2. Preliminaries

Let us introduce the concepts and notions to be used in this paper.

Let P be a poset. A subset A of P is called an upper set (resp., a lower set) if $A = {\uparrow}A$ (resp., $A = {\downarrow}A$ ), where ${\uparrow}A = \{x\in P: x\geq a\ \mathrm{for \ some}\ a\in A\}$ (resp., ${\downarrow}A = \{x\in P: x\leq a\ \mathrm{for \ some}\ a\in A\}$ ). A nonempty subset D of P is said to be directed if for each finite subset $F\subseteq D$ there exists some $d\in D$ such that $F\subseteq {\downarrow}d$ . Then P is directed complete (or a dcpo) if every directed subset D of P has a least upper bound, that is, $\sup D$ exists in P. Let $\sigma(P)$ denote the Scott topology on P, where every U in $\sigma(P)$ , called Scott open, satisfies $U = {\uparrow}U$ and for any directed subset D for which supD exists, supD $\in U$ implies $D\cap U \neq \emptyset$ . Correspondingly, $A\subseteq P$ is Scott closed if A = $\downarrow$ A and for any directed subset D of P contained in A, supD $\in A$ when $\sup D$ exists. Until it is otherwise stated, we always equip posets with the Scott topology, and $cl_{\sigma}(A)$ or $\overline A$ is used to denote the closure of $A\subseteq P$ with respect to the Scott topology.

For a $T_{0}$ topological space X, the partial order $\leq$ $_{X}$ , defined by $x\leq y$ iff x is contained in the closure of y, is called the specialization order. We have that for any $x\in X$ , $\downarrow$ x = $cl(\{x\})$ and a continuous map f between two $T_{0}$ spaces is always order-preserving. X is called a monotone convergence space (or a d-space) if every subset D directed relative to the specialization order has a supremum, and the relation $\sup D\in U$ for any open set U of X implies $D\cap U \neq \emptyset$ . Let C(X) denote the set of all closed subsets of X. The lower Vietoris topology on C(X) is the topology generated by $\{\lozenge U: U\in \mathcal{O}(X)\}$ as a subbase, where $\lozenge U = \{A\in C(X): A\cap U\neq \emptyset\}$ , and the resulting space denoted by $\mathcal H_t(X)$ is called the Hoare power space, here t in the subscript refers to the fact that the construction acts on topological spaces. $A\in C(X)$ is called irreducible if for any $B, C\in C(X)$ , $A\subseteq B\cup C$ implies that $A\subseteq B$ or $A\subseteq C$ . X is called sober if every nonempty irreducible closed set is the closure of a point. From Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Exercise V-4.9) we know that there is a sobrification $(\mathbb S(X), s_X)$ for each $T_0$ space X, where the standard construction for the sobrification is to set

\begin{align*} \mathbb S(X):= \{A\subseteq X: A\ \mathrm{is\ closed\ and\ irreducible}\} \end{align*}

topologized by open sets $U^{s} := \{A\in \mathbb S(X): A\cap U\neq\emptyset\}$ for each open subset U of X and $s_X$ is a topological embedding from X to $\mathbb S(X)$ defined by $s_X(x) = cl(\{x\})$ for each $x\in X$ . We call it standard sobrification.

For a general full subcategory $\mathbf K$ of a category $\mathbf{C}$ , $\mathbf K$ is called reflective if the inclusion functor has a left adjoint, which is then called a reflector and exhibited in the following

For every object K in $\mathbf K$ and every map $f: C\rightarrow K$ in $\mathbf C$ , there is a unique morphism $\tilde{f}: \widetilde{C}\rightarrow K$ in $\mathbf K$ such that $\tilde{f}\circ\mu = f$ :

We call $\widetilde{C}$ together with the universal $\mathbf K$ -ification $\mu$ a $\mathbf K$ -ification of C. It was Keimel and Lawson who first showed that a full subcategory $\mathbf K$ is reflective in $\mathbf{TOP}_{\mathbf{0}}$ if it satisfies $(K_1)$ to $(K_4)$ mentioned in the Introduction. In this sense of universal $\mathbf K$ -ifications, Xu in (2020) provided another approach to constructing $\mathbf K$ -ifications of $T_0$ spaces.

In Xu (Reference Xu2020), Xu called a full subcategory $\mathbf K$ of $\mathbf{TOP}_{\mathbf{0}}$ adequate if for any $T_{0}$ space X, $\mathcal{K}(X)$ , the space obtained by endowing $\mathbf{K}(X)$ with the lower Vietoris topology, is a $\mathbf{K}$ -space. He proved that when $\mathbf K$ satisfies $(K_2)$ and is adequate, the pair $\langle \mathcal{K}(X), \zeta_{X} \rangle$ , where $\zeta_{X}: X\rightarrow \mathcal K(X): x\mapsto cl(\{x\})$ , is a $\mathbf{K}$ -ification of X.

Recently, Ershov called a full subcategory $\mathbf K$ of $\mathbf{TOP}_{\mathbf{0}}$ wide if for any $T_0$ space X there exists an extension $Y\geq X$ such that $Y\in \mathbf K$ . For a wide category $\mathbf K$ , he endowed new definitions for $\mathbf K$ -subspaces and $\mathbf K$ -completions and used them to offer sufficient conditions for the existence of $\mathbf K$ -ification of an arbitrary $T_0$ space (Ershov Reference Ershov2022).

Ershov denoted by $X^{\mathbf K}$ the greatest $\mathbf K$ -subspace for X, whose existence is guaranteed by Ershov (Reference Ershov2022, Theorem 2.2). In addition, he called $X^{\mathbf K}$ a $\mathbf K$ -completion of a $T_0$ space X if $X^{\mathbf K}\in \mathbf K$ .

A $\mathbf K$ -precompletion $(F, \eta)$ is referred to as ample if, for any $T_0$ space X, the fact that $\eta_X: X\rightarrow F(X)$ is an identity map implies the inclusion $X\in \mathbf K$ .

Ershov has proved that for a wide category $\mathbf K$ , the existence of an ample $\mathbf K$ -precompletion guarantees the existence of the $\mathbf K$ -completion of an arbitrary $T_0$ space X (Ershov Reference Ershov2022, Theorem 4.3), in other words, he provided a sufficient condition to make the category $\mathbf K$ reflective in $\mathbf{TOP}_{\mathbf{0}}$ . Next we will show that this condition is also necessary.

1. (1) X is a $\mathbf{K}$ -space.

2. (2) $\mathbf{K}(X)$ = $\{{\downarrow}x: x\in X\}$ , that is, for each $A\in\mathbf{K}(X)$ , there exists an $x\in X$ such that A = $cl(\{x\})$ .

For ( $K_{3}$ ), let $\{X_{i}: i\in I\}$ be a family of $\mathbf{K}$ -subspaces of S, where $\mathbf K$ -subspaces mentioned in $(K_3)$ are subspaces of S that are $\mathbf K$ -spaces in the relative topology. Suppose $A\subseteq\bigcap_{i\in I}X_{i}$ is a $\mathbf{K}$ -set. Then, A is a $\mathbf{K}$ -set of S, which implies that $cl_{S}(A) = cl_{S}(\{a\})$ for some $a\in S$ since the sober space S is a $\mathbf{K}$ -space. Meanwhile, A is also a $\mathbf{K}$ -set of every $\mathbf{K}$ -space $X_{i}$ ; thus, there is an $a_{i}\in X_{i}$ such that $cl_{X_{i}}(A) = cl_{X_{i}}(\{a_{i}\})$ . By Lemma 2.8, we have $cl_{S}(cl_{X_{i}}(A)) = cl_{S}(A)$ , which means $cl_{S}(cl_{X_{i}}(\{a_{i}\})) = cl_{S}(\{a\})$ . By the fact that $cl_{S}(cl_{X_{i}}(\{a_{i}\})) = cl_{S}(\{a_{i}\})$ , we have $cl_{S}(\{a_{i}\}) = cl_{S}(\{a\})$ . Thus, $a_{i} = a$ and $a\in \bigcap_{i\in I}X_{i}$ . Therefore, $cl_{\bigcap_{i\in I}X_{i}}(A) = cl_{\bigcap_{i\in I}X_{i}}(\{a\})$ . So relying on the adequateness, by Lemma 2.7, we have that ${\bigcap_{i\in I}X_{i}}$ is a $\mathbf{K}$ -space.

For ( $K_{4}$ ), let S, T be sober spaces and $f: S\rightarrow T$ a continuous map. Assume that X is a $\mathbf{K}$ -subspace of T. For any $\mathbf{K}$ -set $A\subseteq f^{-1}(X)$ , A is also a $\mathbf{K}$ -set of S. Then, there is an $a\in S$ such that $cl_{S}(A) = cl_{S}(\{a\})$ . Besides, by the continuity of f, we know $f(A)\subseteq X$ is a $\mathbf{K}$ -set of T and one can verify that f(A) is also a $\mathbf{K}$ -set of X. Thus there are two points $a_{1}\in X$ and $a_{2}\in T$ such that $cl_{X}(f(A)) = cl_{X}(\{a_{1}\})$ and $cl_{T}(f(A)) = cl_{T}(\{a_{2}\})$ , respectively. By Lemma 2.8, $cl_{T}(cl_{X}(f(A))) = cl_{T}(f(A))$ , thus $a_{1} = a_{2}$ . Now we have $f(a) = f(\sup_{S} A) = \sup_{T}f(A) = a_{2}$ . As $a_{1} = a_{2}$ and $a_{1}\in X$ , $f(a)\in X$ , that is, $\sup_{S} A = a\in f^{-1}(X)$ . This means $\sup_{f^{-1}(X)}A$ exists and equals to $\sup_{S} A$ . Thus, $cl_{f^{-1}(X)}(A) = cl_{f^{-1}(X)}(\{\sup_{f^{-1}(X)}A \}) = cl_{f^{-1}(X)}(\{\sup_{S}A\}) = cl_{f^{-1}(X)}(\{a\})$ . Therefore, using the adequateness and Lemma 2.7 again, we have that $f^{-1}(X)$ is a $\mathbf{K}$ -subspace of S.

Given that Ershov has shown that a full subcategory $\mathbf K$ of $\mathbf{TOP}_{\mathbf{0}}$ satisfies the properties $(K_1)$ to $(K_4)$ if and only if $\mathbf K$ is wide and $\mathbf K$ -completion exists for each $T_0$ space (see Ershov Reference Ershov2022, Theorem 3.8), then together with Propositions 2.6 and 2.9, we could draw the following conclusion.

1. (1) $\mathbf K$ is a wide category and there exists an ample $\mathbf K$ -precompletion $(F, \eta)$ on the category $\mathbf{TOP}_{\mathbf{0}}$ .

2. (2) $\mathbf K$ satisfies the properties $(K_1)$ - $(K_4)$ .

3. (3) $\mathbf K$ satisfies the properties $(K_1)$ , $(K_2)$ and is adequate.

It can be seen from the above theorem that the $\mathbf K$ -ifications of a $T_0$ space constructed by Keimel and Lawson, Xu, and Ershov respectively are consistent. In our paper, we will mainly use Xu’s description for $\mathbf K$ -ifications. This is because, when constructing an order- $\mathbf K$ -ification monad on DCPO and further examining its algebras, it benefits us a lot if we know concretely what composes such a completion.

3. Categories of Type $\mathrm K^*$

In what follows, a category of type $\mathrm K$ is defined by satisfying Properties $(K_1)$ , $(K_2)$ mentioned in the Introduction and the adequacy property. Categories of type $\mathrm K$ were initially considered by Keimel and Lawson in (2009), and they are also called K-categories in Jia and Mislove (Reference Jia and Mislove2022).

( $K_5$ ) All $\mathbf K^*$ -spaces are monotone convergence spaces.

The following example distinguishes the category of type $\mathrm{K}^{*}$ from that of type K.

Let $\mathbf{K}^{*}(X)$ denote the set of all closed $\mathbf{K}^{*}$ -sets of X. Then, A is a $\mathbf{K}^{*}$ -set iff $cl(A)\in \mathbf{K}^{*}(X)$ . In particular, when $\mathbf{K}^{*}$ is SOB or WF, a $\mathbf{K}^{*}$ -set of a $T_{0}$ space X is indeed an irreducible set or a well-filtered determined set defined by Xu in Xu and Zhao (Reference Xu and Zhao2020), respectively. We will use $\mathbf{WF}(X)$ to denote the family consisting of all closed well-filtered determined sets of X.

Clearly, each directed subset is tapered, which together with the above lemma guarantees the following result:

1. (1) X is a monotone convergence space.

2. (2) For any tapered and closed subset $A\subseteq X$ , $A = cl(\{x_{0}\})$ for some $x_{0}\in X$ .

$(2)\Rightarrow (1)$ : For any $\mathbf{K}$ -set A of X, by Lemma 3.10, A is a $\mathbf{K}^{*}$ -set, thus cl(A) = $cl(\{a_0\})$ for some $a_0\in A$ by (2). So we conclude that X is a $\mathbf{K}$ -space by Lemma 2.7. Similarly, we could show that X is also a monotone convergence space.

Proof. Assume that $\mathcal{A}$ is a closed $\mathbf{K}^{*}$ -set of $\mathbf{K}^{*}(X)$ . We claim that $\bigcup\mathcal{A}$ is a $\mathbf{K}^{*}$ -set of X. Let Y be a $\mathbf{K}^{*}$ -space and $f: X\rightarrow Y$ a continuous map. Then, define a map

\begin{align*} g: \mathbf{K}^{*}(X)\rightarrow \mathbf{K}^{*}(Y): C\mapsto cl(f(C)), \end{align*}

whose rationality is guaranteed by Lemma 3.6. For each open set $\lozenge U$ of $ \mathbf{K}^{*}(Y)$ , where $U\in \mathcal{O}(Y)$ , $g^{-1}(\lozenge U) = \lozenge f^{-1}(U)$ , so g is continuous. Since each $A\in \mathcal{A}$ belongs to $\mathbf{K}^{*}(X)$ , there exists a $y_{A}\in Y$ such that $cl(f(A)) = {\downarrow} y_{A}$ . It follows that $g(\mathcal{A}) = \{{\downarrow} y_{A}: A\in \mathcal{A}\}$ is a $\mathbf{K}^{*}$ -set of $\mathbf{K}^{*}(Y)$ . As Y is a $\mathbf{K}^{*}$ -space, by Lemma 3.11, we could define a map

\begin{align*} h: \mathbf{K}^{*}(Y)\rightarrow Y: E\mapsto \sup E, \end{align*}

one can easily verify that h is continuous. Then, by Lemma 3.6 again, $h(\{{\downarrow} y_{A}: A\in \mathcal{A}\}) = \{y_{A}: A\in \mathcal{A}\}$ is a $\mathbf{K}^{*}$ -set of Y. Hence, $\sup\{y_{A}: A\in \mathcal{A}\} = y_{0}$ exists. Then, we have

\begin{align*} cl(f(\bigcup\mathcal{A})) = cl(\bigcup\{f(A): A\in \mathcal{A}\}) = cl(\bigcup\{{\downarrow}y_{A}: A\in \mathcal{A}\}) = cl(\{y_{0}\}), \end{align*}

which entails that $\bigcup\mathcal{A}$ is a $\mathbf{K}^{*}$ -set. So $\mathcal{A} = cl(\{\bigcup\mathcal{A}\})$ and $\mathcal K_{t}(X)$ is a $\mathbf{K}^{*}$ -space by Lemma 3.11.

Remark 3.14. (Xu 2020, Theorem 5.14) When a category $\mathbf{K}^{*}$ of type $\mathrm K^*$ is specifically taken as WF, $\mathcal K_t(X)$ is $\mathcal{WF}_t(X)$ , i.e., $\mathbf{WF}(X)$ endowed with the lower Vietoris topology, and $\eta_X: X\rightarrow \mathcal{WF}_t(X)$ is defined as

\begin{align*}\mathrm{for any} x\in X, \eta_X(x) = {\downarrow}x.\end{align*}

Then, ( $\mathcal{WF}_t(X), \eta_X)$ is a well-filtered reflection of X.

4. The Order-K-ification Monad

A monad on a category C consists of an endofunctor $\mathcal T$ on C together with natural transformations $\eta : Id_{\mathbf C} \rightarrow \mathcal T$ and $\mu: \mathcal T^2 \rightarrow \mathcal T$ such that $\mu_{A} \circ \mathcal T\eta_A = Id_{\mathcal TA} = \mu_A \circ \eta_{\mathcal TA}$ and $\mu_A \circ \mathcal T\mu_A = \mu_A \circ \mu_{\mathcal TA}$ (Mac Lane Reference Mac Lane1998).

Let $\mathbf K^*$ be a category of type $\mathrm K^{*}$ , as in Remark 3.2, determined by certain category of type $\mathrm K$ . Theorems 3.12 and 3.13 tell us the corresponding reflector $\mathcal K_t$ , sending each $T_0$ topological space X to its $\mathbf{K}^{\mathbf{*}}$ -ification (or the $\mathbf{K}^{\mathbf{*}}$ -space completion), composing the inclusion functor Inc is not only a monad on $\mathbf{TOP}_{\mathbf{0}}$ but can be restricted to $\mathbf {MCS}$ . Now compose them with the pair of functors $\Omega$ and $\Sigma$ :

where $\Sigma$ assigns to each dcpo L the topological space $(L, \sigma (L))$ , and $\Omega$ assigns to each monotone convergence space X the dcpo $(X, \leq)$ with $\leq$ the specialization order on X. We write $\mathcal K_{t}\circ \Sigma$ as $\mathcal K_{d}$ and $Inc\circ \Omega$ as $\mathrm {Inc}$ . Then $\mathcal K_{d}$ is left adjoint to $\mathrm {Inc}$ . By the fact that each adjoint pair determines a monad, one can refer to Borceux (Reference Borceux1994, Proposition 4.2.1), we know the triple $(\mathrm {Inc}\circ \mathcal K_{d}, \eta, \mathrm{Inc}\circ\varepsilon \circ \mathcal K_{d})$ , where $\eta$ and $\varepsilon$ are the unit and counit respectively, turns into a monad on $\mathbf{DCPO}$ .

We denote $\mathrm{Inc}\circ\mathcal K_{d}$ with $\mathcal K$ , for any dcpo L, $\mathcal{K}(L)$ is a dcpo consisting of all closed $\mathbf{K}^{*}$ -sets of $(L, \sigma(L))$ ordered by set inclusion. For each directed family $\mathcal{C}$ of $\mathcal{K}(L)$ , one can verify that $\bigcup \mathcal{C}$ is a $\mathbf{K}^{*}$ -set of L, so the supremum of $\mathcal{C}$ in $\mathcal{K}(L)$ is the Scott closure of $\bigcup \mathcal{C}$ . Meanwhile, we calculate that $\mathrm{Inc}\circ\varepsilon\circ \mathcal K_{d}$ (replaced by $\mu_{L}$ when it works on a dcpo L) is a natural transformation from $\mathcal{K}(\mathcal{K}(L))$ to $\mathcal{K}(L)$ that maps $\mathcal{A}$ to $\sup_{\mathcal{K}(L)}\mathcal A$ .

This lemma tells us for each $\mathcal A\in \mathcal K(\mathcal K(L))$ , $\sup_{\mathcal K(L)}\mathcal{A} = \bigcup\mathcal{A}$ . Now we conclude that

Theorem 4.2. The endofunctor $\mathcal{K}$ together with the unit $\eta$ and the multiplication $\mu$ forms a monad, called an order- $\mathbf{K}$ -ification monad, on $\mathbf{DCPO}$ . Concretely, $\mathcal{K}$ associates with a dcpo L the dcpo $\mathcal{K}(L)$ and with a morphism $f: L\longrightarrow M$ in $\mathbf{DCPO}$ the map $\mathcal{K}(f)$ : $\mathcal{K}(L)\longrightarrow \mathcal{K}(M)$ , which is defined by

\begin{align*} \forall A \in \mathcal{K}(L), \mathcal{K}(f)(A) = \overline{f(A)}; \end{align*}

$\eta_{L}:L\longrightarrow \mathcal{K}(L)$ and $\mu_{L}: \mathcal{K}(\mathcal{K}(L))\longrightarrow \mathcal{K}(L)$ are defined by

\begin{align*} \forall x\in L, \eta(x) = \overline{\{x\}}, \end{align*}

and

\begin{align*} \forall\mathcal{A}\in \mathcal{K}(\mathcal{K}(L)), \mu(\mathcal{A}) = \bigcup \mathcal{A},\end{align*}

respectively.

5. The Eilenberg-Moore Algebras of $\mathcal K$

Recall that a $\mathcal{T}$ ${-algebra}$ (Mac Lane Reference Mac Lane1998) of a monad $(\mathcal{T}, \eta, \mu)$ is a pair $(C, \xi)$ , where C is an object of C and $\xi: \mathcal{T}C\rightarrow C$ is a morphism in C, that satisfies $\xi\circ\mu_{C} = \xi\circ \mathcal{T}\xi$ and $\xi\circ \eta_{C} = id_{C}$ . In this case, $\xi$ is called a structure map. If both $(A, \xi_{A})$ and $(B, \xi_{B})$ are $\mathcal{T}$ -algebras, a $\mathcal T$ -algebra homomorphism from $(A, \xi_{A})$ to $(B, \xi_{B})$ is an arrow $h: A\rightarrow B$ satisfying $ h\circ \xi_{A} = \xi_{B}\circ \mathcal{T}h$ .

For a monad $(\mathcal T, \eta, \mu)$ on the category C, the category of $\mathcal T$ -algebras determines with respect to what algebraic structures the functor $\mathcal T$ can be understood to be universal. More precisely, for each morphism f in $\mathbf C$ which maps A to a $\mathcal T$ -algebra $(C, \xi)$ , there exists a unique $\mathcal T$ -algebra homomorphism $h: \mathcal TA\rightarrow C$ such that $f = h\circ \eta_A$ . Identifying such structures is not only an interesting mathematical problem, but also could be useful for semantics. One can refer to Jia et al. (Reference Jia, Kornell, Lindenhovius, Mislove and Zamdzhiev2022) for example, where the authors proved that the category of algebras of the subprobability valuation monad on the category of domains is isomorphic to the category of so-called continuous Kegelspitzen, and this result plays a crucial role in giving an adequate semantics to variational quantum programming languages.

Let $\mathbf{C}^{\mathcal{T}}$ denote the category of all $\mathcal{T}$ -algebras and their homomorphisms, which is also called the Eilenberg-Moore category of $\mathcal{T}$ over $\mathbf C$ . We proceed to characterize the $\mathcal K$ -algebras over $\mathbf{DCPO}$ .

When $\mathbf K^*$ is $\mathbf {SOB}$ , k-complete posets are precisely strongly complete (Ho et al. Reference Ho, Goubault-Larrecq, Jung and Xi2018, Definition 2.1); when $\mathbf K^*$ is $\mathbf {WF}$ , we call a k-complete poset P w-complete, that is, every Scott closed well-filtered determined subset of P has a supremum in P.

Since directed subsets are tapered, they are $\mathbf{K}^{*}$ -sets by Lemma 3.10. Then the following results are immediately obtained.

Readers will be referred to Escardó (Reference Escardó1998) for the notions mentioned in the following.

A category C is called poset-enriched if the set of its hom-sets is a poset and its composition operation is monotone. A poset-functor between poset-enriched categories is a functor which is monotone on hom-posets. One can easily see that $\mathbf{DCPO}$ is a poset-enriched category and $\mathcal{K}$ is a poset-functor. If there is a pair of arrows $l: X\rightarrow Y$ and $r: Y\rightarrow X$ in C such that $l\circ r\leq {\mathrm{id}}_{Y}$ and $r\circ l\geq {\mathrm{id}}_{X}$ , then l is said to be a left adjoint of r, denoted by $l\dashv r$ . The adjunction $l\dashv r$ is reflective if $l\circ r = {\mathrm{id}}_{Y}$ . A monad $\mathcal{T} = (\mathcal{T}, \eta, \mu)$ on a poset-enriched category $\mathcal{C}$ is called a left KZ-monad if and only if $\mathcal{T}$ is a poset-functor and $ \mu_{C}\dashv\eta_{\mathcal{T}C}$ for all $C\in \mathcal{C}$ . In addition, if $\mathcal{F}: \mathbf{C}\rightarrow \mathbf{C}$ is a poset-functor, we shall say that a map $f: X\rightarrow Y$ in $\mathbf{C}$ is a left $\mathcal{F}$ -embedding if $\mathcal{F}f$ has a right adjoint and the adjunction is reflective.

Proof. We just need to prove that $\mu_{L}\dashv \eta_{\mathcal{K}(L)}$ for any dcpo L. On the one hand, it follows immediately from the monad law that $\mu_{L}\circ\eta_{\mathcal{K}(L)} = {\mathrm{id}}_{\mathcal{K}(L)}$ holds. On the other hand, for any $\mathcal{A}\in \mathcal{K}(\mathcal{K}(L))$ ,

\begin{align*} \eta_{\mathcal{K}(L)}\circ\mu_{L}(\mathcal{A}) = \eta_{\mathcal{K}(L)}(\bigcup\mathcal{A}) = \downarrow\!\{\bigcup\mathcal{A}\}\supseteq \mathcal{A}, \end{align*}

which means $\eta_{\mathcal{K}(L)}\circ\mu_{L}\geq {\mathrm{id}}_{\mathcal{K}(\mathcal{K}(L))}$ . So $\mu_{L}\dashv \eta_{\mathcal{K}(L)}$ ; hence, $\mathcal{K}$ is a left KZ-monad over $\mathbf{DCPO}$ .

We fully characterize the Eilenberg-Moore algebras of the monad $\mathcal K$ in the following theorem.

$(2)\Rightarrow (3)$ : One can derive from the monad law that $\mu_L\circ \mathcal K\eta_L = id_{\mathcal K(L)}$ , which reveals that $\eta_L$ is a left embedding. So there is an extension $m: \mathcal K(L)\rightarrow L$ of the identity of L along $\eta_L$ by (2). This entails that $m\circ \eta_L = id_L$ . Let A be a Scott closed $\mathbf K^{*}$ -set of L. We claim that $\sup A = m(A)$ exists. For each $a\in A$ , $\downarrow\!\!a\subseteq A$ . Since m is monotone, $m(\downarrow\!\!a)\leq m(A)$ , which means $a = m\circ\eta(a)\leq m(A)$ . If b is another upper bound of A, then $A\subseteq \downarrow\!b$ . By the monotonicity of m again, we have $m(A)\leq m(\downarrow\!b) = b$ . Thus, $\sup A = m(A)$ exists and L is a k-complete poset.

$(3)\Rightarrow (1)$ : Since $\sup A$ exists for each $A\in \mathcal{K}(L)$ , we could define a map $\alpha: \mathcal{K}(L)\rightarrow L$ by

\begin{align*} \forall A\in \mathcal{K}(L), \alpha(A) = \sup A.\end{align*}

Then, one can easily verify that $\alpha$ is Scott-continuous, besides, $\alpha\circ\eta = {\mathrm{id}}_{L}$ and $\alpha\circ\mu = \alpha\circ\mathcal{K}\alpha$ hold. So $(L, \alpha)$ is a $\mathcal{K}$ -algebra.

By definitions of the $\mathcal{K}$ -algebra homomorphisms and the k-continuity, the following conclusion is obtained immediately, which will help us characterize the Eilenberg-Moore category of $\mathcal K$ .

Now let $\mathbf{DCPO}^{\mathcal{K}}$ denote the category of all $\mathcal K$ -algebras and all $\mathcal K$ -algebra homomorphisms, that is the Eilenberg-Moore category of $\mathcal K$ . Then obviously, $\mathcal{K}$ produces a monadic adjunction:

\begin{align*} \mathcal{F}: \mathbf{DCPO}\rightarrow \mathbf{DCPO}^{\mathcal{K}}, \qquad \mathcal U: \mathbf{DCPO}^{\mathcal{K}}\rightarrow \mathbf{DCPO},\end{align*}

where $\mathcal{F}$ assigns each dcpo L to $(\mathcal{K}(L), \mu_{L})$ and each map $f: L\rightarrow M$ from L to dcpo M to $\mathcal{K}(f): (\mathcal{K}(L), \mu_{L})\rightarrow(\mathcal{K}(M), \mu_{M})$ , and $\mathcal U$ as a forgetful functor is the right adjoint of $\mathcal F$ . As the characterizations of $\mathcal{K}$ -algebras and their homomorphisms imply that $\mathbf{DCPO}^{\mathcal{K}}$ is equivalent to the category $\mathbf{KCPO}$ , which has the k-complete posets as objects and the k-continuous maps as morphisms, we reach the following conclusion:

In Zhao and Fan (Reference Zhao and Fan2010), Zhao and Fan have proved that the category $\mathbf{DCPO}$ is a reflective full subcategory of the category $\mathbf{POS}_{\mathbf{d}}$ of posets and Scott-continuous maps. Thus combining the above corollary, the following result can be derived immediately:

Let $\mathbf{KCPO}_{\sigma}$ be the full subcategory of $\mathbf{DCPO}$ which has k-complete posets as objects and Scott-continuous maps as morphisms.

It is a truism that Cartesian closed categories give rise to the models of various typed and untyped $\lambda$ -calculi and functional programming languages (Lambek Reference Lambek1985; Scott Reference Scott1976; Streicher Reference Streicher2006). We will see in the following theorem that similar to the category DCPO, $\mathbf{KCPO}_{\sigma}$ is Cartesian closed for each category $\mathbf K^*$ of type $\mathrm K^*$ .

$\mathbf{Claim\ 1}$ : $L\times M$ is a k-complete poset.

Let $A\subseteq L\times M$ be a Scott closed $\mathbf{K}^{\mathbf{*}}$ -set. Then, A is a $\mathbf{K}^{\mathbf{*}}$ -set in $\Sigma L\times \Sigma M$ . Since the projections $\pi_{L}$ and $\pi_{M}$ are continuous, $\pi_{L}(A)$ and $\pi_{M}(A)$ are $\mathbf{K}^{\mathbf{*}}$ -sets of L and M, respectively. The fact that L and M are k-complete posets implies the existence of $\sup(\pi_{L}(A))$ and $\sup(\pi_{M}(A))$ . One can verify that $(\sup(\pi_{L}(A)), \sup(\pi_{M}(A)))$ is the supremum of A. So $L\times M$ is a k-complete poset.

$\mathbf{Claim\ 2}$ : $[L\rightarrow M]$ is a k-complete poset.

Let $\{f_{i}: i\in I\}\subseteq [L\rightarrow M]$ be a $\mathbf{K}^{\mathbf{*}}$ -set. We define $g: L\rightarrow M$ by

\begin{align*} \forall x\in L, g(x) = \sup\{f_{i}(x): i\in I\}. \end{align*}

By Lemma 2-2.8 and Lemma 2-2.9 in Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), we know the map $eval_{x}: [L\rightarrow M]\rightarrow M$ defined by $eval_{x}(h) = h(x)$ is continuous. Thus, $eval_{x}(\{f_{i}: i\in I\}) = \{f_{i}(x): i\in I\}$ is a $\mathbf{K}^{\mathbf{*}}$ -set of M. Since M is k-complete, $\sup\{f_{i}(x): i\in I\}$ exists in M. This means g is well-defined. Obviously, g is Scott-continuous and $g = \sup\{f_{i}: i\in I\}$ . Thus, $[L\rightarrow M]$ is k-complete.

In conclusion, $\mathbf{KCPO}_{\sigma}$ is Cartesian closed.

6. $\mathcal K$ is a Commutative Monad

Recall that a monoidal category (see Borceux Reference Borceux1994, Definition 6.1.1) is a category C equipped with an object $\top$ in $\mathbf C$ called unit, a bifunctor $\otimes: \mathbf C\times \mathbf C\rightarrow \mathbf C$ called tensor product, and the natural isomorphisms $\alpha, r$ and l defined as the following forms (for the objects A, B and C):

1. (i) $ {\alpha_{A, B, C}}: A \otimes (B \otimes C) \rightarrow (A \otimes B) \otimes C$ ,

2. (ii) $r_A: A\otimes \top \rightarrow A$ ,

3. (iii) $l_A: \top\otimes A \rightarrow A$ , such that certain equations hold. It will be a symmetric monoidal category if in addition it has the natural isomorphism s defined as the form:

4. (iv) $ {s_{A, B}}: A\otimes B\rightarrow B\otimes A$ ,

such that certain equations hold. It was Kock who first defined strong monads over symmetric monoidal categories (Kock Reference Kock1972). In this section we will first investigate the strongness of the monad $\mathcal K$ induced by the category $\mathbf K^*$ of type $\mathrm K^*$ on $\mathbf{DCPO}$ , where $\mathbf{DCPO}$ is clearly a Cartesian monoidal category. In this case, the tensor product $\otimes$ is precisely the categorical product and $\top$ is the terminal object, we use $\times$ and 1 to represent them respectively. Moreover, $\alpha(A, B, C), r_A, l_A$ are defined concretely as follows:

1. (i) $\alpha_{A, B, C} = ((\pi_1, \pi_1\circ\pi_2), \pi_2\circ\pi_2))$ ,

2. (ii) $r_A = \pi_1$ ,

3. (iii) $l_A = \pi_2$ ,

where $\pi_{i}(i = 1,2)$ denotes the projection onto the $i{\rm th}$ component.

Definition 6.1. A strong monad over a category $\mathbf{C}$ with an object 1 and finite products is a monad $(\mathcal T,\eta,\mu)$ together with a natural transformation $t'': ({-})\times \mathcal T({=})\rightarrow \mathcal T({-}\times{=})$ such that the following diagrams commute:

in which the natural transformation t” is called tensorial strength. Moreover, $t' = \mathcal Ts\circ t''\circ s: \mathcal T({-})\times (=)\rightarrow \mathcal T({-}\times =)$ is called cotensorial strength.

Proof. Obviously, the direct product $L_{1}\times\cdot\cdot\cdot\times L_{n}$ of finitely many dcpos $L_{1},...,L_{n}$ is still a dcpo. Given $\{L,M,N\}\subseteq \mathbf{DCPO}$ , we define $l_{L}$ as $\pi_{2}$ and $\alpha_{L,M,N}: (L\times M)\times N\rightarrow L\times (M\times N)$ as $(\pi_{1}\circ\pi_{1},(\pi_{2}\circ\pi_{1},\pi_{2}))$ . Define $t''_{L,M}:L\times \mathcal{K}(M)\longrightarrow \mathcal{K}(L\times M)$ by

\begin{align*} \forall (x, A)\in L\times \mathcal{K}(M), t''_{L,M}(x, A) =\ \downarrow x\times A. \end{align*}

$\mathbf{Claim\; 1}$ : $t''_{L,M}$ is well-defined.

It is sufficient to show that $\downarrow$ $x\times A\in \mathcal{K}(L\times M)$ . Obviously, $\downarrow$ $x\times A$ is closed in $\Sigma(L\times M)$ . Now we prove that $\downarrow$ $x\times A$ is a $\mathbf{K}^{*}$ -set in $\Sigma(L\times M)$ . For any $\mathbf{K}^{*}$ -space Y and the continuous map $f: L\times M\rightarrow Y$ , we fix $x\in L$ and define $f_{x}: M\rightarrow Y$ as

\begin{align*} \forall a\in M, f_{x}(a) = f(x,a), \end{align*}

which is well-defined. Now claim that $f_{x}$ is continuous, i.e., $f_{x}^{-1}(U)\in \sigma(M)$ for each $U\in \mathcal{O}(Y)$ . For any $a_{1}\in f_{x}^{-1}(U)$ and $a_{2}\in M$ with $a_{1}\leq a_{2}$ , $f_{x}(a_{1}) = f(x,a_{1})\in U$ , that is, $(x,a_{1})\in f^{-1}(U)\in \sigma(L\times M)$ . The fact $(x,a_{2})\geq (x,a_{1})$ implies $(x,a_{2})\in f^{-1}(U)$ , and so $f_{x}(a_{2})\in U$ . Thus $f_{x}^{-1}(U)$ is an upper set. Now let $D\subseteq M$ be a directed subset with $\sup D\in f_{x}^{-1}(U)$ . Then, $(x, \sup D)\in f^{-1}(U)$ . It follows that there exists a $d\in D$ such that $(x,d)\in f^{-1}(U)$ , that is, $d\in f_{x}^{-1}(U)$ . Thus, $f_{x}$ is continuous, which guarantees that $f_{x}(A)$ is a $\mathbf{K}^{*}$ -set of Y by Lemma 3.6. Since Y is a $\mathbf{K}^{*}$ -space, $\overline{f_{x}(A)} = \overline{\{f(x,a): a\in A\}} = \overline{\{y_{0}\}}$ for some $y_{0}\in Y$ . This means $\overline{f(\downarrow\!\!x\times A)} = \overline{\{y_{0}\},}$ and hence, $\downarrow$ $x\times A$ is a $\mathbf{K}^{*}$ -set of $L\times M$ .

$\mathbf{Claim\; 2}$ : $t''_{L,M}$ is Scott-continuous.

We just need to prove $t''_{L,M}$ is Scott-continuous in each component. Fixed $A\in \mathcal{K}(M)$ , for any directed subset $D\subseteq L$ , we have

\begin{align*} t''_{L,M}(\sup D, A) &= \downarrow\!\!\sup D\times A \\ &= \overline{\bigcup_{d\in D}\downarrow\!\!d}\times A \\ &= \overline{\bigcup_{d\in D}(\downarrow\!\!d\times A)} \\ &= \sup\{t''_{L,M}(d, A): d\in D\}, \end{align*}

where Lemma 6.2 guarantees the third equation. It follows that $t''_{L,M}$ is Scott-continuous in the first component. Next, given $x\in L$ , let $\{A_{i}: i\in I\}$ be a directed family of $\mathcal{K}(M)$ . We have

\begin{align*} t''_{L,M}(x, \sup_{i\in I}A_{i}) &= \downarrow\!\!x\times \sup_{i\in I}A_{i} \\ &= \downarrow\!\!x\times \overline{\bigcup_{i\in I}A_{i}} \\ &= \overline{\bigcup_{i\in I}(\downarrow\!\!x\times A_{i})} \\ &= \sup_{i\in I}t''_{L,M}(x, A_{i}). \end{align*}

So $t''_{L,M}$ is Scott-continuous in the second component, and it is concluded that $t''_{L,M}$ is Scott-continuous.

$\mathbf{Claim\; 3}$ : $t''_{L,M}$ is a natural transformation, that is, the following diagram commutes:

where L’, M’ are dcpos and $f: L\rightarrow L'$ , $g: M\rightarrow M'$ are Scott-continuous maps.

Now pick $(m, A)\in L\times \mathcal{K}(M)$ to prove that $(\mathcal{K}(f\times g)\circ t''_{L,M})(m, A) = (t''_{L',M'}\circ f\times \mathcal{K}(g))(m, A)$ . By the facts that

\begin{align*} (\mathcal{K}(f\times g)\circ t''_{L,M})(m, A) = \mathcal{K}(f\times g)(\downarrow\!\!m\times A) = \overline{(f\times g) (\downarrow\!\!m\times A)} = \overline{f(\downarrow\!\!m)\times g(A)} \mathrm{and},\\ (t''_{L^{'},M^{'}}\circ f\times \mathcal{K}(g))(m, A) = t''_{L^{'},M^{'}}(f(m), \overline{g(A)}) = \downarrow\!\!f(m)\times \overline{g(A)}, \end{align*}

we just need to show $\overline{f(\downarrow\!\!m)\times g(A)} = \downarrow\!\!f(m)\times \overline{g(A)}$ . Obviously, $\overline{f(\downarrow\!\!m)\times g(A)}\subseteq\;\downarrow\!\!f(m)\times \overline{g(A)}$ . On the contrary, for any $x\in \overline{g(A)}$ and $U\in \sigma(L'\times M')$ with $(f(m), x)\in U$ . Set $U_{f(m)} = \{y\in M': (f(m),y)\in U\}$ . Then, $x\in U_{f(m)}$ and $U_{f(m)}\in \sigma({M'})$ . As $x\in \overline{g(A)}$ , $U_{f(m)}\cap g(A) \neq \emptyset$ , which implies that there exists a $y_{0}\in U_{f(m)}\cap g(A)$ , that is, $(f(m), y_{0})\in (f(\downarrow\!\!m)\times g(A))\cap U\neq\emptyset$ . This means $(f(m), x)\in \overline{f(\downarrow\!\!m)\times g(A)}$ , so $\downarrow\!\!f(m)\times \overline{g(A)}\subseteq \overline{f(\downarrow\!\!m)\times g(A)}$ holds.

$\mathbf{Claim\ 4}$ : The four diagrams given in Definition 6.1 commute when replacing $\mathcal T$ with $\mathcal{K}$ and A, B, C with dcpos L, M, N, respectively.

The proof of the following equations is similar to that of the above, so we omit it.

1. (i) For any $A\in \mathcal{K}(L)$ ,

   \begin{align*} (\mathcal{K}l_{L}\circ t''_{\{1\},L})(1, A) = r_{\mathcal{K}(L)}(1, A), \end{align*}
   where 1 is the terminal object in $\mathbf {DCPO}$ .

2. (ii) For any $(a, b)\in L\times M$ ,

   \begin{align*} (t''_{L,M}\circ {\mathrm{id}}_{L}\times \eta_{M})(a, b) = \eta_{L\times M}(a, b). \end{align*}

3. (iii) For any $((a, b), A)\in (L\times M)\times \mathcal{K}(N)$ ,

   \begin{align*} (\mathcal{K}\alpha_{L,M,N}\circ t''_{L\times M,N})((a,b),A) = (t''_{L,M\times N}\circ {\mathrm{id}}_{L}\times t''_{M,N}\circ \alpha_{L,M,\mathcal{K}(N)})((a,b),A) . \end{align*}

4. (iv) For any $(a, \mathcal{A})\in L\times \mathcal{K}(\mathcal{K}(M))$ ,

   \begin{align*} (\mu_{L\times M}\circ \mathcal{K}t''_{L,M}\circ t''_{L,\mathcal{K}(M)})(a, \mathcal{A}) = (t''_{L,M}\circ {\mathrm{id}}_{L}\times \mu_{M})(a, \mathcal{A}). \end{align*}

From the construction of the $s_{L,M}$ above, we can obtain the following result.

Proof. For any $(A, B)\in \mathcal{K}(L)\times \mathcal{K}(M)$ , we define the map $t''^{B}_{L,M}: L\rightarrow \mathcal{K}(L\times M)$ :

\begin{align*} \forall x\in L, t''^{B}_{L,M}(x) = t''_{L,M}(x, B). \end{align*}

By the proof of Proposition 6.3, we know $t''^{B}_{L,M}$ is well-defined and Scott-continuous. Thus, $\overline{t''^{B}_{L,M}(A)} = \overline{\{\downarrow\!\!x\times B: x\in A\}}\in\mathcal{K}(\mathcal{K}(L\times M))$ . It follows that $A\times B = \mu(\overline{\{\downarrow\!\!x\times B: x\in A\}})\in \mathcal{K}(L\times M)$ .

For dcpos L and M, we calculate the cotensorial strength $t'_{L, M}: \mathcal{K}(L)\times M\rightarrow \mathcal{K}(L\times M)$ as $\ t'_{L,M}((C, x)) = C\times {\downarrow}x $ for any $(C, x)\in \mathcal{K}(L)\times M$ . Following the standard categorical terminology (see Kock Reference Kock1970), the strong monad $(\mathcal{K}, \eta, \mu)$ on $\mathbf{DCPO}$ is commutative if $\psi$ and $\widetilde{\psi}$ agree, where $\psi, \widetilde{\psi}: \mathcal{K}(L)\times \mathcal{K}(M) \rightarrow \mathcal{K}(L\times M)$ are defined as follows:

\begin{align*} \psi_{L,M} = \mu_{L\times M}\circ \mathcal{K}t''_{L,M}\circ t'_{L,\mathcal{K}(M)}\\ \widetilde{\psi}_{L,M} = \mu_{L\times M}\circ \mathcal{K}t'_{L,M}\circ t''_{\mathcal{K}(L),M}. \end{align*}

It is folklore that computationally, strongness of a monad together with adequacy of the corresponding denotational semantics can be used to establish contextual equivalences for effectful programs (Moggi Reference Moggi1991; Plotkin and Power Reference Plotkin and Power2001). Particularly, if $\mathcal K$ is commutative, it will carry the structure $\psi$ making $(\mathcal K, \eta, \mu)$ into a symmetric monoidal monad on DCPO. With this stronger commutative property, we would know that it does not matter which order two instances of the effect appear in programs.

Proof. We just need to prove that for any dcpos L, M and $(A, B)\in \mathcal{K}(L)\times \mathcal{K}(M)$ , $ (\mu_{L\times M}\circ \mathcal{K}t''_{L,M}\circ t'_{L,\mathcal{K}(M)}) (A, B) = (\mu_{L\times M}\circ \mathcal{K}t'_{L,M}\circ t''_{\mathcal{K}(L),M}) (A,B)$ , that is,

\begin{align*} \bigcup{\overline{\{\downarrow\!\!a\times B': a\in A, B'\subseteq B\}}} = \bigcup{\overline{\{A'\times {\downarrow}b: A'\subseteq A, b\in B\}}}. \end{align*}

For convenience, let lhs denote the left hand side of the equation and rhs the right hand side. Consider each $a\in A$ and $B'\subseteq B$ . For any $b'\in B'$ , $b'\in B$ . Then, ${\downarrow}a\times {\downarrow}b'\in \{A'\times {\downarrow}b: A'\subseteq A, b\in B\}$ , which implies ${\downarrow}a\times {\downarrow}b'\subseteq rhs$ . It follows that ${\downarrow}a\times B' = \bigcup\{{\downarrow}a\times {\downarrow}b': b'\in B'\}\subseteq rhs$ , in other words, ${\downarrow}a\times B'\in {\downarrow} rhs$ . Then $\overline{\{\downarrow\!\!a\times B': a\in A, B'\subseteq B\}} \subseteq {\downarrow} rhs$ , which means $lhs\subseteq rhs$ . $rhs\subseteq lhs$ can be proved similarly.

We conclude our paper with a brief discussion of the advantage of semantic applications of $\mathcal K$ . We have proved that each $\mathcal K$ is a strong monad over the category DCPO, this gives rise to the structures of $\lambda_c$ -models considered by Moggi (Reference Moggi1989).

Definition 6.7. (Moggi Reference Moggi1989) A $\lambda_c$ -model over a category C with finite products is a strong monad $(\mathcal T, \eta, \mu)$ together with a $\mathcal T$ -exponential for every pair (A, B) of objects in C, i.e., a pair

\begin{align*} \langle {(\mathcal TB)}^A, eval_{A, \mathcal TB}: {(\mathcal TB)}^A\times A\rightarrow \mathcal TB\rangle \end{align*}

satisfying the universal property that for any object C and $f: C\times A\rightarrow \mathcal TB$ , there exists a unique $h: C\rightarrow {(\mathcal TB)}^A$ , denoted by $\Lambda_{A, \mathcal T B}(f)$ s.t.

\begin{align*} f = eval_{A, \mathcal TB}\circ (\Lambda_{A, \mathcal T B}(f)\times Id_A).\end{align*}

From the above definition, we could clearly see that each strong monad on a Cartesian closed category is a $\lambda_c$ -model. So each order-K-ification monad $\mathcal K$ is a $\lambda_c$ -model by its strongness and the Cartesian closedness of $\mathbf{DCPO}$ .

Moggi gave the interpretation of a formal system called $\lambda_c$ -calculus in a $\lambda_c$ -model, which is sound and complete with respect to the interpretation. In our case, $\lambda_c$ -calculus is interpreted as morphisms of the Kleisli category for $\mathcal K$ .

Recall that the Hoare power construction $\mathcal H$ on the category DCPO is useful for modeling the angelic non-determinism. In particular, one can verify that $\mathcal H$ is a strong monad on DCPO, so naturally, it is also a $\lambda_c$ -model. As a refinement of $\mathcal H$ , there are fewer morphisms in the Kleisli category of each monad $\mathcal K$ than that of $\mathcal H$ . So some useless semantic trash could be culled if we consider using the Kleisli category of $\mathcal K$ as semantic categories, to provide a more accurate interpretation for the programming language at hand.

We would end this paper with an example to illustrate that there are indeed fewer morphisms in the Kleisli category of the monad $\mathcal K$ for some particular categories of type $\mathrm K$ , than that of $\mathcal H$ .

Example 6.8. Let’s take $\mathcal K$ as the order-sobrification monad $\mathcal S$ , which assigns $(Irr\Gamma(L), \subseteq)$ to each dcpo L, i.e., the poset of all irreducible closed subsets of $\Sigma L$ with the inclusion order. Clearly, $\mathcal S(L)\subseteq \mathcal H(L)$ . We take a concrete dcpo $L = \mathbb Z\cup \{\perp\}$ with the order $x\leq y$ defined as $x = \perp, y\in \mathbb Z$ or $x= y \in \mathbb Z$ , where $\mathbb Z$ is the set of all integers, and a dcpo $M = \{0, 1\}$ with the usual order $0\leq 1$ . In denotational semantics, L is the semantic of the type int and M is that of the type unit. Note that $\Gamma(L) = \{{\downarrow}A: A\subseteq \mathbb Z\}\cup\{\bot\}$ , $Irr\Gamma(L) = \{{\downarrow}n: n\in \mathbb Z\}\cup\{\bot\}$ . We define a map $f: M\rightarrow \mathcal H(L)$ as

\begin{align*} f(m) =\begin{cases} \bot,& \mathrm{m = 0}\\ {\downarrow}\{1,2\},& \mathrm{m = 1}.\end{cases} \end{align*}

The map f is Scott-continuous since it is monotone and M is finite; hence, it is in $\mathbf{DCPO}_{\mathcal H}$ . However, there is no Scott-continuous map $g: M\to \mathcal S(L)$ in $\mathbf{DCPO}_{\mathcal S}$ such that $f = i\circ g$ , where $i: \mathcal S(L)\rightarrow\mathcal H(L)$ is the canonical inclusion map. This is obvious because the images of $i\circ g$ consist of principal ideals, while f(1) is not a principal ideal.