54 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
The universal characterisation of a free algebra guarantees that all these free algebra constructions yield essentially the
same result on dcpos. In this paperwe investigate the underlying concepts and compare the three approaches beyond dcpos.
In [12], Lawson and Keimel already remark that the semitopological construction generalises Jung, Moshier and Vicker’s
order-theoretic approach. We prove this claim. Furthermore, we show that the notion of algebras in the dcpo-presentation
framework and the compactly-generated framework are compatible. However, we also outlinewhy it is hard to compare the
corresponding free algebra constructions in those frameworks. Finally, we give an example showing that the constructions
of free semitopological algebras and free compactly generated algebras differ outside the world of dcpos.
1.1. Overview
We begin our investigation by fixing our notation and presenting the abstract existence result about free algebras in
domain theory in Section 2. In Sections 3, 4 and 5 we then successively present the free algebra constructions via dcpo-
presentations, semitopological algebras and compactly generated spaces. In each of these sectionswe first recall the relevant
preliminaries and then outline the free algebra construction and the reason why these frameworks yield constructions for
dcpos. In Section 6 we compare the approaches. We first investigate dcpo-presentations from a topological viewpoint,
showing that they faithfully fit into the semitopological setting, and, subsequently, explain the problems of comparing
dcpo-presentations with the compactly generated framework. Finally, we give an example showing that outside dcpos the
free algebra construction of the semitopological and compactly generated frameworks differ. We conclude the paper in
Section 7.
2. Free algebras in the category of dcpos
2.1. Notation
Thecategoryofpreordered sets andmonotonemapsbetween themisdenotedbyPreorder, its full subcategoryofpartially
ordered sets (henceforth simply posets) by Poset. The category of topological spaces and continuous maps is denoted by
Top.
The Scott topology τ≤Xon a poset (X, ≤X) is given by those upward closed subsetsU ⊆ X such that for any directedD ⊆ X
whose supremum∨↑ D exists and is an element of U, it holds that D∩U �=∅. Equivalently the Scott topology can be defined
by its closed sets which are those downward closed subsets C ⊆ X which are closed under all existing directed suprema.
A map f : (X, ≤X)→(Y, ≤Y ) between posets is Scott-continuous if it is continuous with respect to the corresponding Scott
topologies, or equivalently if it preserves all existing directed suprema: f (∨↑ D) = ∨↑ f (D). Notice that Scott-continuity
entails monotonicity, because any totally ordered two-element set is directed.
A dcpo is a directed complete partially ordered set, and the category of dcpos and Scott-continuous maps between them
is denoted by DCPO. Notice that DCPO appears as a non-full subcategory of Poset as well as a full subcategory of Top, when
the objects are equipped with the Scott topology. We will take both views of DCPO in this paper, the order-theoretic view
mainly in Section 3, and the topological view thereafter. When we talk about a dcpo as a topological space we assume it to
carry the Scott topology, unless otherwise stated.
Recall that the specialization preorder τ on a topological space (X, τ ) is defined as x τ y if for all U ∈ τ it holds that
x ∈ U implies y ∈ U. The specialization preorder becomes a partial order if and only if (X, τ ) is a T0-space. A monotone
convergence space is a topological space (X, τ ) such that (X, τ ) is a dcpo and every open set U ∈ τ is Scott-open with
respect toτ . It follows that any dcpo under its Scott topology is a monotone convergence space. The category of monotone
convergence spaces and continuous maps between them is denoted by Mon. In fact it is a full reflective subcategory of
Top. The reflection functor, which we denote by M : Top → Mon, has been introduced by Wyler [20] as the so-called
d-completion. Since most recent texts refer to it as themonotone convergence reflection, so do we.
All these categories introduced above are well-known to be complete and cocomplete, so in particular they have finite
products. Below, we will usually identify a poset, dcpo or topological space simply with its underlying set whenever no
ambiguity can arise, i.e., we may write simply X for the poset (X, ≤X) and so on. However, we will have to be more careful
and use some more cumbersome notation when we compare the different frameworks in Section 6.
2.2. Free dcpo-algebras
The subject of this paper is to compare approaches towards free algebra constructions for inequational algebraic theories
in domain theory. Such free algebra constructions are important for denotational semantics of programming languages since
they provide an elegant way to model many interesting computational effects, see e.g., [17,18]. For instance the classical
powerdomain constructions can be given as free algebras for inequational theories, see Section 6 of [1].
It is well-known that free algebra constructions exist in domain theory, for abstract reasons. We outline the proof below.
However, the abstract result does not give a concrete construction of free algebras. In other words, one cannot construct the
free algebra over a dcpo X simply from the algebraic theory and X itself. We give a concrete example of this problem at the
end of this section.
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 55
Remark. Before we continue our development, let us shortly pause to remark that our viewpoint on domain theory is most
general in this paper, i.e., every dcpo is a domain and therefore domain theory is the study of the category DCPO. Thus,
we are considering free algebra constructions on DCPO. More restricted settings such as continuous or algebraic dcpos and
subcategories thereof have been examined in Section 6 of [1]. The interested reader should also read Koslowski’s note [13],
where it is shown that the free algebra construction in DCPO preserves continuous dcpos. In contrast to free dcpo-algebras,
free algebra constructions in the restricted setting of continuous dcpos can be characterised concretely, via the notion of an
abstract basis, see [1] for details.
Let us recall the definition of an inequational algebraic theory for dcpos. In the subsequent sections this definition will be
applied to more general settings in order to embed DCPO into a framework where it is closed under the corresponding free
algebra construction. However, all of these generalised settings are given by order-enriched categories to which the basic
concept described in the following is easily transferred.
An algebraic signature � is a set of operations symbols {σ ∈ �} each of which has some arity |σ | ∈ N. A dcpo-�-algebra
is a dcpo (X, ≤X) together with, for each σ ∈ �, a Scott-continuous map:
σX : X|σ | → X
Formally, we write (X, ≤X, {σX}σ∈�) for a dcpo-�-algebra, but for simplicity we may also write (X, {σX}).A �-homomorphism between dcpo-�-algebras (X, {σX}) and (Y, {σY }) is a Scott-continuous map f : X → Y such that
for every σ ∈ � the following diagram commutes:
We denote the category of dcpo-�-algebras and homomorphisms between them by DCPO� .
Let {v0, v1, v2, . . . } be a fixed countable set of variables. The set of �-terms is defined inductively using variables and
operations symbols: every variable vi is a �-term, and if t1, . . . t|σ | are �-terms and σ ∈ � is an operation symbol, then
σ(t1, . . . , t|σ |) is a �-term. Each �-term t has an arity |t| ∈ N given by the (number of) distinct variables used in it. A
(�)-inequation is given by a pair of�-terms (t, t′) of the same arity, written as t ≤ t′. We also allow adding dummy-variables
to terms in order to adjust its arity. This comes fromallowing projections of a cartesian product structure and formally results
in a different term, but it allows us to form inequations like σ(v0, v1) ≤ v0 (with v1 being a dummy-variable in the term
v0).
Any �-term t can be interpreted in a dcpo-�-algebra (X, {σX}) as a Scott-continuous map tX : X|t| → X by substituting
the operation symbols and projections by their respective counterparts on X .
Avariable instantiation inadcpo-�-algebra (X, {σX}) is anassignment fromthesetofvariables toelementsx0, x1, x2, · · · ∈X . Observe that every variable instantiation can be extended to a unique term instantiation by propagating it through the
inductive term constructions. The inequation t ≤ t′ is said to be satisfied in (X, {σX}) if for every variable instantiation,
it holds that tX(x0, . . . , x|t|) ≤X t′X(x0, . . . , x|t|). This is equivalent to saying that tX ≤ t′X in the pointwise order for the
corresponding term-maps on (X, {σX}).If I is a set of inequations for �, then the tuple (�, I) is called an inequational theory for DCPO. A dcpo-(�, I)-algebra is
a dcpo-�-algebra (X, {σX})which satisfies all inequations in I . The category of dcpo-(�, I)-algebras and homomorphisms
between them is denoted by DCPO(�,I). Observe that we have DCPO(�,∅) ≡ DCPO� .
For a given inequational theory (�, I) forDCPO andagivendcpoX , the free (�, I)-algebraoverX is a dcpo-(�, I)-algebra(A, {σA}) together with a Scott-continuous inclusion map η : X → A such that for every dcpo-(�, I)-algebra (B, {σB}),every Scott-continuous f : X → B can be uniquely extended to a�-homomorphism f : (A, {σA}) → (B, {σB}) along η as in:
A free algebra for an algebraic theory with no inequations, i.e., I = ∅, is called absolutely free.
Notice that a free (�, I)-algebra exists over any dcpo X if and only if the forgetful functor DCPO(�,I) → DCPO has a left
adjoint. That this is indeed the case can be shown via Freyd’s Adjoint Functor Theorem, see [1]. The category DCPO(�,I) iscomplete with limits computed as in DCPO and a solution set can be constructed for any given dcpo X by considering that
56 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
Fig. 1. The running example.
the cardinality of the algebraic image of X in any dcpo-(�, I)-algebra is bounded by 2|T(X)|, where T(X) denotes the set of
all �-terms with variables taken from X . Since the isomorphism classes of dcpo-(�, I)-algebras with cardinality ≤ 2|T(X)|form a set (and not a proper class), the solution set condition is satisfied.
Theorem 2.1. For any inequational theory (�, I), the forgetful functor DCPO(�,I) → DCPO has a left adjoint. Thus, for each
dcpo X the free dcpo-(�, I)-algebra over X exists.
As we have mentioned, this result does not give any insight on the structure of the free algebra over a given dcpo – it
does not give rise to a concrete construction. However, such concrete constructions are possible in the category of posets
and preorders, into which DCPO naturally embeds. Consider that the definition of algebras for inequational theories readily
generalises to posets and preorders, where the operations are required to bemonotone instead of Scott-continuous. In these
settings the free algebras can be constructed straightforwardly, similar to the classical set-theoretic case, by first constructing
all terms for the signature over the underlying poset and then refining the order to make all operations monotone and all
inequations satisfied. However, even if we disregard the problem of monotonicity versus Scott-continuity in the operations,
we run into problems if we want to extend such a construction to dcpos, because the resulting free poset algebra over a
given dcpo may not be directed complete and it is not clear how to complete it without having additional information. To
make this problem precise consider the following example.
Example 1. Suppose the inequational theory (�, I) consists of a single unary operation symbol σ and the single inequation
v ≤ σ(v). Suppose furthermore that we want to construct the free poset-(�, I)-algebra over the two-element discrete
poset 2 = ({a, b}, =), which naturally is a dcpo. The resulting poset-(�, I)-algebra (P, {σP}) consists of terms of the form
σ n(a) and σ n(b) for n ∈ N, the operations are defined as σP(σn(a)) := σ n+1(a) (and similarly for b) and satisfies the
inequations σ n(a) ≤ σm(a) for n ≤ m (again similarly for b). It is sketched in Fig. 1(a).
Of course, this poset is not a dcpo, because the directed sets {σ n(a)}n∈N and {σ n(b)}n∈N do not have suprema. There are
two obvious dcpo-completions of P, sketched in Fig. 1 (b) and (c), and the algebraic structure extends to both of them, by
defining in P1, σP1(∞) = ∞, and in P2, σP2(∞a) = ∞a and σP2(∞b) = ∞b. In fact the free dcpo-(�, I)-algebra in this
case is given by (P2, {σP2}). However, this cannot be concluded from the term construction process and the given algebraic
theory. It can only be seen by the fact that we have a Scott-continuous homomorphism (P2, {σP2}) → (P1, {σP1}), extendingthe inclusion map, but not vice-versa.
This problem of giving a concrete free algebra construction for dcpos has been solved recently in three different ap-
proaches. The principle idea of each approach is to use some additional structure, intrinsic to dcpos,which can be propagated
through the term construction process and carries sufficient information in order to obtain the “right” completion in the end.
3. Preorder presentations of dcpos
In this section we outline the free algebra construction using dcpo-presentations as given by Jung, Moshier and Vickers
in [10]. The underlying idea stems from locale theory, where a locale (or frame) is presented by a semilattice together with
a cover relation, see [9] II.2.11. This idea can be transferred to dcpos, by presenting them via a similar cover relation on a
preorder. Then the free algebra is constructed in the preorder setting, but with the cover relation propagated through the
term construction process. Moshier, Jung and Vickers argue that this is a natural generalisation of the construction of free
algebras on continuous dcpos via an abstract basis.
3.1. Preliminaries
Definition 3.1. A dcpo-presentation consists of
• a set P of generators;• a preorder ≤ on P;• a relation � between elements of P and directed subsets of P, written a � D. Whenever a � D, D is said to cover a.
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 57
We remark that the notation in [10] differs slightly from ours, in that they take a set C of covers as primitive and the
relation as derived. However, it is easy to see that their definition and ours are equivalent. We shall also use the name cover
related preorders for dcpo-presentations to emphasize that we are primarily interested in the structure at hand instead of
the dcpo that is presented. Intuitively a � Dmeans that a ≤ ∨↑ D if the supremum existed. This intuition also shows how a
dcpo can be viewed as a cover related preorder.
For simplicity we usually write (P, �) instead of (P, ≤, �) for a cover related preorder. Let us define a suitable class of
maps between cover related preorders.
Definition 3.2. Let (P, �) and (P′, �′) be cover related preorders. Then a cover preserving map (P, �) → (P′, �′) is given by
a monotone map f : P → P′ such that a � D implies f (a) �′ f (D).
One of the main results of [10] is that cover related preorders can indeed be used to characterise dcpos. This is done as
follows. Given a cover related preorder (P, �), define a C-ideal to be a downward closed subset I of P so that for any a � D,
D⊆I implies a∈I. Back to our intuition this means that whenever D⊆I then any a≤ ∨↑ D should also be contained in I, i.e.,
I corresponds to a Scott-closed subset of P. We write C − Idl(P, �) for the poset of C-ideals ordered by subset inclusion. It
is easy to see that C − Idl(P, �) is closed under arbitrary intersections in the powerset lattice P(P). Hence, it becomes a
complete lattice itself. In fact, one can assign to an arbitrary subset S ⊆ P the smallest ideal 〈S〉 containing S and thismap is a
closure operator on P(P). Jung, Moshier and Vickers have shown that C−Idl(P, �) is the free sup-lattice generated by (P, �).Notice that the inclusion map i : (P, �) → C − Idl(P, �), given by a �→ 〈{a}〉, is monotone and preserves covers, in the
sense that whenever a � D, we get that a ∈ 〈D〉 because of D ⊆ 〈D〉, and hence i(a) = 〈{a}〉 ⊆ 〈D〉. Moreover, it is easy to
check that for any D ⊆ P one gets 〈D〉 = ∨x∈D i(x).
Now let (P, �) be the smallest sub-dcpo of C − Idl(P, �) which contains i(P). Then Jung, Moshier and Vickers get the
following:
Theorem 3.3. (P, �) is the free dcpo generated by (P, �), i.e., whenever there exists a monotone map f : P → D into a dcpo D
such that a � D implies f (a) ≤ ∨↑x∈D f (x), then there exists a unique Scott-continuous map f : (P, �) → D which makes the
following diagram commute:
Thus, it makes sense to say that (P, �) is presented by (P, �).Notice that a dcpo may be presented by different cover related preorders, and this concerns both, the involved preorder
and the cover relation. Consider for instance the containment cover relation �, given by ∈. Then (P, �) is isomorphic to the
“usual” ideal completion, Idl(P), i.e., the poset consisting of downward closed, directed subsets of P ordered by inclusion,
and it is well known that there exist posets P � P′ with Idl(P) ∼= Idl(P′). Also, if the cover relation is given by {x�{x}|x ∈ P},then (P, �) ∼= Idl(P), hence the presented dcpo is the same as with the containment relation.
Considering the presenting preorder in (P, �) we cannot expect to have uniqueness for a given dcpo, and we do not
even desire it. In fact, it is quite natural to have a variety of different preorders (or posets) which present the same dcpo
(intuitively they can be completed to the same dcpo), e.g., in [10] an inductive construction of (P, �) is considered and it is
natural to demand each step to correspond to a presentation of (P, �). However, it seems less natural for a given preorder P
to have different cover relations which lead to the same dcpo. If one considers cover-preserving maps as a natural definition
of morphism between cover related preorders this means that the identity map (P, �) → (P, �) need not be a morphism
although the same dcpo is presented. This was also a problem for the authors of op. cit., and we use their suggested solution
to remedy the situation.
Definition 3.4. A cover relation � on a preorder P is called saturated if all of the following hold:
1. ∀x ∈ P.x � {x}2. ((x′ ≤ x � D) ∧ (↓ D ⊆↓ D′)) ⇒ (x′ � D′)3. ((x � D) ∧ (∀y ∈ D.y � E)) ⇒ x � E.
We call a cover related preorder (P, �) with saturated cover relation a saturated cover related preorder.
Let us remark that every cover relation on a preorder can be extended to a saturated one, along the lines of Definition 5.2
of [10]. In fact, one can gradually close � first by condition (1), then by condition (2) and finally by condition (3) to obtain the
58 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
saturation of �. Notice that our definition differs from the one given in op. cit. in that we added condition (1). This ensuresthe uniqueness of a saturated cover related preorder which we will show next.
Theorem 3.5. Let (P, �) and (P, �) be saturated cover related preorders. Then they present the same dcpo if and only if � =�.
For the proof we will use the following lemmas.
Lemma 3.6. Let (P, �) be a saturated cover related preorder. Define T : P(P) → P(P) by:
T(D) := {x ∈ P| ∃E ⊆↓ D.x � E}.Then the following hold:
(i) T(D) =↓ T(D),(ii) D ⊆ T(D),(iii) if D is directed then T2(D) ⊆ T(D).
Proof. For (i), assume that x ≤ x′ ∈ T(D). Thus, there exists a directed E ⊆ D with x � E. By Definition 3.4 (2), it follows
that x′ � E, hence x′ ∈ T(D), and so ↓ T(D) ⊆ T(D).For (ii), observe that for any x ∈ D, we have x � {x} ⊆ D by Definition 3.4 (1), hence x ∈ T(D).Finally for (iii), assume D is directed and x ∈ T2(D). Then there exists a directed E ⊆ T(D) such that x � E. But E ⊆ {z ∈
P|∃G ⊆↓ D.z�G}, and by Definition 3.4 (2) this is equivalent to E ⊆ {z ∈ P|z� ↓ D}. It follows that x�E and∀z ∈ E.z� ↓ D,
hence by Definition 3.4 (3), we get x� ↓ D, showing that x ∈ T(D). �
Lemma 3.7. If (P, �) is a saturated cover related preorder, then a � D holds if and only if i(a) ≤ ∨↑x∈D i(x) in (P, �).
Proof. That a � D implies i(a) ≤ ∨↑x∈D i(x) follows directly from the fact that the inclusion i : (P, �) → (P, �) preserves
covers, as observed before Theorem 3.3. For the converse assume that a � D for some directed D. We claim that T(D) as
defined in the previous lemma forms a C-ideal. This is seen as follows: by the first part of the previous lemma, T(D) is
downward closed. So suppose E ⊆ T(D) is directed and y � E. Then we get y ∈ T2(D) and, by the third part of previous
lemma, T2(D) = T(D), showing that T(D) is indeed a C-ideal. In fact it follows that T(D) = 〈D〉. But then a � D yields
a /∈ T(D), because � is saturated. Hence, i(a) � T(D) = 〈D〉, as required. �
Now we can prove Theorem 3.5.
Proof. Clearly if�=�, then (P, �)=(P, �). So supposewithout loss of generality thata � Dbuta � D. Let i:(P, �)→(P, �)
and i∗ : (P, �) → (P, �) denote the inclusion maps. By the previous lemma we have that i(a) ≤ ∨↑x∈D i(x) but i∗(a) �∨↑
x∈D i∗(x). Hence, the following diagram cannot commute for any Scott-continuous map i∗:
showing that (P, �) �= (P, �). �
Thus, we have solved the problem of different cover relations on the same preorder giving rise to the same dcpo, and
henceforth we shall assume all our cover relations to be saturated.
Definition 3.8. The category CRP is given by cover preserving maps between saturated cover relating preorders.
3.2. Free preorder presentation algebras
Now we explain how Jung, Moshier and Vickers use the above concept of cover related preorders to characterise free
dcpo-(�, I)-algebras.As we have mentioned before, the free (�, I)-algebra in the category of preorders and monotone maps over a given
preorder P can be constructed in an inductive way: first one constructs the set T (P) of all �-terms over the elements of P,
then one considers the order relation� obtained as the transitive closure of the join of the original order on P lifted through
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 59
all operations with the order imposed by the inequations also lifted through all operations. Then the free preorder-(�, I)-algebra over P is (T (P), �, {σT (P)}σ∈�).
Jung, Moshier and Vickers have shown that the free preorder algebra construction naturally extends to the category CRP.
Here (�, I)-algebras are defined as follows.
Definition 3.9. A crp-(�, I)-algebra is given by a tuple (P, �, {σP}σ∈�) such that
• (P, �) is a saturated cover related preorder,• for each σ ∈ �, σP : P|σ | → P is a monotone map which is cover preserving in each argument separately,• all inequations are satisfied in the underlying preorder algebra (P, {σP}σ∈�).
A homomorphism between crp-(�, I)-algebras is a homomorphism between the underlying preorder-(�, I)-algebraswhich is cover preserving.
The category CRP(�,I) is the category of crp-(�, I)-algebras and homomorphisms between them.
One of the main results of [10] is then the following.
Theorem 3.10. For every saturated cover related preorder (X, �X) the free crp-(�, I)-algebra (A, �A, {σA}σ∈�) over (X, �X)exists and it is characterised as follows:
(i) (A, {σA}σ∈�) is the free preorder-(�, I)-algebra over X,
(ii) �A is the coarsest compatible saturated cover relation, i.e., the coarsest saturated cover relation on A making the inclusion
map η : X → A cover preserving and all operations σA : A|σ | → A cover preserving in each argument separately
Proof. The characterisation above is well-defined, yields a crp-(�, I)-algebra (A, �A, {σA}σ∈�) and a cover preserving
inclusion map η : X → A, so only the universal property of the free algebra needs to be verified. Suppose (B, �B, {σB}σ∈�)is a crp-(�, I)-algebra and f : X → B is monotone and cover preserving. Since (A, {σA}) is the free preorder algebra over
X , there exists a unique monotone homomorphism f : (A, {σA}) → (B, {σB}) extending f along η, and it remains to show
that f is cover preserving. But this is easily seen from the fact that the cover relation � on A defined as a � D whenever
f (a) �B f (D) is saturated and fulfills the compatibility criteria of (ii). Since �A as defined above must have fewer covers than
� all of them are preserved by f . �
This yields the following concrete construction of the free dcpo-(�, I)-algebra over a given dcpo (X, ≤X). As a dcpo,
(X, ≤X) carries the intrinsic cover relation �X given by x �X D if and only if x ≤X
∨↑ D. Of course, the resulting cover related
preorder (X, �X) is a dcpo-presentation of its underlying dcpo. The above theoremyields an concrete construction of the free
crp-(�, I)-algebra (A, �A, {σA}σ∈�) over (X, �X) which has an underlying dcpo-presentation of (A, �A). Every operation
σA : A|σ | → A can be extended to a Scott-continuous operation σ(A,�A): (A, �A)
|σ | → (A, �A) which follows essentially
from a parametrised version of Theorem 3.3. The same theorem shows that the resulting dcpo-(�, I)-algebra fulfills the
universal property of a free algebra.
Let us sketch this construction on our Example 1. The cover relation on 2 is given by a � {a} and b � {b}. Thus, the cover
relation � on its free preorder algebra P (of Fig. 1(a)) is given by σ n(a) � D if and only if there exists some m ≥ n with
σm(a) ∈ D (and similarly for b). This is so, because the so-defined � is saturated and compatible in the sense of Theorem
3.10 (ii), and one can show by induction on n that each of this coverings must hold in order for σ to be cover preserving.
Thus, it is the coarsest compatible cover relation on P and so (P, �, {σP}) is the free crp-(�, I)-algebra over 2. It is not hard
to see that the dcpo-reflection (P, �) is P2, cf. Fig. 1(c), which is indeed the underlying dcpo of the free dcpo-(�, I)-algebraover 2. Thus, the cover relation approach allows us construct the free algebra concretely, i.e., just given the data of the dcpo
2 and the inequational theory (�, I).
Remark. We could have developed the above results as well in the realm of posets instead of preorders. In that case one
simply has to factor by the equality relation induced by the preorder in every step. The straightforward translation of the
results to posets is left to the inclined reader.
4. Semitopological algebras
Next we sketch the approach of Keimel and Lawson [12] who characterise the free algebra over a dcpo as the monotone
convergence reflection of the free semitopological algebra over it. From now on we will consider a dcpo (X, ≤X) as a
topological space, i.e., equipped with the Scott topology.
60 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
4.1. Preliminaries
The category DCPO is a full subcategory of Top. However, it is well-known that the embedding DCPO ↪→ Top does
not preserve finite products. Thus, studying dcpo-algebras in a topological framework does not seem to be a very promis-
ing approach at first sight, because a dcpo-algebra (as defined in Section 2) need not be a topological algebra due to the
incompatible product structure. Nevertheless, the following observation is the starting point for the development here.
Lemma 4.1. Let X and Y be dcpos and Z be a monotone convergence space. Then a map f : X × Y → Z is Scott-continuous if
and only if it is separately continuous.
Proof. Lemma II-2.8 of [8] shows the claim for the case that Z carries the Scott topology, as well. This slightly more general
result follows easily from the observation that dcpos form a full coreflective subcategory of Mon. �
This motivates an investigation of algebras where the operations are separately continuous instead of jointly continuous,
they are called semitopological algebras.
Definition 4.2. Let (�, I) be an inequational algebraic theory (like introduced for dcpos in Section 2). Then a semitopological
(�, I)-algebra is a tuple (X, τ, {σX}σ∈�) such that:
• (X, τ ) is a topological space,• for every σ ∈ �, σX : X|σ | → X is a separately continuous map (with respect to τ ),• the specialization preorder τ satisfies all inequations of I .
A homomorphism between semitopological (�, I)-algebras is a continuous map between the underlying spaces which
commutes with the operations accordingly.
Thus, with Lemma 4.1 every dcpo-(�, I)-algebra becomes a semitopological (�, I)-algebra. From the viewpoint of
category theory the fact that the operations need not be morphisms of the ambient category is not really a problem, since
the semitopological algebra structure can simply be seen as a further structure on the underlying topological space. Thus,
we can talk about the category STA(�,I) of semitopological (�, I)-algebras and homomorphisms between them. Then
DCPO(�,I) becomes a full subcategory of STA(�,I) and both categories live faithfully in the universe of topology since the
homomorphisms are simply required to be continuous maps.
4.2. Free semitopological algebras and completions
Now let us see how free dcpo-(�, I)-algebras can be constructed via free semitopological algebras. First, we show the
existence of this construction and give a characterisation of it, analogously to Theorem 3.10 above.
Theorem 4.3. Let (�, I) be an inequational algebraic theory. Then for any topological space X, the free semitopological (�, I)-algebra (A, τA, {σA}σ∈�) over X exists and it can be characterised as follows:
(i) with the specialization preorder, the underlying preorder algebra (A, τA , {σA}σ∈�) is the free preorder-(�, I)-algebraover X,
(ii) the topology τA is the finest compatible topology, i.e., the finest topology on Amaking the inclusionmap X ↪→ A continuous,
all operations σA : A|σ | → A separately continuous, and for which the order on the free preorder-(�, I)-algebra is the
specialization order.
Proof. The proof proceeds exactly as the one for Theorem 3.10 above. One shows that the so-characterised (A, τA, {σA}σ∈�)fulfills the universal property of a free algebra by showing that for any semitopological (�, I)-algebra (B, τB, {σB}σ∈�) and
continuous f : X → B, the topology on A obtained by f−1
(τB) (where f is the unique preorder�-homomorphism extending
f ) is a compatible topology in the sense of (ii). �
Remark. Notice that this theorem yields a way to concretely construct the free algebra from the underlying space X alone.
One first constructs the free preorder (�, I)-algebra over the specialization preorder on X , and then equips this set with the
finest compatible topology in the sense of (ii). Of course, to find the finest such topology is in general not a computationally
feasible process. In particular, it is not an inductive process. Nevertheless, abstractly it is possible with the given data.
The condition on the specialization order in (ii) is necessary for the inequations to be satisfied by (A, τA, {σA}σ∈�). Forinstance, consider an inequational theory with two constants c1, c2 and the inequation c1 ≤ c2. Without the condition on
the specialization preorder, the singletons {c1} and {c2} would be open in every free algebra.
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 61
If one starts with a dcpo X (equipped with the Scott topology), then of course the free semitopological (�, I)-algebraover X will not be a dcpo in general, because the specialization order need not be directed complete, as e.g., in Example 1.
Nevertheless, Keimel and Lawson [12] have shown that the algebraic structure can be extended to themonotone convergence
reflection of the free (�, I)-algebra over X , and that this yields the free dcpo-(�, I)-algebra over X .
Theorem 4.4. Let (A, τA, {σA}σ∈�) be a semitopological (�, I)-algebra. Then all operations can be extended to separately con-
tinuousmaps on itsmonotone convergence reflection such that the resulting semitopological algebra (M(A), τM(A), {σM(A)}σ∈�)satisfies all inequations of I .
Moreover if X is a dcpo with its Scott topology and (A, τA, {σA}σ∈�) is the free semitopological (�, I)-algebra over X, then
(M(A), τM(A), {σM(A)}σ∈�) carries the Scott topology and hence is the free dcpo-(�, I)-algebra (M(A), τM(A), {σM(A)}σ∈�)over X.
To prove this result is quite nontrivial due to the separate continuity of the operations. First of all, one cannot simply
use the fact that the reflection functor M : Top → Mon preserves finite products for showing that the operations extend
to the monotone convergence reflection. Secondly, and more seriously, the separate continuity of operations has the effect
that term maps, as defined in Section 1, may not be separately continuous. This happens particularly in the case that the
operations are not linear, i.e., use a variable more than once. Keimel and Lawson give examples of term maps which are not
separately continuous in [12]. This observation leads to the problem of nonlinear inequations, i.e., inequations inwhich both
terms are nonlinear in the same variable, whose satisfaction cannot be extended to certain completions of a topological
space. Only the special nature of themonotone convergence reflection allows for them to be extended through it, and so the
monotone convergence reflection of a semitopological algebra satisfies the same inequations as the original one.
Keimel and Lawson argue that Theorem 4.4 is a genuine extension of Theorem 3.10 in that the cover related preorders can
be equipped with a topology such that the completion process is equivalent to the monotone convergence reflection. This
carries through the free algebra construction as described above. We will make this claim very precise in Section 6 below.
Let us finish this section with explaining how Theorem 4.4 is applied to our running Example 1. The free semitopological
(�, I)-algebra over2has asunderlying setP fromFig. 1(a)with the topology τ generatedbyopen sets of the form {σ k(a)|k ≥n} and {σ k(b)| k ≥ n}. This topology is clearly compatible in the sense of (ii) and any set B ⊆ P which is not open in this
topology has the property that
∀k ∃n ≥ k (σ n(a) /∈ B) and ∀k ∃n ≥ k (σ n(b) /∈ B)
and thus the specialization order for a topology containing B cannot satisfy the required inequation. But the monotone
convergence reflection M((P, τ )) is the dcpo P2, from Fig. 1(c), with the Scott topology.
5. Compactly-generated algebras
Nowwe turn towards the third approachof constructing free dcpo-algebras, proposedbyBattenfeld in [3]. Again it is given
by a topological framework, but this time one works in the category of compactly generated spaces, also called k-spaces,
and continuous maps.
5.1. Preliminaries
Let us recall the basic definition of k-spaces.
Definition 5.1. A topological space is a compactly generated space (or k-space) if it can be obtained as a topological quotient
of a locally compact Hausdorff space. The category of continuous maps between k-spaces is denoted by kTop.
We will use the notation k-space rather than compactly generated space in the subsequent development to avoid the
very clumsy notation in the context of algebras.
The category kTop is well-known to be a full coreflective subcategory of Top and has a rich structure. Furthermore, in [7]
it is shown that all dcpos with their Scott topology are k-spaces.
Theorem 5.2. The category kTop is cartesian-closed, complete and cocomplete, and contains all dcpos equipped with the Scott
topology.
We remark that limits in kTop may differ from limits in Top. More precisely, the k-product (i.e., the cartesian product
structure in kTop) of two k-spaces is obtained by coreflecting the topological product of those spaces, and the subspace
construction works similarly. Of course, this fact implies that kTop-algebras for a given signature need not be topological
algebras, and also that the free algebra constructions in these categories differ. Evenmore is known, for equational algebraic
theories, the free k-algebra construction is not simply given by the coreflection of the corresponding free topological algebra,
see [14] for an example.
62 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
Again we give special attention to the monotone convergence spaces in this setting, for they form the category in which
dcpo-algebras can be constructed.
Definition 5.3. A k-predomain (or compactly generated predomain) is a k-spacewhich is also amonotone convergence space.
The category of continuous maps between k-predomains is denoted by kP.
Notice that the monotone convergence reflection can be restricted to k-spaces, since it preserves the property of being a
k-space, see [4]. Thus, kP becomes a full reflective exponential ideal of kTop and the reflection preserves finite products.
It is not hard to see that every map between k-predomains is necessarily Scott-continuous, see e.g., Lemma 3.1.2 of [6]. It
follows that DCPO is a full coreflective subcategory of kP. A crucial result for us is Proposition 5.1 in [4], which shows that
the inclusion functor DCPO ↪→ kP does preserve finite products, in contrast to the embedding of dcpos into topological
spaces. It does not preserve infinite products in general, which is easily seen by considering a countable product of a discrete
space with more than one element.
Let us summarize the crucial results of our discussion in the following theorem:
Theorem 5.4.
1. The category kP forms a full reflective exponential ideal in kTop with reflection functor M : kTop → kP. The reflection
functor M preserves finite products.
2. The category DCPO is a full coreflective subcategory of kP with coreflection S : kP → DCPO.
3. The inclusion functor DCPO ↪→ kP preserves finite products (and hence so does the inclusion functor DCPO ↪→ kTop).
It follows that kP provides a good framework to study dcpo-(�, I)-algebras, because it has the right product
The notion of algebras for inequational theories is easily adjusted to the compactly-generated framework.
Definition 5.5. Let (�, I) be an inequational algebraic theory. Then a k-(�, I)-algebra is a tuple (X, τ, {σX}σ∈�) such that:
• (X, τ ) is a k-space,• for every σ ∈ �, σX : X|σ | → X is a morphism in kTop (i.e., a continuous map with respect to the k-product topology
on X|σ |),• the specialization preorder τ satisfies all inequations of I .
A homomorphism between k-(�, I)-algebras is a continuous map between the underlying k-spaces which commutes
with the operations accordingly.
If the underlying space of a k-(�, I)-algebra is a k-predomain, it is called a kp-(�, I)-algebra.
We obtain the categories kTop(�,I) and kP(�,I) of k-(�, I)-algebras and homomorphisms between them, respec-
tively its full subcategory of kp-(�, I)-algebras. In [3,6] it is shown that in both cases free algebra functors exist, and that
the free kp-algebras are obtained by completing the corresponding free k-algebras via the monotone convergence reflec-
tion.
Theorem 5.6. The forgetful functors kTop(�,I) → kTop and kP(�,I) → kP both have left adjoints FkTop and FkP . Moreover, it
holds that FkP = M ◦ FkTop.
Remark. The treatment in [3,6] deals with a more general notion of algebraic theories, namely parametrised equational
theories. In a parametrised equational theory, algebraic operations may have a parameter space besides their arity. Inequa-
tional theories as used in the presentwork are recovered by using the Sierpinski space as parameter and introducing suitable
equations. More details on how to do this can be found in op. cit..
Taking closer look at the free algebra constructions in the compactly-generated framework, we get similar results to
Theorems 3.10 and 4.3. The proofs can again be found in [3,6].
Theorem 5.7. Let (�, I) be an inequational algebraic theory and X be a k-space. Let furthermore T (X) denote the set of all
�-terms over X (which carries a set-theoretic �-algebra structure) and ≈ denote the congruence on T (X) generated by the
equations which can be derived from the inequations. Then the following hold:
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 63
• the absolutely free k-�-algebra over X is given by (T (X), {σT (X)}) equipped with the finest compatible topology, i.e., the finest
topologymaking the inclusionmap X ↪→ T (X) and all operations T (X)|σ | → T (X) continuous (of course the product carriesagain the k-product structure, which in general is finer than the product topology),
• the free k-(�, I)-algebra over X is given by (T (X)/ ≈, {σT (X)/≈}) equipped with the finest compatible topology, i.e., it is
compatible in the sense of the previous point and its specialization order satisfies the inequalities. Hence its underlying set
equipped with the specialization order is the free preorder (�, I)-algebra over X.
Thus, one can argue that also this construction is concrete.
Remark. Notice that like in the semitopological setting, we do not get an inductive construction of free k-(�, I)-algebras,since we have to calculate a finest topology, with the additional difficulty of having to work with k-products.
However, in special cases, like countably-based spaces and quotients of them [5], the construction of free k-(�, I)-algebras comes very close to being inductive. Clearly, the construction of terms can be given inductively. Moreover, in [2],
an inductive construction of the topology of the absolutely free topological �-algebra is given, which for a countably-based
space is also the absolutely free k-�-algebra. Finally, if one works with a corresponding parametrised equational theory,
one can establish the free k-(�, I)-algebra as the quotient of the absolutely free parametrised k-algebra by the congruence
obtained from the inequations. Details can be found in [6].
Theorem 5.4 above shows that dcpo-(�, I)-algebras are kp-(�, I)-algebras. Furthermore Theorem 4.10 of [3] shows
that DCPO is closed under the free kp-(�, I)-algebra construction. Thus, free dcpo-(�, I)-algebras can be constructed via
the free algebra construction in compactly-generated predomains. The only step remaining to get the free algebras in the
category DCPO is the monotone convergence reflection.
Let us sketch this construction again for our running Example 1. The situation here is exactly as in the semitopological
case above. The free k-(�, I)-algebra over 2 has as underlying set P from Fig. 1(a) with the topology τ generated by open
sets of the form {σ k(a)| k ≥ n} and {σ k(b)| k ≥ n}. This topology is compactly generated and, as in the semitopological
case, clearly compatible. Since the monotone convergence reflectionM((P, τ )) is the dcpo P2, from Fig. 1(c), with the Scott
topology, we have obtained the required result in the compactly generated framework.
6. Comparing the approaches
Nowwecompare thesedifferent approaches toconstruct freedcpo-algebras, i.e.,wecompare the freealgebra construction
in the frameworks of cover related preorders, semitopological algebras and k-spaces. First we embed the cover related
preorders into the category of topological spaces and show that the semitopological algebra construction is a genuine
extension of the one presented in Section 3. After that we investigate the difficulties in comparing the frameworks of cover
related preorders and k-spaces. In the final part of this sectionwe compare the two topological approaches and show that the
semitopological setting is the more general one and that the free algebra constructions in k-spaces and the semitopological
setting differ outside the world of dcpos.
6.1. A topological view on cover related preorders
We start by showing that cover related preorders can be equipped with a topology, as already observed by Keimel and
Lawson [12]. Subsequently we show that under its topological characterisation CRP becomes a coreflective subcategory in
Top.
Definition 6.1. Let (X, �) be a cover related preorder. Then the �-induced topology on X , denoted by τ�, consists of thoseupward closed sets U such that whenever x ∈ U and x � D then it holds that D ∩ U �= ∅.
Let us show that this definition is invariant under the saturation on a cover relation.
Proposition 6.2. Let (X, �) be a cover related preorder and � be the saturation of �. Then the �- and �-induced topologies
coincide.
Proof. It trivially holds that τ� ⊆ τ�, so we only have to show the converse. For this let U ∈ τ� be given. We show that
U ∈ τ� by induction along the transitive closure of a cover relation under properties (1)–(3) of Definition 3.4.
(1) is clear, so let us look at (2). We suppose that x′ � D′ is obtained in the closure step
(x′ x � D) ∧ (↓D ⊆↓D′) ⇒ (x′ � D′).Let x′ ∈ U ∈ τ�, then also x ∈ U, sinceU is upward closed. Thus,U∩D �= ∅ has already been established by the induction
hypothesis. But this entails ↓D ∩ U �= ∅, hence ↓D′ ∩ U �= ∅, and so D′ ∩ U �= ∅, because U is upward closed.
64 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
(3) is a bit more tricky: suppose that x � D′ is obtained in the closure step
(x � D) ∧ (∀y ∈ D.y � D′) ⇒ (x � D′).Furthermore, assume x ∈ U ∈ τ�. We have to show that U ∩ D′ �= ∅, which we conclude as follows. By the induction
hypothesis, U ∩ D �= ∅ has already been established. Thus, we can pick any y ∈ U ∩ D, for which we have that y � D′, andso D′ ∩ U �= ∅ must have been established before the current closure step.
Thus, we have shown that a closure step of a cover relation according to Definition 3.4 does not change the induced
topology. But � is the transitive closure (hence ω-closure) of �, and so the claim follows. �
Corollary 6.3. If (X, �) and (X, �) are cover related preorders for which the saturations coincide, it holds that τ� = τ�.
Remark. This result again hints at the special role of the saturation when it comes to study cover related preorders. One
may view the saturation as topologisation, as will be made precise below.
After assigning a topology to a cover related preorder, we now show that this assignment is functorial. For this, we need
the following lemma.
Lemma 6.4. Let (X, �) be a saturated cover related preorder. Then x � D if and only if for all U ∈ τ� containing x it holds that
U ∩ D �= ∅.Proof. The “only if” direction follows from the definition of τ�. For the converse, observe that T(D) := {x ∈ X| x � D}, asdefined in Lemma 3.6, is closed in τ�, since for any x ∈ X \ T(D) and x � D′, one uses (3) of Definition 3.4 to obtain that
D′ � T(D), hence D′ ∩ X \ T(D) �= ∅. But then x � D implies the existence of some U ∈ τ� which contains x and satisfies
U ∩ D = ∅, which shows the claim. �
Now we can show the desired result.
Theorem 6.5. Let (X, �) and (Y, �) be saturated cover related preorders. Then a map f : (X, �) → (Y, �) is cover preservingif and only if it is continuous as a map (X, τ�) → (Y, τ�).
Proof. Suppose f : (X, �) → (Y, �) is cover preserving and V ∈ τ�. Then for any x ∈ f−1(V) and directed D with x � D,
we get f (x) � f (D), hence f (D) ∩ V �= ∅, and thus D ∩ f−1(V) �= ∅. Lemma 6.4 yields that f−1(V) is open in τ�.Conversely, suppose f : (X, τ�) → (Y, τ�) is continuous and x � D. Then for any V ∈ τ� containing f (x), we have
f−1(V) ∩ D �= ∅, hence f (D) ∩ V �= ∅, and so again Lemma 6.4 gives f (x) � f (D). �
Notice that oncewedrop the condition of the cover related preorders being saturated,we only can use the “only if”-part of
theLemma6.4, andhenceonlyget the“only if”-partof the theorem.Thus, ingeneral, a cover relatingmap f : (X, �) → (Y, �)between cover related preorders is continuous as a map (X, τ�) → (Y, τ�), but not necessarily vice-versa for the same
reasons we commented on in Section 3.
Nevertheless, Theorem 6.5 makes the assignment of a topology to a cover related preorder, as given in Definition 6.1,
functorial and if restricted to saturated cover relating preorders this functor becomes full and faithful.
Corollary 6.6. The assignment of Definition 6.1 yields a full and faithful functor T : CRP → Top whose image is closed under
isomorphisms.
Definition 6.7. A space in the image of the functor T is called a crp-space.
We get that the categories of continuous maps between crp-spaces and cover preserving maps between saturated cover
related preorders are equivalent. Next we want to give the crp-spaces a topological characterisation, which we start by
defining a converse to the functor T , motivated by Lemma 6.4.
Definition 6.8. For a topological space (X, τ ), the τ -induced cover relation on X (equippedwith the specialization preorder),
denoted by �τ , is defined by x �τ D whenever for all U ∈ τ containing x it holds that U ∩ D �= ∅.We get the following technical results.
Lemma 6.9. Let (X, τ ) be a topological space, respectively (X, �) be a cover related preorder. Then the following hold:
(i) τ ⊆ τ�τ ,
(ii) � ⊆ �τ� ,(iii) (X, �τ ) is saturated,
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 65
(iv) �τ� is the saturation of �,(v) τ� = τ�τ� ,(vi) �τ = �τ�τ
,
(vii) τ�τ is the Scott topology whenever (X, τ ) is a monotone convergence space.
Proof. The claims (i) and (ii) follow directly from the definitions of τ� and �τ .
The claim (iii) is easily shown along the lines of the proof of Proposition 6.2, and left to the reader.
For (iv), observe that by (iii) and Proposition 6.2 it suffices to show that for a saturated cover related preorder (X, �), itholds that x � D if and only if for all U ∈ τ�, x ∈ U implies U ∩ D �= ∅. But this is exactly the result of Lemma 6.4.
The claim (v) now follows from (iv) and Proposition 6.2.
The claim (vi) follows from (iii) and (iv).
For (vii), observe that if (X, τ ) is amonotone convergence space, then x�τ D if and only if x ∨↑ D. But then byDefinition
6.1, τ�τ is the Scott topology. �
We get our desired characterisation of saturated cover related preorders as topological spaces.
Corollary 6.10. A topological space (X, τ ) is a crp-space if and only if τ = τ�τ
Let us show that also the assignment of Definition 6.8 is functorial.
Lemma 6.11. A continuous map f : (X, τ ) → (Y, τ ′) is cover preserving as a map (X, �τ ) → (Y, �τ ′). The converse holds if
(X, τ ) is a crp-space.
Proof. Suppose x�τ D. We have to show that f (x)�τ ′ f (D), i.e., for any open V ∈ τ ′, it holds that f (x) ∈ V ⇒ f (D)∩V �= ∅.So let f (x) ∈ V , then x ∈ f−1(V), hence by continuity of f we have D ∩ f−1(V) �= ∅, and so f (D) ∩ V �= ∅, as required.
The converse, i.e., that a cover preservingmap f : (X, �τ ) → (Y, �τ ′) is continuous as amap (X, τ ) → (Y, τ ′), whenever
(X, τ ) is a crp-space follows from Theorem 6.5 and Lemma 6.9 (iii). �
Thus, identifying CRP with its topological counterpart, i.e., the category of continuous maps between crp-spaces, we get
the following.
Theorem 6.12. CRP is a full coreflective subcategory of Top.
Proof. The assignment (X, τ ) �→ (X, τ�τ ) is functorial by Theorem6.5, Lemma 6.9 (iii) and Lemma 6.11.Moreover the latter
yields that we have a natural isomorphism between the hom-sets Top((Y, τ ′), (X, τ )) and Top((Y, τ ′), (X, τ�τ )), if (Y, τ ′)is a crp-space. The unit of the coreflection is simply the identity map on the underlying set which is continuous by Lemma
6.9 (i). �
Observe that we also get the following property of crp-spaces.
Proposition 6.13. If (X, τ ) is a crp-space then its monotone convergence reflection carries the Scott topology.
Proof. Suppose (X, τ ) is a crp-space. Let (Y, τ ′) be amonotone convergence space and f : (X, τ ) → (Y, τ ′) be continuous.Then, by Theorem 6.5 and Lemma 6.11, it is continuous as a map (X, τ ) ≡ (X, τ�τ ) → (Y, τ�τ ′ ), where the latter carries
the Scott topology. The claim now follows by the universal property of the monotone convergence reflection. �
We remark that the converse of this result is not true in general, i.e., X need not be a crp-space although M(X) carries
the Scott topology. This can be seen with the following example.
Example 2. Let Z be the poset given in Fig. 2. We can identify Z as (N × (N ∪ {∞})) ∪ {∞, ∞} with the order defined as
(i, j) (i′, j′) if either of the following holds:
• i = i′ and j ≤ j′, or• j′ = ∞ and i ≤ i′, or• (i′, j′) = (∞, ∞).
The element (∞, ∞) is denoted by y0 in the figure, and we will identify it so in the following considerations. This poset
clearly forms a dcpo and so we assume it to be equipped with its Scott topology, which has a basis given by the sets of the
form:
Un0,f := {(i, j) ∈ N × (N ∪ {∞})| i ≥ n0 and j ≥ f (i)} ∪ {y0}
66 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
Fig. 2. The dcpo Z.
for n0 ∈ N and f : N → N.
Consider the topological subspace X ⊆ Z given by (N ×N)∪ y0, and let τ denote the subspace topology. Then it is easily
seen that M(X, τ ) ≡ Z. However, any directed subset D ⊆ X which does not contain y0 is constant in its first component,
hence we can find U ∈ τ with y0 ∈ U and U ∩ D = ∅. Thus, for a directed D ⊆ X , y0 �τ D implies y0 ∈ D, hence {y0} ∈ τ�τ .
But {y0} /∈ τ , because every Scott-open subset of Z intersects N×N. This yields that (X, τ ) forms a topological space which
is not a crp-space, but for which M(X) carries the Scott topology.
It follows from Lemma 6.9 (vii) that DCPO is the largest common full subcategory of CRP and Mon. We also know
that for an inequational algebraic theory (�, I) we can construct the free dcpo-(�, I)-algebra by either constructing the
free crp-(�, I)-algebra over it and then completing it to the presented dcpo, or by constructing the free semitopological
(�, I)-algebra over it and then applying themonotone convergence reflection. In a categorical picture this looks as follows:
We want to show that the middle of this picture forms a commuting square, generalising the situation for dcpos:
By Definition 3.9 and Theorem 6.5 we know that every crp-(�, I)-algebra (A, τ�, {σA}) forms a semitopological algebra.
Moreover, Theorems 3.10 and 4.3 yield that for a given crp-space (X, τX) the underlying sets of the free crp-(�, I)-algebraand the free semitopological (�, I)-algebra over it coincide. Thus, the universal property of the free semitopological algebra
guarantees the identity map FSTAX → FCRPX to be continuous. The converse follows from the following lemma.
Lemma 6.14. Let (A, τ, {σA}) be a semitopological (�, I)-algebra. Then the crp coreflection (A, τ�τ , {σA}) is a crp-(�, I)-algebra.
Proof. It needs to be shown that all operations A|τ | → A are cover preserving in each argument with respect to �τ , which
is equivalent to them being separately continuous with respect to the topology τ�τ , by Theorem 6.5. But separate continuity
of these maps is equivalent to continuity of the compositions:
A −→ A|σ | σA−→ A
where the map on the left hand side is given by x �→ (x1, . . . , xi−1, x, xi+1, . . . , x|σ |) for fixed 1 ≤ i ≤ |σ | and
(x1, . . . , xi−1, xi+1, . . . , x|σ |) ∈ A|σ |−1. That these compositions are indeed continuous follows from the fact that they are
continuous with respect to τ , by assumption, and Theorem 6.5 and Lemma 6.11. �
This yields our generalisation of the free algebra construction for dcpos.
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 67
Corollary 6.15. For any crp-space (X, τ�) the constructions of the free crp-(�, I)-algebra and the free semitopological (�, I)-algebra coincide.
Thus, one can say that the semitopological framework faithfully generalises the framework given by cover related pre-
orders.
6.2. The problem of crp-spaces in the compactly generated framework
We want to investigate whether the free algebra constructions for cover related preorders and compactly generated
spaces also coincide for a larger class than classical dcpos. This quickly raises the question in how far the categories CRP and
kTop are related. We know that both of them are coreflective subcategories of Top. However, it is unclear whether any of
them is closed under the coreflection into the other.
Unfortunately, we do not know the answer to either case. The coreflections themselves, i.e., the characterisation of the
additional open sets, seem to be of different nature. One may try to use the fact that crp-spaces are subspaces of classical
dcpos. However, attempts to use the ideas of the proof of Theorem4.7 in [7], which shows every dcpo under its Scott topology
is a k-space, for investigating whether kTop is closed under the crp-coreflection are bound to fail. The reason for this is that
this proof is by induction and uses a decomposition of directed suprema into a (possibly uncountable) chain of directed
suprema of lower cardinality. Once we leave dcpos and enter the world of crp-spaces those suprema of lower cardinality
need not exist, because these spaces may have ”holes”. Consequently, the inductive proof cannot work here. Finally, it is
not clear how to use this embedding into dcpos for showing that CRP is closed under the k-coreflection, because regular
subobjects in kTop may have a finer topology than the subspace topology.
In fact we do not even know the answer to the more general question whether there exist crp-spaces which are not
compactly generated. Jimmie Lawson has suggested that the following space might serve as a counterexample for this. 1
Let ω1 denote the set of all countable ordinals and its supremum, the first uncountable ordinal, equipped with the Scott
topology with respect to its usual ordering. Let Y ⊂ ω1 be its subspace obtained by removing all countable limit ordinals.
Then Y is obviously a crp-space, but possibly not compactly generated - in fact, at this point we do not know whether it is
compactly generated, we only know that the intuitive approaches towards showing that it is compactly generated fail.
Remark. Observe thatourExample2abovecannotbeused toanswer thesequestions, because the spaceX constructed in this
example is not compactly generated. In fact, it serves as an interesting example of a space for which M(k(X)) �= k(M(X)),which answers anopenquestion raisedbyAlex Simpson in the context of his topological domain theory researchprogramme.
Thus, we are left to consider our original question whether the free algebra constructions for cover related preorders and
compactly generated spaces coincide for a larger common class of spaces.
Definition 6.16. kCRP denotes the largest common full subcategory of CRP and kTop, i.e., it is the category of continuous
maps between topological spaces which are both crp-spaces and k-spaces.
The following result is used to show that in kCRP the notions of k-(Sigma, I)-algebras and crp-(Sigma, I)-algebrascoincide.
Lemma 6.17. Suppose X1, X2 are k-spaces whose monotone convergence reflections carry the Scott topology. Then a map f :X1 × X2 → Y from the k-product X1 × X2 into another k-space Y is jointly continuous if and only if it is separately continuous.
Proof. Of course, we only need to show the “if” direction. Notice that we may assume the spaces in question to be T0-
spaces, since the T0-reflection and k-coreflection commute and a map X → Y , which preserves the T0-equivalence classes,
is continuous if the corresponding map X0 → Y0 on the T0-reflections is continuous.
Thus, suppose f : X1 × X2 → Y is a separately continuous map between T0-k-spaces whose monotone convergence
reflections carry the Scott topology. Recall that by Theorem 5.4, the monotone convergence reflection preserves k-products
and k-products preserve the Scott topology on dcpos. Let ι denote the unit of the monotone convergence reflection. By
Theorem 5.6 of [11], the map f extends uniquely along ιX1 × ιX2 to a Scott-continuous map
M(X1) × M(X2) ∼= M(X1 × X2)f−→M(Y)
which is (jointly) continuous, as it is a map between dcpos under their Scott-topology.
But this yields that ιY ◦ f ≡ f ◦ (ιX1 × ιX2) is (jointly) continuous. Since Y is a (topological) subspace ofM(Y), it follows
that f must have already been continuous. �
Of course, this result holds for all objects of kCRP, hence we can conclude the following.
1 In private communication.
68 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
Corollary 6.18. Let (�, I) be an inequational algebraic theory. For a kCRP-space A and a family {σA} of �-operations on A the
following are equivalent:
1. (A, {σA}) is a crp-(�, I)-algebra.2. (A, {σA}) is a semitopological (�, I)-algebra.3. (A, {σA}) is a k-(�, I)-algebra.
Thus,we get awell-defined and unambiguous notion ofkCRP-(�, I)-algebras, and the categorykCRP(�,I) of continuoushomomorphisms between them embeds fully and faithfully into both CRP(�,I) and kTop(�,I).
Unfortunately the general problems of comparing CRP and kTop prevent us from concluding that any of the free algebra
constructions in the cover related preorder framework or the compactly-generated framework ends up in kCRP(�,I). In fact,
we do not even know if kCRP supports a free algebra construction, i.e., whether the forgetful functor kCRP(�,I) → kCRP
has a left adjoint. The problem that arises here is that we cannot use an Adjoint Functor Theorem, because we do not know
whether kCRP is complete. It is certainly not closed under infinite products in kTop. This is easily seen by considering the
Cantor space 2N which is the countable k-product of the two element discrete space 2, see Section 5 of [7]. The Cantor space
is a non-discrete Hausdorff space, but clearly a Hausdorff-space is a crp-space if and only if it is discrete. Thus, the k-product
2N is not an object of kCRP.
6.3. Semitopological algebras in the compactly generated framework
We finally want to investigate how semitopological and k-algebras compare on a broader scale. Of course, we have to
start our investigation by examining whether they can be compared at all outside the dcpo-setting. That this is the case
follows from the two following results.
Letusassumean inequational theory (�, I)beinggiven.Webeginbyshowing thatk-algebrasembed into semitopological
algebras.
Proposition 6.19. Every k-(�, I)-algebra (X, {σX}) is a semitopological (�, I)-algebra.
Proof. We need to show that all operations σX : X|σ | → X are separately continuous. For |σ | ≤ 1 this is clear, so we may
assume |σ | > 1.
For y = (y1, y2, . . . , y|σ |−1) ∈ X|σ |−1 and 1 ≤ i ≤ |σ | we denote by ιy,i the map X → X|σ | given by:
x �→ (y1, . . . , yi−1, x, yi, . . . , y|σ |−1).
Then the map σX : X|σ | → X is separately continuous if and only if for all y ∈ X|σ |−1 and all 1 ≤ i ≤ |σ | the composite
σX ◦ ιy,i : X → X is continuous. So let y ∈ X|σ |−1 and 1 ≤ i ≤ |σ | be given. Since (X, {σX}) is a k-(�, I)-algebra,σX ◦ ιy,i : X → X is a morphism in kTop, and so continuous. It follows that (X, {σX}) is indeed a semitopological �-algebra.
The satisfaction of all inequations is immediate. �
Our second result shows that semitopological algebras are closed under the compactly generated coreflection.
Proposition 6.20. For every semitopological (�, I)-algebra (X, {σX}), the operations σX constitute a semitopological structure
on the compactly generated coreflection k(X). In other words, also (k(X), {σX}) is a semitopological (�, I)-algebra.
Proof. We have to show that all operations σX are separately continuous as maps k(X)|σ | → k(X). For |σ | ≤ 1 this is
immediate from the property of a coreflection. So we assume |σ | > 1 and adopt the definition of ιy,i from the proof of
Proposition 6.19. Since k is a coreflection functor we only need to show that for all y ∈ X|σ |−1 and all 1 ≤ i ≤ |σ |, thecomposite σX ◦ ιy,i is continuous as a map k(X) → X . This follows from the fact that (X, {σX}) is a semitopological algebra
and so all the maps in the following composition are continuous:
k(X)id−→ X
ιy,i−→ X|σ | σX−→ X. �
Let us denote by (FSTX, {σFX}) and (FkX, {σFX}) the free semitopological and k-(�, I)-algebras over X , respectively.
Observe that the characterisation Theorems 4.3 and 5.7 show that the underlying sets and set-theoretic operations of them
coincide. With the previous results we can conclude that semitopological algebras and k-algebras are comparable and that
the semitopological framework is the more general one.
Corollary 6.21. The free semitopological algebra construction for inequational theories preserves k-spaces. Moreover, for every
given k-space X, the identity map FSTX → FkX yields a continuous homomorphism extension of the corresponding free algebra
inclusions.
I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70 69
Of course, simple examples fromabasic calculus (or analysis) course show that semitopological algebras forma genuinely
more general class. Let f : R2 → R be a separately continuous but not jointly continuous map, probably the simplest
example is the map:
(x, y) �→{0 if x = y = 02xy
x2+y2otherwise.
This map is separately continuous, because for every fixed x0 ∈ R, the map y �→ f (x0, y) is easily seen to be continuous.
However, it fails to be (jointly) continuous in the point (0, 0), since in every neighbourhood of this point one can finds some
( 1n, 1n) with f ( 1
n, 1n) = 1. Thus, (R, {f }) is a semitopological algebra for a signature of a single binary operation. However,
because R is a countably-based space the k-product topology on R2 is the usual product topology (see [7]) and so (R, {f })is not a k-algebra for this signature.
Nevertheless, the question remains whether the free algebra constructions in the two settings can differ. The following
example shows that even in simple cases they do.
Example 3. Let N+ denote the one point compactification of the natural numbers, i.e., the space whose underlying set is
N ∪ {∞} and an open neighbourhood of ∞ includes all k ≥ n0 for some n0 ∈ N. Let � be a signature consisting of a single
binary operation ∗, where we write x∗y instead of ∗(x, y). In the absolutely free topological �-algebra F(N+), the terms
x∗y for x, y ∈ N+ form an open subset, see [6], hence they must also form an open subset in the respective absolutely
free semitopological and k-�-algebras. Moreover, in the absolutely free k-algebra the sequence {n∗n}n∈N must converge to
∞∗∞ by continuity of ∗. However, this is not the case in the free semitopological algebra, where the sequence {n∗n}n∈Nitself is sequentially closed, because the convergence is here given by component-wise sequence convergence. In otherwords
the set
{∞∗∞} ∪ ⋃n∈N
({k∗n|k ≥ n + 1} ∪ {n∗k|k ≥ n + 1})
is open in the free semitopological�-algebra, since ∗ : F(N+)× F(N+) → F(N+) is component-wise continuous with
respect to it.
It follows that free algebra constructions generally differ in the semitopological and compactly generated frameworks.
Of course the space N+ is a non-discrete Hausdorff space and hence not a crp-space. Thus, this example does not give any
insight for the relation between free crp-algebras and free k-algebras.
7. Conclusions
We have investigated three approaches towards free algebra constructions for dcpos. One order-theoretic approach
suggested by Jung, Moshier and Vickers, and two topological approaches suggested by Keimel and Lawson and Battenfeld.
We have shown that the order-theoretic approach embeds faithfully into Keimel and Lawson’s approach via semitopological
algebras. Furthermore,we have established that there is a connection between the order-theoretic approach and Battenfeld’s
approach, as both frameworks share are well-defined notion of algebraic structure. However, determining how close this
connection really is turns out be a subtle problemwhich remains future work. Finally, we have given examples proving that
the two topological approaches differ outside the world of dcpos (or crp-spaces).
Summarizing one can say that Keimel and Lawson’s construction of free semitopological algebras provides the most
general framework, since every crp-algebra and also every k-algebra fits in here. It is also a very classical approach, since
it is a construction on the category of topological spaces and continuous maps. However, there are some problems in this
approach arising from the fact that the operations do not use the categorical structure of the underlying category, but have
to be viewed as additional structure. This is reflected by the fact that one has to be very careful when introducing term
maps, as shown by Keimel and Lawson in [12]. On the opposite end, Jung, Moshier and Vicker’s order-theoretic approach
uses an unstudied categorical framework and potentially provides the most restricted setting. However, in spirit it is closest
to the construction of free algebras for continuous dcpos via abstract bases. Finally, the approach via compactly generated
spaces provides a very neat categorical framework capturing both classical domain theory and Simpson’s topological domain
theory [5]. It does not suffer the term map problems of semitopological algebras. However, outside restricted classes such
as quotients of countably-based spaces it seems to be the least concrete approach, since it involves calculating k-products.
Moreover, we do not know to what extent it is compatible with cover related preorders.
From the viewpoint of denotational semantics, cover relations and topologies seem to be opposite sides when it comes to
addingmore structure to ordered sets. By this wemean that cover relationsmay be interpreted as a notion of approximation
whereas topology provides a notion of observation. In a categorical sense they formopposite sides, because theymeet exactly
in the realm of crp-spaces, respectively saturated cover related preorders, and, as is not hard to see, the saturation yields a
reflection functor on the category of all cover related preorders, whereas crp-spaces form a coreflective subcategory in Top,
70 I. Battenfeld / Journal of Logic and Algebraic Programming 82 (2013) 53–70
as shown in Section 6. The question naturally arises whether there exists a better topological characterisation of crp-spaces
on the one hand, andwhether one can extend the definition of cover relations to capture a wider class of spaces on the other
hand. For the latter problem one also has to consider the question of what would be an appropriate generalisation which
in spirit still describes a covering system. Finally, interpreting covers as a means to describe approximation one might also
consider attempting to formulate a notion of computability from it.
Wewant to endour conclusionsby remarking that there arewider classes of algebraic theories of interestwhen it comes to
modelling computational effects.One such class is givenby theparametrisedequational theoriesmentioned inSection5. This
class providesmeans to give an algebraic theory for probabilistic computations and to recover the probabilistic powerdomain
construction, as Battenfeld has shown in [6]. It is known that free parametrised dcpo algebras can be constructed in the
compactly generated framework, and it is expected that this also holds for the frameworks of cover related preorders and
semitopological algebras. Interestingly, for such theories the category of continuous dcpos need not support a free algebra
construction and we do not know necessary and sufficient conditions under which it does. Another class of useful algebraic
theories is obtained by increasing the choice of the arities of the operations. In the simplest case one allows the arity to be
countable, in more complex cases one allows them to be arbitrary dcpos (or other objects of the ambient category). Even for
the simple case, we do not know if any of the approaches above can be used to characterise the respective free dcpo algebras.
The semitopological approach appears to be more promising than the compactly generated one, because Theorem 5.4 does
not extend to infinite products.
Acknowledgements
I have greatly benefited from discussions and suggestions from Ernst-Erich Doberkat, Achim Jung, Klaus Keimel, Jimmie
Lawson, Matthias Schröder, Christoph Schubert and Alex Simpson. Also I wish to thank the anonymous referees for many
helpful comments.
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