ with $A := \mathbb{R}$ , $B := (0,1]$ and $x \preceq y$ if and only if $x \leq y$ with $x,y \in B$ or $x \in A$ and $y \in B$ , where $\mathbb{Q} \cap B$ is a Debreu dense subset and $\text{K}(P)=A$ is uncountable). Strengthening the order density assumption in Proposition 7, we can improve upon Proposition 5 and construct countable bases instead of weak bases, as we show in Proposition 8.

Proposition 8. If $P$ is a continuous dcpo with a countable order dense subset $D\subseteq P$ , then $D$ is a countable basis for $P\setminus \text{min}(P)$ . Furthermore, if $P$ also has a countable Debreu upper dense subset, then $D \cup \text{min}(P)$ is a countable basis for $P$ .

Proof. We begin with the first statement. In order to do so, we start by showing the following lemma.

Lemma 9. If $P$ is a continuous dcpo with a countable order dense subset $D\subseteq P$ , then

\begin{equation*} \text {K}(P) = \text {min}(P). \end{equation*}

Proof. $({\subseteq})$ If $y \in P\setminus \text{min}(P)$ , then we can construct an increasing chain $(d_n^{\prime})_{n\geq 0} \subseteq D$ (defined as in Lemma 5, (6)) with $D\subseteq P$ a countable order dense subset chosen w.l.o.g. with the property $D \cap \text{min}(P)=\emptyset$ , such that $\sqcup (d_n^{\prime})_{n\geq 0}=y$ and $y \not \in (d_n^{\prime})_{n\geq 0}$ . Thus, $y \not \in \text{K}(P)$ by Lemma 7 and, hence, $\text{K}(P) \subseteq \text{min}(P)$ . $({\supseteq})$ This is the case since $P$ is continuous and, hence, we must have, for all $x \in \text{min}(P)$ , for some basis $B \subseteq P$ . That is, $x \in \text{K}(P)$ .

We finish noticing $Q := P\setminus \text{min}(P)$ is a dcpo where $K(Q)=\emptyset$ . Thus, by (11), $D \subseteq Q$ is a countable basis for $Q$ . For the second statement, we notice, if $P$ has a countable Debreu upper dense subset, then $\text{min}(P)$ is countable by Lemma 4 and $\text{K}(P)$ is also countable, as $\text{K}(P) = \text{min}(P)$ by Lemma 9. Thus, by Proposition 7, $P$ has a countable basis. In fact, we have that $D \cup \text{min}(P)$ is a countable basis for $P$ .

Remark 5 (Implication for computability). By Proposition 8, we can define computable elements and functions (in the sense of Definitions 15 and 16) on a continuous dcpo $P$ with a countable order dense subset $D$ and a countable Debreu upper dense subset whenever $D \cup \text{min}(P)$ is effective.

As we did with Proposition 6 for Proposition 5, we complement Proposition 8 with Proposition 9, where we substitute order separability by a weaker property, namely, the existence of a countable Debreu dense subset where each element has a finite number of immediate successors. Notice, unlike Proposition 8, Proposition 9 holds for the Cantor domain.

Proposition 9. If $P$ is a continuous dcpo with a countable Debreu dense subset $D \subseteq P$ whose elements have a finite number of immediate successors, then $(D \cup \text{K}(P)) \setminus \text{min}(P)$ is a countable basis for $P\setminus \text{min}(P)$ . Furthermore, if $P$ also has a countable Debreu upper dense subset, then $D \cup \text{K}(P)$ is a countable basis for $P$ .

Proof. The first statement can be shown by emulating Lemma 5 (as in the proof of Proposition 6) to obtain, after applying Lemma 7 as in Lemma 9, that $\text{K}(P) \subseteq \text{min}(P) \cup Q_1$ , with $Q_1$ defined as in (7), that is,

\begin{equation*} Q_1 = \{ y \in P\, |\, \exists x \in P \text { s.t. } (x,y)=\emptyset \}. \end{equation*}

Since $Q_1$ is countable, we follow the proof of Proposition 7 and conclude $P\setminus \text{min}(P)$ has a countable basis. For the second statement, we rely again on Lemma 4 and obtain that $\text{min}(P)$ is countable. We conclude that $\text{K}(P)$ is also countable and, thus, that $D \cup \text{K}(P)$ is a countable basis for $P$ .

Remark 6 (Implication for computability). By Proposition 9, we can define computable elements and functions (in the sense of Definitions 15 and 16) on a continuous dcpo $P$ with a countable Debreu dense subset $D$ whose elements have a finite number of immediate successors and a countable Debreu upper dense subset whenever $D \cup \text{K}(P)$ is effective.

As we show in Proposition 10, we cannot improve upon Proposition 9 eliminating the assumption about the finiteness of the immediate successors of $D$ . In particular, there are continuous Debreu upper separable dcpos where $\text{K}(P)$ , and, hence, any basis, is uncountable. Actually, the result remains false if we add the assumption that $x \preceq y$ implies $x \ll y$ $\text{for all } x,y \in P$ . We will return to this assumption in Section 5.2.

Proposition 10. There exist continuous Debreu upper separable dcpos $P$ where $x \preceq y$ implies $x \ll y$ $\text{for all } x,y \in P$ , but for which $\text{K}(P)$ is uncountable and, thus, no countable bases exist.

Proof. Take the counterexample from the proof of Proposition 4, $P$ . We will show every element in $P$ is compact, $\text{K}(P)=P$ . If $x \in \Sigma ^\omega$ and $x \preceq \sqcup A$ for some directed set $A$ , then $x=\sqcup A$ , as there is no other element above $x$ . Thus, $x \in A$ since, as we showed in Proposition 4, $A$ would not be directed in the opposite case. Hence, $x \in \text{K}(P)$ and $\Sigma ^\omega \subseteq \text{K}(P)$ . If $x \in \Sigma ^*$ and $x \preceq \sqcup A$ for some directed set $A$ , then either $\sqcup A=x$ or $\sqcup A \in \Sigma ^\omega$ . If $\sqcup A=x$ , then $A=\{x\}$ as there is no other element below $x$ . If $\sqcup A \in \Sigma ^\omega$ , then $\sqcup A \in A$ , as we argued above, and we have $x \in \text{K}(P)$ . Thus, $\Sigma ^*\subseteq \text{K}(P)$ . We conclude $\text{K}(P)=P$ . To finish, we only need some remarks. $\text{K}(P)=P$ implies $P$ is continuous, as we can take the directed set $\text{for all } x \in P$ . Moreover, since $\text{K}(P) \subseteq B$ for any basis $B$ and $\text{K}(P)$ is uncountable, $P$ is not $\omega$ -continuous. Notice, also, if $x \preceq y$ , then we have $x \in \text{K}(P)$ , which means $x \ll x \preceq y$ and $x \ll y$ $\text{for all } x,y \in P$ .

Since we tried to construct countable bases in Propositions 8 and 9 using order density properties, we intend now to do the converse. Notice, by Proposition 2, $\omega$ -continuous dcpos have countable Debreu upper dense subsets. However, there are $\omega$ -continuous dcpos which are not Debreu separable. To see this, we can take the counterexample in Proposition 3 (as the countable weak basis defined there is actually a basis). Alternatively, we can use either majorization for any $n\geq 3$ or the interval domain since, as we show in Lemma 6 in the Appendix A.2.2, both are $\omega$ -continuous and any of their Debreu dense subsets has cardinality $\mathfrak{c}$ . We summarize this paragraph in the following statement.

Proposition 11. If $P$ is a dcpo with a basis $B \subseteq P$ , then $B$ is a Debreu upper dense subset of $P$ . However, there are $\omega$ -continuous dcpos which are not Debreu separable.

5.2 Conditionally connected Debreu upper separable dcpos

Although Debreu upper separability and $w$ -continuity are not equivalent for general continuous dcpos (cf. Propositions 10 and 11), they do coincide for the Cantor domain $\Sigma ^\infty$ . In this section, we use one of its properties, conditional connectedness, to show, in Theorem 2, Debreu upper separability and $w$ -continuity are equivalent for any conditionally connected dcpo. We begin by defining conditional connectedness.

Definition 17 (Conditionally connected partial order). A partial order $P$ is conditionally connected if, for any pair $x,y \in P$ for which there exists some $z \in P$ such that $x \preceq z$ and $y \preceq z$ , we have $x$ and $y$ are comparable, that is $\neg (x \bowtie y)$ .

Note that conditional connectedness amounts to comparable elements conforming to equivalent classes. We could have named partial orders fulfilling Definition 17 upward conditionally connected to distinguish them from downward conditionally connected, which has been used in order-theoretical approaches to thermodynamics (Giles Reference Giles2016; Landsberg Reference Landsberg1970; Roberts and Luce Reference Roberts and Luce1968) and is referred to as conditionally connected. Note that it is straightforward to characterize conditionally connected partial orders as those where every directed set is a chain (see Lemma 12 in the Appendix A.1).

An example of a conditionally connected partial order is the Cantor domain since, if $x,y,z \in \Sigma ^{\infty }$ are three strings such that $x$ and $y$ are prefixes of $z$ , then it is clear that either $x$ is a prefix of $y$ or vice versa. A negative example comes from majorization $\text{for all } n\geq 3$ , as one can pick, for example, the pair $x,y \in \Lambda ^n$ where $x=(0.6,0.2,0.2,0,\ldots,0)$ and $y=(0.5,0.4,0.1,0,\ldots,0)$ , and, although $x,y \preceq (1,0,\ldots,0) \in \Lambda ^n$ holds, we have $x \bowtie y$ . Note that conditional connectedness weakens a common property in order theory and its applications, namely, totality (Bridges and Mehta Reference Bridges and Mehta2013; Debreu Reference Debreu1954; Lieb and Yngvason Reference Lieb and Yngvason1999).Footnote 3

Even though conditionally connected dcpos may be more suitable to be interpreted in terms of information, many relevant order-theoretical models of uniform computability are not conditionally connected, like majorization, and have no conditionally connected straightforward variation. Take for example the interval domain (2). It is not conditionally connected since, for any $x \in \mathbb{R}$ and $0\lt \varepsilon _i,\varepsilon^{\prime}_i$ $i=1,2$ with $\varepsilon _1\gt \varepsilon^{\prime}_1$ and $\varepsilon _2 \lt \varepsilon^{\prime}_2$ , we have $a := [x-\varepsilon _1,x+\varepsilon _2] \sqsubset x$ and $b := [x-\varepsilon _1^{\prime},x+\varepsilon _2^{\prime}] \sqsubset x$ although $a \bowtie b$ . However, in case $\varepsilon _1+\varepsilon _2 \lt \varepsilon^{\prime}_1+\varepsilon^{\prime}_2$ , one would say $a$ contains more information than $b$ about any $y \in a \cap b$ , which is not reflected by $\sqsubseteq$ although it is meant to represent information. The natural modification of the interval domain, which takes care of this, is, however, not a dcpo. We would like to define the binary relation $(\mathscr{I},\sqsubseteq _2)$ , where $[a,b] \sqsubseteq _2 [c,d] \iff [a,b] \cap [c,d] \neq \emptyset$ and $\ell ([a,b]) \geq \ell ([c,d])$ with $\ell ([a,b]) := b-a$ being the length of the interval. Notice in order to turn $\sqsubseteq _2$ into a partial order, we need to identify all intervals which share some intersection and have the same length. One can easily see, however, that transitivity does not hold, hence, $\sqsubseteq _2$ is not even a partial order.

In the remainder of this section, we study the properties of conditionally connected dcpos to end up showing, in Theorem 2, the equivalence of Debreu upper separability and $w$ -continuity for them. From Proposition 10, we know Debreu upper separability is insufficient to assure $\text{K}(P)$ is countable in a continuous dcpo, which is necessary in order to hope for a countable basis $B \subseteq P$ , since $\text{K}(P) \subseteq B$ . Nevertheless, as we will see in Proposition 14, $\text{K}(P)$ is countable for conditionally connected Debreu separable dcpos. In order to arrive at that result, we first characterize both the way-below relation (Proposition 12) and $\text{K}(P)$ (Proposition 13) for conditionally connected dcpos.

Proposition 12 (Conditionally connected characterization of $\ll$ ). If $P$ is a conditionally connected dcpo, then we have, for all $ x, y \in P$ ,

\begin{equation*} x \ll y \iff \begin {cases} x \prec y \text {, or}\\ x = y \text { and there is no directed set } A \subseteq P\setminus \{x\}\\ \text { such that } \sqcup A = x. \end {cases} \end{equation*}

Proof. Take $x,y \in P$ . If $x \ll y$ , then by definition we have $x \preceq y$ and either $x \neq y$ , implying $x \prec y$ , or $x \in \text{K}(P)$ and there is no directed set $A \subseteq P\setminus \{x\}$ such that $\sqcup A = x$ by Lemma 7. For the converse, we first show, if $P$ is a conditionally connected dcpo, $x\preceq y$ and $A$ is a directed set such that $y \preceq \sqcup A$ , then either there exists $a \in A$ such that $x \preceq a$ or $x=y=\sqcup A$ . Consider, thus, a directed set $A$ such that $y \preceq \sqcup A$ and $x \preceq y$ . We have $x \preceq \sqcup A$ by transitivity and $\neg (x \bowtie a)$ $\text{for all } a \in A$ by conditional connectedness. If there exists some $a \in A$ such that $x \preceq a$ , then we have finished. Otherwise, we have $a \prec x$ $\text{for all } a \in A$ , thus, $\sqcup A \preceq x$ by definition of supremum and $x=y= \sqcup A$ by antisymmetry. We can now use this property to finish the proof. If $x \prec y$ , then, given any directed set $A$ such that $y \preceq \sqcup A$ , we get there exists some $a \in A$ such that $x \preceq a$ by the property we proved above, since we have $x \neq y$ . As a consequence, $x \ll y$ by definition of $\ll$ . Consider now some $x \in P$ such that there is no directed set $A \subseteq P\setminus \{x\}$ fulfilling $\sqcup A = x$ . Given some directed set $A$ such that $x \preceq \sqcup A$ , we get $\neg (x \bowtie a)$ $\text{for all } a \in A$ by conditional connectedness. If there exists $a \in A$ such that $x \preceq a$ , then $x \ll x$ . If the contrary holds, then $a \prec x$ $\text{for all } a \in A$ and $\sqcup A \preceq x$ by definition of supremum. Thus, $x = \sqcup A$ by antisymmetry and $x \in A$ by assumption, contradicting the fact $a \prec x$ $\text{for all } a \in A$ .

Note that the characterization of compact elements in Proposition 12 also holds under the hypothesis that $P$ is continuous, which is, as we show in Proposition 15, weaker than conditional connectedness. However, $x \prec y$ does not necessarily imply $x \ll y$ in case $P$ is just continuous. (To convince ourselves about this, we can take, for example, majorization and compare (3),

\begin{equation*} x \preceq _M y \iff (\text {for all } i\lt n)\text { } s_i(x) \leq s_i(y), \end{equation*}

with (10),

\begin{equation*} x \ll _M y \iff x = \perp \text { or } s_k(x)\lt s_k(y) \text { } \text {for all } k\lt n.) \end{equation*}

Notice, also, there is no non-essential information in conditionally connected dcpos, that is, whenever a process converges to some $x\in P$ , it must eventually gather the information of any element which contains information about $x$ , since $y \prec x$ implies $y \ll x$ $\text{for all } x,y \in P$ . Using Proposition 12, we can characterize the compact elements $\text{K}(P)$ for conditionally connected dcpos. We do so in Proposition 13, for which we need another definition. We say $x \in P$ is isolated if there exists some $v_x \in P$ such that $v_x \prec x$ and, $\text{for all } y \in P$ , we have $y \preceq v_x$ provided $y \prec x$ . We denote the set of isolated elements by $\text{I}(P)$ and note $I(\Sigma ^\infty)= \Sigma ^*\setminus \{\perp \}$ .

Proposition 13 (Conditionally connected characterization of $\text{K}(P)$ ). If $P$ is a conditionally connected dcpo, then

\begin{equation*} \text {K}(P)= \text {I}(P) \cup \text {min}(P). \end{equation*}

Proof. $({\supseteq})$ $\text{min}(P) \subseteq \text{K}(P)$ holds by the characterization of $\text{K}(P)$ in Proposition 12 since, if $x \in \text{min}(P)$ , the only directed set $A$ such that $\sqcup A=x$ is $A=\{x\}$ . Take $x \in \text{I}(P)$ and assume $x \not \in \text{K}(P)$ . Then, there exists a directed set $A \subseteq P\setminus \{x\}$ such that $\sqcup A=x$ by Proposition 12. Since $a \preceq v_x$ $\text{for all } a \in A$ by definition of an isolated element, we would have $x= \sqcup A \preceq v_x$ by definition of supremum, a contradiction. Thus, $\text{I}(P) \subseteq \text{K}(P)$ . $({\subseteq})$ Take $x \not \in \text{I}(P) \cup \text{min}(P)$ and define $A_x:= \{z \in P| z \prec x\}$ . Since $x \notin \text{min}(P)$ , we have $A_x \neq \emptyset$ and, by conditional connectedness, $A_x$ is a chain. In particular, $\sqcup A_x$ exists and, since $a \prec x$ $\text{for all } a \in A_x$ , $\sqcup A_x \preceq x$ by definition of supremum. If $\sqcup A_x = x$ , then $x \notin \text{K}(P)$ by Proposition 12, since $x \not \in A_x$ . Conversely, $\sqcup A_x \prec x$ would contradict the fact $x \not \in \text{I}(P)$ as, by definition of $\sqcup A_x$ , any $y \in P$ such that $y \prec x$ would fulfill $y \preceq \sqcup A_x$ . Thus, we could take $v_x := \sqcup A_x$ and conclude $x \in \text{I}(P)$ , a contradiction. In summary, $x \in \text{K}(P)$ implies $x \in \text{I}(P) \cup \text{min}(P)$ .

Notice, although $\text{I}(P) \cup \text{min}(P) \subseteq \text{K}(P)$ , $\text{K}(P) \not \subseteq \text{I}(P) \cup \text{min}(P)$ if we weaken the hypothesis from conditionally connected to continuous in Proposition 13 (take, e.g., the dcpo in Proposition 10). As we already know from Lemma 4, $\text{min}(P)$ is countable whenever countable Debreu upper dense subsets exist. Moreover, for conditionally connected dcpos, the set of isolated elements $\text{I}(P)$ is countable as well if Debreu upper separability holds. Because of the characterization of the compact elements in Proposition 13, these two facts result in Proposition 14, where we show conditionally connected dcpos which are Debreu upper separable have a countable number of compact elements.

Proposition 14. If $P$ is a conditionally connected and Debreu upper separable dcpo, then $\text{K}(P)$ is countable.

Proof. Since $\text{K}(P)=\text{min}(P)\cup \text{I}(P)$ by Proposition 13 and $\text{min}(P)$ is countable whenever there exists some countable Debreu upper dense set by Lemma 4, we only need to show $\text{I}(P)$ is countable to get the result. Take $D \subseteq P$ a countable Debreu dense subset and $D^{\prime}:= \text{I}(P)\setminus D$ , where $\text{I}(P)$ the set of isolated points. If $x \in D^{\prime}$ , then there exists some $d \in D$ such that $v_x \preceq d \preceq x$ , where $v_x \in P$ has the property that $\text{for all } y \in P$ such that $y \prec x$ we have $y \preceq v_x$ . By definition of $v_x$ and since $x \not \in D$ , $v_x =d$ holds. We define now a map

\begin{alignat*}{3} f: \text{ } &D^{\prime} &&\rightarrow && \text{ }D\\ &x &&\mapsto &&\text{ }v_x. \end{alignat*}

If we show $f$ is injective, then we get $D^{\prime}$ is countable and, thus, $\text{I}(P)$ also, since we have $\text{I}(P) \subseteq D^{\prime} \cup D$ . Take, thus, $x,y \in D^{\prime}$ such that $f(x) = f(y)$ and assume $x \neq y$ . Notice $x \prec y$ contradicts the definition of $v_y$ , since it would mean $v_y = v_x \prec x \prec y$ . Equivalently, we can discard $y \prec x$ by definition of $v_x$ . If $x \bowtie y$ , then, by Debreu upper density, there exists some $d \in D$ such that $x \bowtie d \preceq y$ . Since $y \not \in D$ , we have $x \bowtie d \prec y$ . Thus, $d \preceq v_y = v_x \prec x$ which contradicts, by transitivity, the fact that $x \bowtie d$ . Since $x \neq y$ leads to contradiction, we conclude $f(x) = f(y)$ implies $x = y$ $\text{for all } x,y \in D^{\prime}$ and, hence, $f$ is injective.

To conclude $\text{I}(P)$ is countable in Proposition 14, it is sufficient for $P$ to be a partial order instead of a dcpo. Notice, also, that we do not need the elements of $D$ in its proof to have a finite number of immediate successors as in Proposition 9. In fact, there exist conditionally connected dcpos where any such a $D$ has elements with an infinite number of immediate successors (take, e.g.,, $P$ the dcpo defined analogously to (1) but using the natural numbers as alphabet $\Sigma := \mathbb{N}$ and note that any such $D\subseteq P$ will have elements with a countably infinite number of immediate successors). Actually, Proposition 14 shows any conditionally connected and Debreu upper separable partial order has a countable number of jumps, since the set of jumps is equinumerous to the set of isolated elements for conditionally connected dcpos (one can see that the map

\begin{alignat*}{3} \phi : \text{ } &\text{J}(P) &&\rightarrow && \text{ }\text{I}(P)\\ &(x,y) &&\mapsto &&\text{ }y \end{alignat*}

is bijective, where $\text{J}(P)$ denotes the set of jumps of $P$ ). We improve, thus, on the classical result that the number of jumps is countable for any Debreu separable total order (Debreu Reference Debreu1954) (see also Bridges and Mehta Reference Bridges and Mehta2013, Proposition 1.4.4), as total partial orders are conditionally connected and Debreu separability coincides with Debreu upper separability for total orders. Notice, if we eliminate conditional connectedness, the number of jumps could be uncountable even if Debreu upper separability holds (take, for example, the dcpo in the proof of Proposition 4). In fact, there are continuous Debreu upper separable dcpos where $\text{K}(P)$ is uncountable (see Proposition 10) and conditionally connected dcpos where $\text{K}(P)$ is uncountable (for example, any uncountable set with the trivial partial order).

Before proving Theorem 2, we show, in Proposition 15, that conditionally connected dcpos are continuous. This fact plus the countability of $\text{K}(P)$ from Proposition 14 and some results from Sections 3 and 5.1 will allow us to derive Theorem 2.

Proposition 15. If $P$ is a conditionally connected dcpo, then $P$ is a basis for $P$ .

Proof. We will show $B := P$ is a basis whenever $P$ is conditionally connected. We need to show there exists a directed set such that $x = {\sqcup}B_x$ $\text{for all } x \in P$ . If $x \in \text{K}(P)$ , we can take $B_x := \{x\}$ . If $x \not \in \text{K}(P)$ , then, by Proposition 12, there exists a directed set $B_x \subseteq P\setminus \{x\}$ such that ${\sqcup}B_x = x$ . Thus, $b \prec x$ and, again by Proposition 12, we have $b \ll x$ $\text{for all } b \in B_x$ .

Notice, if $P$ is conditionally connected, then, by the characterization of $\text{K}(P)$ in Proposition 13 plus the fact it has a basis $B\subseteq P$ (Proposition 15), any $x \in P\setminus (\text{I}(P) \cup \text{min}(P))$ is a non-trivial element of $B$ . If we also assume $P$ has a countable Debreu dense subset $D \subseteq P$ , we can then find an increasing sequence $(x_n)_{n\geq 0} \subseteq D$ such that $x = \sqcup (x_n)_{n\geq 0}$ and $x \not \in (x_n)_{n\geq 0}$ (see Lemma 3) and we can easily see $x_n \ll x$ $\text{for all } n \geq 0$ . Thus, if we assume $P$ is a conditionally connected Debreu upper separable dcpo, we can construct a countable basis using $\text{K}(P)$ (countable by Proposition 14) and a countable Debreu dense subset. We show this and its converse in Theorem 2.

Theorem 2. If $P$ is a conditionally connected dcpo, then the following statements are equivalent:

1. 1. $P$ is Debreu upper separable.

2. 2. $P$ is $\omega$ -continuous.

3. 3. $P$ is Debreu separable and $\text{K}(P)$ is countable.

Proof. We begin showing $(1)$ implies $(2)$ . Given that $\text{K}(P)$ is countable by Proposition 14, we can apply Proposition 7, since conditionally connected dcpos are continuous by Proposition 15, and get the result. We proceed now to show $(2)$ implies $(3)$ . If $B \subseteq P$ is a countable basis, then $\text{K}(P)\subseteq B$ is also countable. Take $x,y \in P$ such that $x \prec y$ . We can follow Proposition 2 to get some $b_0 \in B$ such that $b_0 \preceq y$ and $\neg (b_0 \preceq x)$ . Since we have both $b_0 \preceq y$ and $x \prec y$ , then $\neg (x \bowtie b_0)$ holds by conditional connectedness. As a result, we have $x \prec b_0 \preceq y$ . Thus, $B$ is a countable Debreu dense subset of $P$ . We finish showing $(3)$ implies $(1)$ . We will show $D^{\prime} := D \cup \text{K}(P)$ is a countable Debreu upper dense subset, where $D\subseteq P$ is a countable Debreu dense subset. Given any pair $x,y \in P$ such that $x \bowtie y$ , we need to show there exists some $d \in D^{\prime}$ such that $x \bowtie d \preceq y$ . If $y \in \text{K}(P)$ , then we take $d := y$ and have finished. If $y \not \in \text{K}(P)$ , then there exists a directed set $A \subseteq P$ such that $\sqcup A = y$ and $y\not \in A$ by Proposition 12. By Lemma 3, thus, there is a directed set $D_y \subseteq D$ such that $\sqcup D_y = y$ and $y \not \in D_y$ . Analogously to the proof of Proposition 2, we can conclude there exists some $d \in D_y$ such that $d \prec y$ and $\neg (d \preceq x)$ . Since $x \prec d$ would imply $x \prec y$ by transitivity, which is a contradiction, we obtain $x \bowtie d \prec y$ .

Remark 7 (Implication for computability). The proof of Theorem 2 shows that we can introduce uniform computability (for both elements and functions) in conditionally connected dcpos which have a countable subset $D$ , which is both Debreu dense and Debreu upper dense provided the countable basis $D \cup \text{I}(P) \cup \text{min}(P)$ is effective.

We conclude from Theorem 2 that the coincidence between Debreu upper separability and $w$ -continuity is not a specific fact of the Cantor domain but it holds for any conditionally connected dcpo. Note that we cannot substitute conditional connectedness by the weaker pair of hypotheses, which includes both continuity of $P$ and the fact $x \preceq y$ implies $x \ll y$ $\text{for all } x,y \in P$ . In that scenario, $(1)$ does not imply $(2)$ in Theorem 2 (see Proposition 10). This is relevant, since the second hypothesis in the pair is a rather strong property of conditionally connected dcpos.

In Corollary 2, we refine Theorem 2, characterizing a stronger order density property for conditionally connected dcpos. We say a partial order is upper separable if there exists a countable subset $D \subseteq P$ , which is order dense and such that, for any pair $x,y \in P$ where $x \bowtie y$ , there exists some $d \in D$ such that $x \bowtie d \prec y$ (Ok et al. Reference Ok2002). Note that the Cantor domain is not upper separable, as it is not order separable. Moreover, if $s \in \Sigma ^*$ and $\beta, \gamma \in \Sigma$ $\beta \neq \gamma$ , then we have $s\beta \bowtie s\gamma$ , although, $\text{for all } t \in \Sigma ^*$ such that $t \prec s\beta$ , we have $t \prec s\gamma$ .

Corollary 2. If $P$ is a conditionally connected dcpo, then the following statements are equivalent:

1. 1. $P$ is upper separable

2. 2. $P$ is $\omega$ -continuous and either $\text{K}(P)=\{\perp \}$ or $\text{K}(P)=\emptyset$ holds.

Proof. We begin by showing $(1)$ implies $(2)$ . Take $D \subseteq P$ a countable upper dense subset. By Theorem 2, we have $P$ is $\omega$ -continuous and, by Lemma 9, we obtain that $\text{K}(P) = \text{min}(P)$ . We finish by noticing, if we have $x,y \in \text{min}(P)$ $x \not = y$ , then $x \bowtie y$ by definition and there exists $d \in D$ such that $x \bowtie d \prec y$ by upper separability, contradicting the fact $y \in \text{min}(P)$ . Thus, we have $|\text{min}(P)| \leq 1$ . Assume there exists some $x \in \text{min}(P)$ and take $y \in P\setminus \text{min}(P)$ . Notice $y \prec x$ contradicts the fact $x \in \text{min}(P)$ and so does, by upper separability, $y \bowtie x$ . Thus, we have $x \prec y$ and $x=\perp$ . Hence, we either have $\text{K}(P)=\{\perp \}$ , or $\text{K}(P)=\emptyset$ . We finish showing $(2)$ implies $(1)$ . Take $B \subseteq P$ a countable basis and $x,y \in P$ such that $x \prec y$ . We can follow the argument in Theorem 2 ( $(2)$ implies $(3)$ ) to get some $b_0 \in B$ such that $x \prec b_0 \ll y$ . Since $y \neq \perp$ , we have $y \not \in \text{K}(P)$ and, thus, $x \prec b_0 \prec y$ . Take now $x,y \in P$ such that $x \bowtie y$ . Since we have a countable basis, we fulfill the hypotheses in Theorem 2 $(3)$ and we can follow the argument there (when proving $(3)$ implies $(1)$ ) to conclude there exists some $b_0 \in B$ such that $x \bowtie b_0 \prec y$ .

Notice, a dcpo to which Corollary 2 applies can be found in the proof of Proposition 16. To finish this section, we notice the Cantor domain is not only $\omega$ -continuous but $\omega$ -algebraic. A dcpo $P$ is algebraic if it has a basis $B\subseteq P$ consisting of compact elements $B = \text{K}(P)$ and $\omega$ -algebraic if such a basis is countable (Abramsky and Jung Reference Abramsky and Jung1994). The Cantor domain is actually $\omega$ -algebraic, as $B := \{\perp \} \cup \Sigma ^*$ is a countable basis and $B=\text{min}(P) \cup \text{I}(P) = \text{K}(P)$ . Nevertheless, we cannot substitute $\omega$ -continuous by $\omega$ -algebraic in Theorem 2, as we show in Proposition 16.

Proposition 16. There exist conditionally connected Debreu upper separable dcpos which are not $\omega$ -algebraic.

Proof. We can consider $P$ the interval $[0,1]$ with the usual order $\leq$ . Clearly, it is conditionally connected (it is total) and Debreu upper separable (because of $\mathbb{Q} \cap [0,1]$ ). However, while $B := \mathbb{Q} \cap [0,1]$ is a countable basis, we have $\text{K}(P)=\{0\}$ . Thus, $P$ is not $\omega$ -algebraic.

To summarize, the main results in this section are Propositions 7, 8, and 9, and Theorem 2. In the first one, we show how one can profit from Debreu density and the countability of $\text{K}(P)$ to introduce computability on a continuous dcpo. In the second, we avoid the requirement on $\text{K}(P)$ and use the stronger property of order separability to introduce computability on all non-minimal elements. In the third one, we show that we can achieve the same results as in Proposition 8 by asking for the existence of a countable Debreu separable subset whose elements only have a finite number of immediate successors. Moreover, we extend computability to the whole dcpo in Propositions 8 and 9 by also requiring the existence of a countable Debreu upper dense subset. Lastly, in Theorem 2, we use conditional connectedness to show the equivalence between having a countable basis and being Debreu upper separable.