Electronic Notes in Theoretical Computer Science 40 (2001)

URL: http://www.elsevier.nl/locate/entcs/volume40.html 21 pages

Uniform Completion versus Ideal Completion of Posets with Projections

Ralph Kummetz 1

Institut für Algebra Technische Universität Dresden D-01062 Dresden, Germany 2

Abstract

Posets with (/, <)-indexed projections are triples V = (D, <, (pi)i£j)- They consist of a partially ordered set (D, <) and a monotone net (pi)i£j of projections on D with respect to a fixed directed index set (I, <). In the present paper we prove that there are natural "completions" C(£>) = (C(D), <, (pi)iei) and J(T>) = (J(-D), <, (Pi)iei) of V. They are complete with respect to the uniformity induced by the kernels of all fi and pi, respectively. Moreover, they satisfy a universal property concerning the extension of special mappings ("homomorphisms"). If supieJpi = idr>, then V can be viewed as a substructure of C(£>) and of J(V). Further, D is dense in C(D); whence C(D) appears as the uniform completion of D. On the other hand, J(-D) can be obtained as the ideal completion of a suitable subset of D. A comparison shows that the completion C(£>) may be seen as a substructure of the completion J(£>). We also investigate under which conditions both completions coincide.

1 Introduction

Completions of mathematical structures emerge in many branches of mathematics and theoretical computer science. Well-known are completions of metric spaces, uniform spaces, the ideal completion of partially ordered sets, and so forth. In a recent paper, Bonsangue, van Breugel, and Rutten [3] investigated the completion of generalized metric spaces. This yields a generalization both of the chain completion of (pre)ordered sets and of the metric Cauchv completion.

In a certain sense, a completion of a structure is unique and "small". It is the "smallest" structure satisfying certain nice properties such that the original structure is contained in it as a substructure.

1 This work was supported by the German Research Foundation (DFG).

2 Present affiliation: 3SOFT GmbH, Frauenweiherstrafie 14, D-91058 Erlangen, Germany.

©2001 Published by Elsevier Science B. V.

The mathematical objects we deal with concern both partially ordered sets and uniform spaces induced by some family of projections. Such structures appear in mathematics and in theoretical computer science.

For instance, Plotkin introduced SFP-domains in [14]. They are inverse limits of w-ehains of finite pointed posets and form a convenient mathematical model for the semantics of nondeterministic programming languages, Gunter [6] and Jung [7] extended SFP-domains to bifinite domains (also called profinite ([6]) or FB-domains ([7])), These are limits of directed systems of finite posets. They can be characterized as dcpo's admitting a monotone approximating net of Scott-continuous and image-finite projections (cf, [6,7]), Moreover, bifinite domains turn out to form a maximal cartesian closed category of algebraic domains. Such maximal categories were completely classified by Jung [7].

On the other hand, Baier and Majster-Cederbaum [2,12] investigated pseudo rank ordered posets. They are posets (£>, <) together with a monotone sequence of projections. These projections define a canonical pseudo-ultrametric on D. In some situations the projections are induced by a weight. This is a special mapping from D into the set of natural numbers with infinity. Intuitively, the elements of D are regarded as "processes" and the weight of d E D is the maximal number of steps that is needed for an execution of d. In [12] a comparison of the metric completion with the ideal completion of a weighted poset is established for finitarv weights.

Recently, Spreen [15] used rank orderings on dl-domains to obtain several models of the untyped A-ealeulus,

A general approach comprising both bifinite domains and pseudo rank ordered posets can be found in [9,10], where posets with projections (pop's) are introduced. These posets carry a directed family of projections. The kernels of the projections form a basis for a uniformity: the pop uniformity. This uniformity and its induced topology are closely related to the poset structure, see [9,10] for details.

In the present paper we consider completions of posets with projections. Since we study posets and uniform spaces at the same time, we can form either the ideal completion of the poset (or a suitable subset of it) to obtain an algebraic domain or the uniform completion with respect to the pop uniformity to get a Hausdorff and complete uniform space that contains the original one as a dense subspace. We investigate how both completions are related.

The reader is assumed to have some basic knowledge in topology, especially in the theory of uniform spaces. In the books by Bourbaki [4] and Kelley [8] he or she will find much more than we actually need here. For notions from and a survey on domain theory we recommend Abramsky and Jung [1].

2 Basic facts on posets with projections

In this section we recall some facts concerning posets with projections. Throughout this paper let (£>, <) be a partially ordered set.

For .1 C I) lei Aln := Aj. := {d E D | 3o E A: d < a}. A lower set is a subset A C D with A = . We shorten ill := {d\l for any d E D. A set A C J~) is directed provided that it is non-empty and for all x, y E D there is some z E D with z > x,y. A non-empty subset A C D is bounded if there is some d E D with a < d for all a E A. A dcpo (directed complete partial order) is a poset in which all directed subsets have a supremum. Similarly, a bcpo (bounded complete partial order) is a poset where each bounded set admits a supremum. An ideal is a directed, lower subset of D. Let (ld(D), C) be the ideal completion of (£>, <),

An element d E D is called compact if for all directed subsets AC D with sup A > d there is some a E A with a > d. Let K(D) denote the set of all compact elements of (D, <), The poset (D, <) is algebraic provided that for all d E D the set K(D) fl dl is directed and d = sup (K(D) fl dl).

Let / : D —y E be a mapping between posets (D, <) and (E, <). Then / is Scott-continuous if it preserves suprema of directed sets (in case these suprema exist), A mapping p : D —y D is a projection if p is monotone, idempotent, and p <\dn with respect to the pointwise ordering of mappings, i.e. p(d) < d for all d E D. The kernel of p is the set kerp := {(d,e) e D2 | p(d) =p(e)}.

Next, we define the objects we deal with. Recall that a net is a mapping whose domain is a directed set.

Definition 2.1 Let (J, <) be a directed index set, let (D, <) be a poset, and let (Pi)iei be a monotone net of projections on D. Then we call the triple V = (D, <, {pi)i£i) a poset with (J, <)-indexed projections or (J, <)-pop. We say that V is approximating provided that supi£lPi{d) = d for all d E D. The net (j>i)i£i is the projection net of V. It is Abelian if Pi o pj = pj o ^ for all

hi e IFor any (J, <)-pop V = (£>, <, (pi)i^j), the kernels kerpit form a basis for a uniformity Ur> on D, see [9,10] for details. We call Ur> the pop uniformity of V. The induced topology r-p is the pop topology of V. A neighbourhood basis of an element d E D is given by the sets {e E D \ Pi(d) = Pi(e)} with i E I. We say that V is Hausdorff, complete, ... if this is true for the underlying uniform space (D,Ut>) (topological space (D,Tn), respectively).

We list some known facts on (J, <)-pop's that will be used subsequently without further reference:

Proposition 2.2 Let V = (D,<, (pi)iei) be an (I,<)-pop, let (dn)neN be a net in D, and let d E D.

(1) For all i,jEl we have Pi < pj if and only if Pi = Pi o pj if and only if

Pi = Pj ° Pi (see e-9■ [9], Lemma 2.1).

(2) (dn)neN converges to an element d E D in the pop topology if and only if, for all i E I, there is an index rii E N such that Pi{dn) = Pi{d) for all n> rii-

(3) {dn)n£N is a Cauchy net with respect to the pop uniformity if and only if, for all i E I, there is an index rii E N such, that Pi{dn) = Pi{dn.) for all n> rii ■

(4) The net (pi(d))iei converges to d with respect to the pop topology. This follows from [9], Prop. 2.8(1).

(5) V is approximating if and only if < is closed in D2 (Lemma 2.9 in [9]). Clearly, V is Hausdorff in this case.

(6) If (D, <) is a dcpo and pi is Scott-continuous for all i E I, then D is complete with respect to the pop uniformity (see Prop. 3.1 in [9]). □

Substructures of (J, <)-pop's are defined as follows:

Definition 2.3 Let V = (I). <, (/>()(. ,) be an (J, <)-pop, Let X C I). We say that X induces a suhpop of V if ^¿pf] C X for all i E I. Then, together with the induced order and the restricted projections, we call X = (X, <, (pi\x)iei) a suhpop of V. Moreover, X is a full suhpop of V provided that Pi[D] C X for all i E I.

Clearly, any subpop X = (X, <, (pi\x)iei) of an (J, <)-pop is an (J, <)-pop itself. If V is approximating, then X is also approximating.

Example 2.4 Let V = (D, <, (pj)ieJ) be an (J, <)-pop. The set (JieIPi[D] induces a full subpop of V. Obviously, this is the least full subpop of V. Note that it is approximating. We denote this subpop by (JieIPi[D].

Full subpop's of V are precisely those subpop's that are dense in D with respect to the pop topology. This is the subject of the next lemma. Its easy proof is left to the reader.

Lemma 2.5 Let V = (£>, <, (pi)i^j) be an (I,<)-pop and let X = (X, <, (pi\x)ier) be a subpop of V. Then X is a full subpop if and only if X is dense in (D,tv). □

Next, we define structure preserving mappings between (J, <)-pop's.

Definition 2.6 Let V = (D, <, (pj)ieJ) and £ = (E, <, (qj)ieJ) be (/,<)-pop's and let f : D —Y E.

(1) The mapping / commutes with all projections if % o / = / o pi for all i E I.

(2) We call / a (pop) homomorphism provided that / is monotone and commutes with all projections,

(3) We say that / is a (pop) embedding if / is an order embedding and a pop homomorphism,

(4) Finally, / is a (pop) isomorphism if / is both an order isomorphism and a pop homomorphism. The (/,<)-pop's V and £ are said to be (pop) isomorphic if there exists a pop isomorphism between them.

Intuitively, the image Pi(d) can be seen as the "i-th approximation" of the element d E D. Hence, the property of / : D —y E to commute with all projections means that / preserves all levels of approximation: the i-th approximation of f(d) coincides with the image of the i-th approximation of d.

Clearly, any homomorphism is uniformly continuous with respect to the pop uniformities. Further, there is an obvious connection between subpop's and pop embeddings: let V = (D, <, (pj)ieJ) be an (J, <)-pop. If .1' = (X, <, (pi\x)iej) is a subpop of V, then the inclusion map idx,d is a pop embedding, Conversely, let £ = (E, <, (ft)ie/) be an (J, <)-pop and let f : D —y E be a pop embedding. Then f[D] induces a subpop of £ that is pop isomorphic to V.

A natural question is when all projections Pi of an (J, <)-pop (D, <, (pj)j,ej) are also homomorphisms, Abelian projection nets give the answer:

Lemma 2.7 Let V = (D, <, (pi)i^j) be an (J, <)-pop. Then the following are equivalent:

(i) (Pi)iei is Abelian.

(ii) pi is a homomorphism for all i E I.

(iii) Pi[D] induces a subpop ofV for all i El. □

Again, the easy proof is left to the reader. Note that the composition of projections need not be Abelian, whence in general the projection net (jpi)^ is not Abelian, However, if (J, <) is a chain, then (Pi)i£i is always Abelian because of Proposition 2,2(1),

3 Existence and uniqueness of the pop completion

This section is devoted to a universal object that we call "pop completion". We show that each (J, <)-pop V has such a completion C(X>) and, furthermore, that it is uniquely determined up to a unique pop isomorphism. Any homomorphism from V to a complete approximating (J, <)-pop £ can be extended uniquely to a homomorphism from C(X>) to £. IfV is approximating, then we can pop embed V as a full approximating subpop into its completion. The pop uniformity of C(X>) is the uniformity of the uniform completion of

(D,UV).

The following (J, <)-pop turns out to be the proper candidate for the pop completion:

Proposition 3.1 Let V = (D, <, (pi)i^j) he an (I, <)-pop. Let

Doc ■= {(>/,),, i e Ul>'iy Ie 1 : i<3 =>di= Pi(dj)} iei

be equipped with the product order. For all i E I define ri : D00 —y D^ by ri((dj)jei) : = (pj(di))jei. Then V^ := (D^, <, (ri)ie/) «« a complete approximating (I,<)-pop. If (pi)iei Abelian, then (ri)iei is Abelian and ri((dj)jej) = (pi(dj))jei for all i E I and all (dj)jei £ A»- -if ^ ^ Scott-continuous for all i E I, then ri is Scott-continuous for all i E I.

Proof (Sketch). For all? e I let fi : Pi[D] —y D^ be defined by /¿(d) := (pj(d))j£r- Note that fi is well-defined. Let gi : D^ —y Pi[D] be the canonical projection from D^ onto Pi[D], Recall that (fi, gi) is an embedding projection pair. In particular, fi is Scott-continuous, By Theorem 3,3(2) in [9] we obtain (Dqo, <,(fi o gi)i£i) to be a complete approximating (J, <)-pop. Further, {fi ° 9i){{dj)jei) = fi{di) = (Pj(di))ja = ri((dj)jei) f°r all i E I.

Now let {pi)i£i be Abelian and let i,j E I. Choose some k El with k > i,j and recall that di = Pi(d,k) and dj = Pj(d,k). Then Pj(di) = pj(pi(d,k)) = Pi{Pj{dk)) =Pi{dj). As a consequence, (ri)ieJ is Abelian,

Finally, let Pi be Scott-continuous for all i E I. It is well-known that then gi is also Scott-continuous for all i E I. Consequently, n = fi o gi is Scott-continuous for all i El. □

Next, we formulate our existence and uniqueness theorem of the "pop completion". The approach to obtain the completion using an inverse limit construction is similar to the one given by Ehrig et al. [5], Theorem 1,14, This result states the existence and uniqueness of a universal completion of so-called projection spaces. These are sets together with a sequence of idempotent self-maps satisfying certain conditions. They do not carry any order relation. For instance, it is easy to see that if (D, <, (pn)nen) is an (N, <)-pop, then (D, (pn)n&i) is a projection space. For further connections between projection spaces and (N, <)-pop's the reader is referred to [10], Section 4,2,

Theorem 3.2 Let V = (D, <, (pi)iej) be an (I, <)-pop.

(1) There exist a complete approximating (J, <)-pop C(V) = (C(£>), <, (pi)i£i) and a pop homomorphism i/j : D —y C (D) with the following universal property:

For any complete approximating (J, <)-pop (E, <, (qi)iei) and any homomorphism f : D —y E there is a unique homomorphism f : C (£>) —y E with f o ip = f.

(2) Let V = (£>',<', (p'i)i£i) be a complete approximating (I,<)-pop and let <f> : D —y D' be a homomorphism such that (V, (j>) fulfils the universal property of (1). Then there exists a unique pop isomorphism $ : C(D) —y D' with $oijj = (j).

(3) xp[D] induces a full subpop of C(V) and is dense in (C(D),tq^).

(4) (a) Let d,e E D. Then ip{d) < -0(e) if and only if Pi(d) < pi(e) for all

i e I.

(b) ^\Pi[D] is an order isomorphism from Pi[D] onto Pi[C(D)] for alii E I.

(5) i/j is a pop embedding ofV into C(V) if and only if V is approximating.

(6) (a) If (pi)iei is Abelian, then so is (pi)iei-

(b) If pi is Scott-continuous for all i E I, then i/j and pi are Scott-continuous for all i E I.

(7) C(V) is pop isomorphic to V^ (cf. Proposition 3.1). More precisely, there is a unique pop isomorphism ^ : C(£>) —y V^ with o xp)(d) = {Pi{d))i£i for all d E D.

Proof. Let C(V) := V^, i.e. (C(£>), <) = (D0c, <) and pi = r« for all i E I. Proposition 3,1 tells us that C(X>) is a complete approximating (J, <)-pop.

Let tj) : D —y C(D) be defined by ip(d) := (Pi(d))ier■ Clearly, ip is monotone. Let V be the uniformity on D^ that is induced by the product uniformity of the family (pi[D],Ud-ls)iej, where Wdis is the discrete uniformity on Pi[D], Let i E I and let ^ : D^ —y Pi[D] be the canonical projection from £>00 onto Pi[D], One easily sees that x gi)^l[¡dPj[Dj] = kerpim We conclude that V = Uq{v)- Therefore, we can apply Lemma 3,2 in [9] to deduce that ip[D] is dense in C(£>). Given i E /, we have Pi(ip(d)) = ri((pj(d))j^r) = {Pj{Pi{d)))j£i = il>(jpi(d))\ whence ip is a homomorphism. In particular, ip[D] induces a subpop of C(X>), It is full in view of Lemma 2,5, which proves (3), Furthermore, let ip(d) < ip(e). Then, by definition of ip, we have Pi{d) < Pi(e) for all i E I. Conversely, if Pi{d) < Pi{e) for all i E /, then Pi{i){d)) = ii{Pi{d)) < ip(pi(e)) = Pi(ip(e)) for all i E I. As C(X>) is approximating, we conclude that ip(d) < ip(e). This proves (4)(a). To verify (4)(b), let i E I. Since tj) is a homomorphism, we have Vfet-D]] = C pi[C(D)]. On

the other hand, let d E C(£>). Then pi(d) E ip[D] by (3), whence Pi(d) E pt[ip[D]] = ip[pt[D]]. We obtain ip[pt[D]] = pt[C(D)]. As ip\Pi[D] is monotone and, in addition, order-reflecting by (4)(a), it is an order isomorphism from Pi[D] onto pt[C(D)].

Let tj) be a pop embedding, let d,e E D, and let Pi(d) < e for all i E I. Then Pi(ip(d)) = ip(pi(d)) < ip(e) for all i E /; whence ip(d) < ip(e) since C(X>) is approximating. Then d < e and thus V is approximating. Conversely, if V is approximating, then if) is an order embedding by Lemma 3,2 in [9] and therefore a pop embedding. This shows (5),

(6) results from Proposition 3,1 (the proof that ip be Scott-continuous is deferred to Proposition 3,7 below).

To prove the universal property, let £ = (E,<,(qi)ieI) be a complete approximating (J, <)-pop and let / : D —y E be a homomorphism. Let (di)iei E C(D) and let i,j E I with j > i. Then qi(f(dj)) = =

<li(f(Pi(dj))) = qi(f(di)). Consequently, (f(di))i^r is a Cauchv net in (E,U£). As £ is complete Hausdorff, let f((di)ieJ) := \imi£lf(di). Now let (di)ieJ,

(di)iei E C(D) with di < di for all i E I. Then f{dj) < f{dj) for all i E I and thus f((di)iei) < /((¿¿)iei) because the partial order of £ is closed. Hence, / is monotone. Note that since di E pi[D], we have di = Pj(di) for all j > i. Consequently, qi(f(dj)) = f(pi(dj)) = f(di) = f(pj(di)) for all i,j E /, I > i. We conclude qt{f{{dj)ja)) = f{(pj{di))jei) = /(MrfiW)); whence / commutes with all projections and thus is a homomorphism. Furthermore, f(ip(d)) = f{{pi{d))ia) = Yimia f(pi(d)) = f(d) because (pi(d))i£l —> d and / is continuous. As £ is Hausdorff, we infer from (3) that / is the unique homomorphism with / o ip = f. This proves (1),

Uniqueness of (C(X>), ip) (i.e. (2)) can be shown by exploiting the universal property in the usual way. Finally, (7) is true by definition and (2), □

Definition 3.3 For any (J, <)-pop V we call the complete approximating (J, <)-pop C(V) of Theorem 3,2 the pop completion of V. The mapping : D —> C (£>) is the canonical homomorphism.

Notice that (C(D),Uc(v)) is the uniform completion of the space (D,Ut>)-

Corollary 3.4 Let V = (£>, <, (pi)i^i) he an (J, <)-pop with pop completion C(V) = (C(£>),<, (pi)i£i) and canonical homomorphism i/j : D —> C(£>). Then i-IAlJifziPilD] a P°P isomorphism from the suhpop (JieIPi[D] onto the suhpop (Ji£lpi[Q(V)]. If, furthermore, (pi)i£i is Abelian, then tp\pi[d] ^ a P°P isomorphism from the suhpop induced by Pi[D] onto the suhpop induced by

Proof. By Theorem 3,2(1) and (4), ip is a homomorphism with ip[[Ji£iPi[D]] = Uj£/pi[C(D)]. Recall that ip(d) < ip(e) if and only if Pi(d) < Pi{e) for all i E I (3,2(4)), Thus, given i,j, k E I with k > i,j, we infer that ip(pi(d)) < ip(pj(e)) implies pi(d) = Pk{Pi{d)) < pk(pj(e)) = Pj(e).

Now let (pi)i£i be Abelian, Lemma 2,7 and Theorem 3,2(6)(a) tell us that Pi[D] and pi\C(D)] induce subpop's for all i E I. They are isomorphic in virtue of 3.2(4). □

Let V = (£>, <, (pi)iei) be an approximating (J, <)-pop. In light of Theorem 3,2(5) we may identify V with the subpop induced by ip[D], Thus, we assume that I) C C(£>), <|d = <, ip = ¡d/i.c;/)!- and f>) /> = p, for all i E I. Note that for all i E I we have fi[C(D)] = Pi[D] by 3,2(4), Furthermore, V is a full subpop of C(X>) and D is dense in C(D). Summing things up, we obtain:

Corollary 3.5 Let V be an (J, <)-pop. Then the following are equivalent:

(i) V is approximating.

(ii) V is a full subpop of a complete approximating (J, <)-pop. □

Corollary 3.6 Let V be an approximating (J, <)-pop. Then V = C(V) (more precisely: i/j is a pop isomorphism) if and only if V is complete in its pop uniformity.

Proof. This results from the fact that both D is dense in (C(D), tqv)) and (D,Ux>) is complete; whence D is closed in (C(D),rc^) because the latter is Hausdorff, Alternatively: the canonical homomorphism ip is an order isomorphism due to Lemma 3,2 in [9] and thus a pop isomorphism, A further alternative is to apply Theorem 3,3(1) in [9] and Theorem 3,2(7), □

Let (D, <, (pi)iei) be an approximating (J, <)-pop. We show that whenever a subset A has a supremum or an infimum in (D, <) that is preserved by each projection pi, then it has a supremum or an infimum in (C(£>), <) which coincides with the one formed in (D, <), More precisely:

Proposition 3.7 Let V = (D,<, (pi)i^j) be an (I,<)-pop and let C(X>) = (C(D), <, (pi)i£i) be its pop completion. Let i/j : D —y C(D) be the canonical homomorphism. Let AC D such that sup A exists and Pi (sup A) = sup £>¿[/4] for all i E I. Then we have ^(sup A) = sup^[A] and j^(^>(sup A)) = sup ^[V* [AD for all i E I. In particular, ifV is approximating, then sup^ A = suPc(D) Similarly for the infimum.

Proof. We may assume C(X>) = V^ and ip(d) = (j>i(d))iei for all d E D, cf. Theorem 3,2(7), Clearly, ip[A] < ip(supA). Let (ej)je/ E D00 with ip[A] < (ei)iei. Then Pi(a) < e« for all a E A and all i E /; whence />((Mip A) = supp^A] < ei for all i E I. Therefore, c(Mip A) = sup^[A] and J^(^(supA)) = ip(pi(sup A)) = sup ^[^¿[A]] = suppi[^[A]] for all i E I. □

We mention the following two facts without proof:

Proposition 3.8 Let V = (£>, <, (pi)i^j) be any (J, <)-pop with pop completion C(V) = (C(£>), <, {pi)i£i). Let £ = (E, <, (ft)ie/) be a complete approximating (J, <)-pop. Let f : D —y E be a homomorphism with unique extension f : C(£>) —y E. If f, pi, and qi are Scott-continuous for all i E I, then so is

7- □

Proposition 3.9 Let V = (D,<, (pi)i^j) be an (I,<)-pop and let X = (X, <, (pi\x)ier) be a full subpop of V. Let £ = (E,<,(qi)iei) be a complete approximating (I,<)-pop and let f : X —y E be a homomorphism. Then there exists a unique homomorphism f : D —y E with f\x = f- If furthermore, f, pi, and qi are Scott-continuous for all i E I, then so is f. □

Proposition 3.10 Let V = (D, <, (pi)ieJ) be an (I,<)-pop and let X be a full subpop of V. Then the pop completions C(X) and C(V) are isomorphic. In particular, C(V) is the pop completion of the subpop (JieIPi[D]-

Proof. Let X =: (X, <, (pi\x)iei)- Since X is a full subpop of V, we obtain Pi[X] = Pi[D] for all i E I. Theorem 3,2(7) yields the assertion, □

Especially, if V is a complete approximating (J, <)-pop with full subpop X, then V is (pop isomorphic to) the pop completion C(X) of X (Corollary 3,6),

Proposition 3.11 Let V = (£>, <, (pi)i^j) be an (J, <)-pop. Let E be a "property" that is invariant under monotone mappings (such as e.g. bounded or

directed). Suppose all subsets of D with property E have a supremum that is preserved by each pi. Then all subsets of C(D) with property E have a supremum which is preserved by each pi.

Proof (Sketch). Let A C C(D) have property E and let i e I. Recall that ip\Pi[D] is an order isomorphism from Pi[D] onto pi\C(D)]. Let Ajt := (^1 Pi[D])^l\Pi[A\\. Then Ai has property E; whence Mip;,. 1( exists, Mip;,. 1( = supPi[D]Aj, =: Mip Aj. and pi(sup Aj) = supp^A^] f°r h3 e I- Moreover, one can show that Pi(supp}[A]) = pi(suppi[A]) for all j > i. Therefore, (supj^[A])iej is a Cauchv net. Let d e C(D) with (supp^A]^/ —y d. Then it turns out that d = sup A and Pi(d) = supp^A], □

Corollary 3.12 Let V = (£>, <, (pi)i^i) be an (I,<)-pop whose underlying poset is a bcpo such that for all i e I the projection pi preserves suprema of bounded sets. Let C(V) = (C(D), <, (pi)i^i) be the pop completion ofV. Then (C(D), <) is a bcpo and Pi preserves suprema of bounded sets for all i e I. □

If property E means directed, then we can improve Proposition 3,11:

Proposition 3.13 Let V = (D, <, (pi)iej) be an (J, <)-pop whose underlying poset is a dcpo and whose projections Pi are Scott-continuous for all i e I. Let A(D) := {supi£iPi(d) | d e D}. Then A(D) induces a full subpop A(V) = (A(D), <, (pi\A(D))iei) °f D suc'h that A(V) is approximating, (A(£>),<) is a dcpo, and Pi\.A(D) is Scott-continuous for all i El. Moreover, A( V) is the pop completion ofV. It coincides with V if and only ifV is approximating.

Proof. Clearly, the pointwise supremum := supi£lPi is a projection with Pi ° £ = £ °Pi = Pi for all i G I. Hence, A(V) is an approximating full subpop of V. Now A(D) = £[£>] and (A(D), <) is therefore a dcpo with sup^) C = sup^C for any directed subset C C A(D). Thus, Pi\.A(D) is Scott-continuous for all i e I. As a consequence, A(V) is complete in its pop uniformity (2,2(6)), Therefore, A(V) = Q(A(V)) = C(V) by Corollary 3,6 and Proposition 3,10, It is obvious that A(V) = V if and only if V is approximating, □

Suppose that V = (D, <, (pi)i^j) is an approximating (J, <)-pop such that (.D, <) is a dcpo or even a complete lattice and each pi preserves suprema of all (directed) subsets of D. Then, due to the previous proposition, C(X>) = V. This does not hold anymore if the projections Pi do not preserve suprema of directed sets, cf, the example given in [9] after Prop, 3,1,

Let V = (£>, <, (pi)iei) and £ = (E, <, (ft)iej) be (J, <)-pop's such that {qi)i£i is Abelian, Let [D —y E]hom :={/:£> —y E \ f is a homomorphism}. For all i e J define Qi : [D —y E]hoia [D —y E]hoia by Qt(f) := qt o f. Notice that Qi is well-defined because qj o Qi(f) = qj o qjt o f = q{ o qj o f = Qi° f °Qj = Qi(f) °Pj- Clearly, Qi is a projection. Endow [D —y E]hom with the pointwise order. Then [D —> £]hora := ([D —y E]hom, <, (Qt)ta) is an (J, <)-pop. It is easy to see that [D —y £]hom is approximating if and only if £ is approximating,

If, moreover, ft is Seott-eontinuous for all i E /, then let [D —y _gjShom ._ {/ : D —y E | / is a Seott-eontinuous homomorphism} and observe that Qiif) E [D —y E]Shom for all f e [D E]Shom and all i E I. Hence, [p £]Shom ;= shom^ (Qt)ta) is also an (J, <)-pop. (By abuse

of language we write Qt : [D —► E]Shom —y p —> E]Shom.) clearly, Q, is Scott-continuous for all i E I.

Now we calculate the pop completions of [D —y £]hom and [D —y £]Shom, respectively:

Theorem 3.14 Let V = (D, <, (pj)ieJ) be an (I,<)-pop and let £ = (E, <, (qi)i£i) be an approximating (J, <)-pop with Abelian projection net. Let C(V) = (C(£>), <, {pi)i£i) andC(S) = (C(E), <, (ft)iei) be the pop completions ofV and£, respectively. Then

(1) [D —y C(£)]hom, [C(V) —y C(£)]hom, and the pop completion of [D —y £]hom are pairwise pop isomorphic.

(2) If pi and qi are Scott-continuous for all ¡El. then an analogous result holds for the pop completion of [D —y £]Shom.

Proof. We only show (1); (2) is proven similarly, (Use Theorem 3,2(6)(b) and Proposition 3,8 for (2),) As usual, we view £ as a subpop of C(£). Then, clearly, [D —y £]hom can be seen as a subpop of [D —y C(£)]hom, For all /' E I and all g E [D —► C(E)]hoia we have ft o g E [D —► £]hom; that is, [P £fora ig a fuU subpop of [p C(£)]h0m.

In order to show that [D —y C(£)]hom is complete, let (fn)neN be a Cauchv net in [D —y C(£)]hom, Let i E I and let rii E N such that ft o fn = ft o fn. for all n > Let d E D. Since ft(/„(cf)) = qi(fni(d)) for all n > u,. we obtain (fn(d))neN to be a Cauchv net in C(£). Hence, it is convergent. Let f(d) := lim„eJV fn{d). Then ft (/(d)) = ft(lim„eJV fn{d)) = lim„ejvft(/„(d)) = Y\mn£N qi{fni{d)) = qi(frH(d)). We conclude that ft(/„(d)) = ft (/(d)) for all a > u, and all d E D, i.e. Qi(fn) = Qi(f) for all n > Hence, (fn)n£N converges to / with respect to the pop topology. Moreover, qi(f(d)) = ft(lim„eJV/„(d)) = lim„eJVft(/„(d)) = limn£N fn(Pi(d)) = f(Pi(d)); that is, / commutes with all projections. Let d < e. Then fn(d) < /„(e) for all u E A and thus f{d) < fie) because < is closed. Therefore, / is a homomorphism. As a consequence, [D —y C(£)]hom is complete.

Since Ci£) is approximating, [D —y C(£)]hom is also approximating. We infer from Corollary 3,6 and Proposition 3,10 that [D —y C(£)]hom is the pop completion of [V —y £]hom.

Let ip : D —y C (£>) be the canonical homomorphism from V to C(X>), For each gE[D —y C(£)]hom let g be the unique element of [C(D) —y C(£)]hom with gorp = g (Theorem 3.2(1)). Clearly, if 9l,g2 E [D —y C(£)]hom with 9i < 92, then giiipid)) < git^d)) for all d E D. As ip[D] is dense in C(£>) and Ci£) is approximating, we deduce that ~g{ < ~gi. Let i E I. Then (ft og) o ip = ft o g. By uniqueness, ft og = ft o g. We conclude that the mapping g i—y g

is a homomorphism from [V —> C(£)]hom to [C(I>)^—► C(£)]hom. If gl<gi for any gi, g2 E [D —y C(i?)]hom, then =7f[oip < ]h ° = fh- whence we have an embedding. Obviously, if h E [C(£>) —y C(i?)]hom and g := h o ip, then we have hotp = g = goip and thus h = g by uniqueness. Consequently, [D —y C(£)]hom and [C(X>) —y C(£)]hom are isomorphic. □

Remark 3.15 Let V = (£>, <, (pi)i^j) and £ = (E, <, (qi)i^j) be (J, <)-pop's such that £ is approximating, (E, <) is a dcpo, is Abelian, and qjt is

Scott-continuous for all i E I. Then it is easy to see that [D —y l^]Shom is a dcpo with respect to the pointwise order. The supremum of any directed subset of [D —y i^]Shom is taken pointwise. Moreover, the projection net of [D —y £]Shom consists of Scott-continuous projections. As a consequence, [D —y £]Shom is complete in its pop uniformity by 2.2(6) and thus coincides with its pop completion (Corollary 3.6).

4 Domain completion and ideal completion

In general the underlying poset of the pop completion is not a dcpo. This section deals with another completion of a given (J, <)-pop: the "domain completion". We show that each (J, <)-pop V admits an approximating (J, <)-pop J(X>) whose partial order is a directed complete partial order and whose projection net consists of Scott-continuous projections. The completion J(X>) also satisfies a universal property with regard to the extension of homomorphisms. Similarly to the pop completion we can pop embed V into J(X>) provided that V is approximating.

Since the ideal completion of a suitable subset of D will be the candidate for the domain completion, we need some facts on the ideal completion of a poset. We cite the following well-known statement from Marrowsky and Rosen [13], Theorem 2.7 (cf. also Lawson [11], Section I; Abramsky and Jung [1], Prop. 2.2.24.):

Proposition 4.1 Let (D, <) be a poset and let (ld(D), C) be the ideal completion of (D, <). Let (E, <) be a dcpo and let f : D —y E be a monotone 'mapping. Let (pn : D —y ld(D), (po(d) := d\., be the canonical order embedding of (D, <) into (Id(D), C). Then f* : Id(D) —y E, defined by f*(A) := sup f[A], is the unique Scott-continuous mapping such that f*o(pD = f. □

For any (J, <)-pop (D, <, (pj)j,ej) we endow the ideal completion (ld(D), C) of (D, <) with a natural (J, <)-pop structure:

Proposition 4.2 Let V = (£>, <, (pi)i^j) be an (J, <)-pop. Let (ld(D), C) be the ideal completion of (D, <).

(1) For alii E I define a mapping pi : ld(D) —y ld(D) by Pi(A) := pi[A]l for all A E Id(D). Then \d(V) := (ld(D), C, {pi)i£i) is a complete (J, <)-pop with Scott-continuous projections Pi.

(2) The mapping (pn : D —y ld(D), defined by (po(d,) := d\., is a pop embedding ofV into \d(V).

(3) Let (E,<,(qi)iei) be an (I,<)-pop such that (E,<) is a dcpo and qi is Scott-continuous for all i E I. Let f : D —y E be a homomorphism and define f* : ld(£>) —y E by f*(A) := sup f[A], Then f* is the unique Scott-continuous homomorphism with f* o (pD = f.

(4) ld(X>) is approximating if and only if D = {jieJPi[D].

Proof. (1) Let i E I. Clearly, pi is Seott-eontinuous, Let A E ld(D) and let j E I with i < j. Then ^¿[^¿[A]^]^ = ^¿[^¿[A]]^ = Pi[A]l; whence pi is idempotent. Obviously, C A. As C Pj[A]l, we have that ld(T>)

is an (J, <)-pop, By recalling that (ld(D), C) is an (algebraic) dcpo, we infer that ld(X>) is complete with respect to its pop uniformity by 2,2(6),

(2) We know that ipD is an order embedding of (D,<) into (ld(D),C). Moreover, Pi(d)l = Pi[dl]l = Pi(dl) for all i E I and all d E D.

(3) We already know that f* is the unique Scott-continuous mapping such that f* o ipD = f (Proposition 4,1), Let A E ld(£>) and let i E I. Then qt(f*(A)) = qt (sup f[A]) = sup qt[f[A]] = sup f\pt[A]] = sup f\pt[A]i] =

(4) Observe that ld(X>) is approximating if and only if A = \Ji£iPi{A) for all A E Id(£>). Consequently, if ld(X>) is approximating, then dl = [JieiPi[d>l]l = [jieIPi(d)l for all d E D. Thus, there is some i E I with d < Pi(d), i.e. d = Pi(d). Conversely, if D = (JieIPi[D] and A E ld(£>), then we certainly obtain A = {jiaP%[A]l. □

Definition 4.3 We call ld(X>) the ideal completion of P.

Next, we formulate the existence and uniqueness theorem of the "domain completion". Observe the analogy to Theorem 3,2,

Theorem 4.4 Let V = (£>, <, (pi)i^j) be an (J, <)-pop.

(1) There exist an approximating (I,<)-pop J(V) = (J(£>),<, (^¿ej) with (J (£>), <) a dcpo and pi Scott-continuous for all i E I and a pop homomorphism i : D —y J (£>) with the following universal property:

For any approximating (J, <)-pop (E, <, (ft)iej) with (E, <) a dcpo and qi Scott-continuous for all i E I and any homomorphism f : D —y E there is a unique Scott-continuous homomorphism f * : J (D) —y E with

f*OL = f.

(2) Let V = (£>', <', (p'^iei) be an approximating (J, <)-pop such that (Df, <') is a dcpo and p\ is Scott-continuous for all i E I. Let <f> : D —y D' be a homomorphism such that (V, (j>) fulfils the universal property of (1). Then there exists a unique pop isomorphism $ : J (£>) —y D' with $ o i = (j).

(3) (J(£>), <) algebraic with K(J(D)) = i[VielPi[D]] = \JieIPi[i[D]\- Moreover, S(T>) is complete in its pop uniformity.

(4) (a) Let d.t E I). Then ¿(d) < ¿(e) if and only if Pi{d) < Pi{e) for all

i e I.

(b) i\Pi[D] is an order embedding of Pi[D] into Pi[J(D)] for all i E I.

(5) ¿ is a pop embedding ofV into J (V) if and only ifV is approximating.

(6) J(V) is pop isomorphic to Id(LJiejPi• More precisely, there is a unique pop isomorphism ^ : J (D) —y wiih i)(d) = (Jiei Pi(d>)l for all d E D.

Proof. Let J(X>) := Id ([J ieIPi[D]). Then J(X>) is approximating by Proposition 4,2(4), Let ¿ : D —y J(D) be defined by i(d) := \Ji£iPi{d)^. Clearly, ¿ is (well-defined and) monotone. Let i E I and let d E D. Then Pi(i{d)) = Pi[{JjerPj(d)l]l = [Jj>iPi(Pj(d))i = Pi(d)i = t(pi(d)). Therefore, i is a homo-morphism.

Let i(d) < ¿(e) and let i E I. By definition of ¿ we obtain Pi(d) E [Jj>iPj(e)i, whence Pi(d) < Pj(e) for some j > i. This implies Pi(d) < Pi(Pj(e)) = Pi(e)- Conversely, if Pi(d) < Pi{e) for all i E /, then Pi(i{d)) = L(Pi(d)) < i(jpi(e)) = Pi(¿(e)) for all i E I. As J(X>) is approximating, we obtain ¿(d) < ¿(e). This proves (4),

Let ¿ be a pop embedding and let d,e E D with Pi(d) < e for all i E I. Then Pi(d)\. = i(pi(d)) C ¿(e) = (J for all i E /; whence (JiejPi(d)l C

UieiPi(e)l- As ¿ is an order embedding, we conclude d < e. Thus, d = supi£iPi{d) and V is approximating. Conversely, if V is approximating, then ¿ is an order embedding by (4) (a) or by [9], Prop, 4,7, This shows (5),

Now we prove the universal property. Let £ = (E, <, (%)iej) and / be as in (1), Let f * be as in Proposition 4,2(3) applied to Id({JieJPi[D]), i.e. f*(A) := sup f[A] for all A E Id([J^j[£>]), Then f* is a Scott-continuous homomor-phism (4,2(3)), Let i E I and let d E D. Scott-continuitv implies that f*(\Jj>iPj(d)l) = SUP rMf*(Pj(d)l) = sup j>iqj(f*(dl)) = sup j>iqj(f(d)). Hence qi{f*{i{d))) = qi(supj>iqj(f(d))) = qi(f(d)). As £ is approximating, we conclude that f*(t(d)) = f(d).

Let g : J (D) —y E be a Scott-continuous homomorphism with g o ¿ = /, Since ¿|(j. Pi[D] coincides with the canonical embedding d i—y dl of (JieIPi[D] into ld((J\apAD}) = J(D), we infer that f* = g (Proposition 4.2(3)). This proves (1).

(2) follows from the universal property and (6) holds by definition and (2). It is well-known that (J(£>),<) is algebraic with K(J(D)) = {pi(d)l \ i E I, d E D} = {i{pt{d)) \ i E I, d E D} = i[1JieIPi[D]] = Finally,

J(X>) is complete in its pop uniformity by Proposition 4.2(1); whence (3) is true. □

Definition 4.5 We call J(X>) the domain completion of P. By abuse of language, we call the mapping ¿ : D —y J (£>) the canonical homomorphism, too.

The advantage of the domain completion J(X>) as opposed to the pop

completion C(X>) is that J(X>) always yields an algebraic domain. However, J(X>) has some disadvantages compared to C(X>), For instance, t[D] need not induce a full subpop of J(X>) and i\Pi[D] need not be an isomorphism from Pi[D] onto pi [J (D)]. In Theorem 5,6 we will investigate pop's where these problems do not occur.

Moreover, J(J(X>)) need not be isomorphic to J(X>) whereas C(C(X>)) is always isomorphic to C(X>) (cf. Corollary 3,6), For example, let V = (N0, <, (id)iej). Then, with the abbreviation uj = (No, <) and the usual ordinal number arithmetic, the underlying posets of J(X>) and J(J(X>)) are uj + 1 and lu + 2, respectively.

In contrast to the pop completion, the canonical homomorphism of the domain completion need not preserve suprema. Instead, we have the following result, whose proof is left to the interested reader.

Proposition 4.6 Let V = (D, <, (pi)i^j) be an (J, <)-pop with domain completion J (V) = (J (£>), <, (j>i)i£i). Let i : D —y J (£>) be the canonical homomorphism. Then the following are equivalent:

(i) i is Scott-continuous.

(ii) Pi[D] C K(D) for all IE I.

(iii) For all i e I we have that Pi is Scott-continuous and Pi[A] has a greatest element for all directed subsets AC D that have a supremum.

(iv) For any approximating (I,<)-pop (E,<,(qi)iei) with (E,<) a dcpo and qi Scott-continuous for all i e I and any homomorphism f : D —y E we have that f is Scott-continuous. □

Remark 4.7 Let V = (£>, <, (pi)iej) be an approximating (J, <)-pop. Corollary 3,6 tells us that the canonical homomorphism ip : D —y C(D) is an isomorphism if and only if V is complete in its pop uniformity. The question arises when the canonical homomorphism i : D —y J (£>) is surjeetive. The answer is given in Theorem 4,8 in [9] (together with Theorem 4,4(5) above): i is an isomorphism if and only if (£>, <) is a dcpo with K{D) = \JieIPi[D], In this case V is complete in its pop uniformity; whence V, J (X>), and C(X>) are pairwise isomorphic, A more detailed analysis when J(X>) and C(X>) are pop isomorphic is given in Section 5,

Remark 4.8 Analogously to Proposition 3,10 we have the following. Let V = (£>, <, (pi)iei) be an (J, <)-pop and let X be a full subpop of V. Then the domain completions S{X) and J(X>) are pop isomorphic. In particular, J(X>) is the domain completion of the subpop {jieI Pi[T>].

5 Comparison of the completions

It is natural to ask how these completions are related. This is the subject of the present section. We demonstrate how the pop completion embeds into the

domain completion and give some criteria to get equality. Finally, we show when the completions yield a bifinite domain.

We need the following well-known lemma to extend monotone mappings between posets to mappings between their respective ideal completions.

Lemma 5.1 Let (£>, <) and (E, <) be posets and let f : D —Y E be monotone. Define J : ld(L>) —► ld(£) by f(A) := f[A]l. Then f is a Scott-continuous mapping. In addition, we have the following:

(1) / is an order embedding if and only if f is an order embedding.

(2) / is an order isomorphism if and only if f is an order isomorphism. In this case, J-\C) = f~l[C]i = J^HC) for all C e \d(E). □

Lemma 5.2 Let V = (D,<, (pi)iei) and £ = (E, <, be (J, <)-pop's

and let f : D —Y E be a monotone mapping. Let f : Id (£)) —Y ld(i?) be the Scott-continuous 'map defined by f(A) := /[A]4-. Then f is a homomorphism if and only if f is a homomorphism.

Proof (Sketch). The "onlv-if'-part follows from Proposition 4,2(3) because / = (ipE o /)*, where ipE is the canonical order embedding e i—Y el of (E, <) into (\d(E), C), To prove the converse, use principal ideals, □

For any poset (D,<) let cpn ■ D —y ld(£>), d i—y dl, be the canonical order embedding.

Proposition 5.3 LetV = (£>, <, (pi)i^j) be an (J, <)-pop. Letip : D —Y C(D) be the canonical homomorphism. Then we have the following:

(1) J(V) is isomorphic to the subpop of ld(X>) induced by {(Ji£lpi{A) \ A E Id(D)}.

(2) ld(X>) is isomorphic to J(V) if and only if D = (JieIpi[D].

(3) The mapping ip : ld(£>) —Y ld(C(£>)), defined by ip(A) := V'tAI-i-? i-s a Scott-continuous pop homomorphism. It is a pop embedding if and only ifV is approximating. Furthermore, (pc(D) = H>d- Hence, we have commutative diagrams as in Figure 1.

Proof (Sketch). (1) Let A(\d(D)) := {(Jt£lpt(A) \ A e \d(D)}. Since the inclusion map idy Pi[D],D is a pop embedding of (JieIPi[D] into V, we deduce that the mapping (j> : ld((Jje7]?«[£)]) —Y ld(D), defined by <j>(A) := . 14/»- is also a pop embedding (Lemmas 5,1 and 5,2), We leave it to the reader to show that (j> is in fact a pop isomorphism from J(X>) = Id ((J ieIPi[D]) onto A(ld(P)).

(2) If \d(V) is isomorphic to J(V), then \d(V) is approximating and thus D = U ieiPi[D] by Proposition 4,2(4), The converse is clear by Theorem 4,4(6),

(3) The mapping ip is a Scott-continuous homomorphism by Lemma 5,2, Furthermore, V is approximating if and only if ip is a pop embedding if and only if ip is a pop embedding (cf. Theorem 3,2(5) and Lemma 5,1),

'pc(d)

ld(C(£>))

ld(£>)

U ziPi[D]

^Ue/PiP]

Fig. 1. Canonical embeddings.

Let (I e I). Then (pc(D)(ip(d)) = tp(d)l = tp[dl]l = tp(dl) = tp((pD(d)). □

At first sight, the following result might surprise the reader - especially in consideration of the previous proposition. It states that the pop completion C(X>) can be embedded into J(X>) and thus into ld(X>), Moreover, it turns out that the domain completion J(X>) can actually be obtained as the pop completion of ld(X>):

Theorem 5.4 Let V = (D, <, (pj)j,ej) be an (J, <)-pop. Let ip : D —y C(D) and i : D —y J (£>) be the canonical homomorphisms.

(1) The following (I, <)-pop's are pairwise pop isomorphic: J(V), J(C(P)), C(J(Z>)), C(ld(Z>)), and C(ld(C(X>))).

(2) Let T : C(D) —y J(£>) be the unique homomorphism with To i/j = t. Then T is a pop embedding of C(V) into J(V). Assuming J(V) = Id [£>]),

we have T(d) = {Ji^jitplpiiD])^1 (Pi,(d))i for d £ C(D).

Fig. 2. The pop completion embeds into the domain completion.

Proof. (1) By Corollary 3.4, ip0 := ip|y lPi[D] is a pop isomorphism from UiEiPtlv] onto (Ji£lpt[C(V)]. Therefore, ip0 ■ J(D) = \d(\Jt£lpt[D]) —y

Id((JieiPi[C(-£*)]) = J(C(X>)), A i—y is a pop isomorphism, cf. Lem-

mas 5,1 and 5,2,

Since J(X>) is complete in its pop uniformity (Theorem 4,4(3)), the canonical homomorphism xp : J(£>) —y C(J(X>)) is a pop isomorphism by Corollary 3,6,

Let A(\d(D)) := {(JteJpt(A) | A e Id(£>)}. Then we infer from Proposition 3,13 that C(ld(X>)) is isomorphic to the subpop induced by v4.(ld (£>)). Since the latter is isomorphic to J(X>) by Proposition 5,3(1), we obtain J(X>) to be isomorphic to C(ld(X>)), When switching from V to C(X>) we get that J(C(X>)) is isomorphic to C(ld(C(X>))).

(2) Consider again the pop isomorphism xp0 : J(D) —y J(C(X>)) from (1), Its inverse is given by B i—y ip^lB]^ (where B e ld([J^Jpi[C(£>)])), Let T : C(D) —y Id([J^j[C(£>)]) = J(C(X>)) be the canonical homomorphism from C(D) to J(C(X>)), With regard to Theorem 4,4(5), Tis a pop embedding.

Thus, ipo oTis a pop embedding. Given d e D, we deduce xpQ (T(ip(d))) = V'oMUi £i^P(d))i]i = U eIipolbP(Pt(d))]l = U eIpt(d)i = t(d). Hence, (V'o o T) o i/j = t. By uniqueness we infer T = xpo o T. □

Consider an (J, <)-pop (D,<, (pi)iei) with D = \JieIPi[D], Then we know by Proposition 5,3 that ld(X>) = J(X>). Therefore, the pop embedding (fin d i—y dl of V into ld(P) coincides with the embedding i and thus extends to a pop embedding of C(X>) into ld(X>) by the previous theorem. Hence, the following corollary generalizes Theorem 3,13 in Majster-Cederbaum and Baier [12], where a similar statement is proven for pointed posets with weight functions. Notice that weight functions induce (N0,<)-pop's whose projections map ideals to ideals, see [10], Section 4,2, for details. Note further that only metric completions and isometric embeddings are considered in [12], whereas we deal with pop completions and pop embeddings.

Corollary 5.5 Let V = (D,<, (pi)iei) be an (I,<)-pop such that D = (JieIpi[D], Then the pop embedding ipo d i—y d\. of V into ld(P) extends uniquely to a pop embedding Tpo of C(V) into ld(P). □

Next, we investigate when C(X>) coincides with J(X>).

Theorem 5.6 Let V = (£>, <, (pi)i^j) be an (J, <)-pop with pop completion C(V) = (C(£>),<, (pi)i£i) and domain completion S(T>) = (J(D),<, {pi)i^i)-Let xp : D —y C (D) and i : D —y J (£>) be the respective canonical homomor-phisms. Let T : C(D) —y J(£>) be the unique homomorphism with Jo xp = i. Then the following are equivalent:

(i) Each monotone net in D is a Cauchy net with respect to the pop uniformity.

(ii) For all directed subsets A C D and for all i e I we have that Pi[A] has a greatest element.

(iii) i[D] induces a full subpop of S{V).

(iv) i\Pi[D] is an order isomorphism from Pi[D] onto Pi[J(D)] for all i E I.

(v) fclui€iPi[D] is a pop isomorphism from \JiaPi[T>] onto [Ji£lPi[J(V)].

(vi) T is a pop isomorphism from C(V) onto J(V).

In this case we obtain K(J(D)) = (Ji£l pi{S(D)\. In other words, (C(£>), <) is an algebraic dcpo with K(C(D)) = (JieIpi[C(D)].

Proof, (i)—>(ii): Let A C D be directed. Then ( o>)a£A is a monotone net, whence Cauchy by (i). Therefore, for given i E I we find some a, E .1 such that Pi (a) = Pi(a,i) for all a, E A with a > a,. Let b E A and choose an element a E A such that a, > b,ai. Then pi(b) < Pi(a) = Pi(a,i). As a consequence, Pi [A] < pi(a,i) and Pi{a,i) is the greatest element of Pi[A],

(ii)—>-(vi): Theorem 5,4(2) tells us that I is a pop embedding. Let T : C(D) —y Id([J^j[C(£>)]) = J(C(X>)) be the canonical homomorphism from C(D) into J(C(X>)), We know that Tis a pop embedding and that I = <f>oT for some pop isomorphism (j> from J(C(X>)) onto J(V), see the proof of Theorem 5,4(2), Therefore, it suffices to show that T is surjective. In order to prove this, let i E I and let A C C(D) be directed. Then fi[A] is a directed subset of pi\C(D)]. Since ip\Pi[D] is an order isomorphism from Pi[D] onto Pi[C(D)] (Theorem 3,2(4)), the set B := {^\Pi[D})^l\pi{A]\ '1S a directed subset of Pi[D]; whence B = Pi[B] and Pi[B]] has a greatest element b by condition (ii). Therefore, ip(b) is the greatest element of Pi[A], We have thus proven that Pi,[A] has a greatest element for all directed subset A C C (D) and all i E I. Since C(X>) is complete and approximating, we can apply Theorem 4,8 in [9] to obtain that T is surjective. Moreover, the same result tells us that (C(D),<) is an algebraic dcpo and K(C(D)) = (JieIpi[C(D)]. This vields

Kim)=AK(c(D))] = ueJ^[cp)]] = iwfflcp)]] = {j^mmi

(vi)—>-(iii): ip[D] induces a full subpop of C(X>) (Theorem 3,2(3)), By (vi), l[D] = ¿[V4-D]] induces a full subpop of 5{V).

(iii)—>-(iv): Let i E /, We already know that i\Pi[D] is an order embedding of Pi[D] into Pi[J(D)] (Theorem 4,4(4)(b)), Let A E J(D). By (iii) there is an element d E D such that Pi(A) = t(d); whence Pi(A) = Pi{i{d)) = i{pi{d)) E

[£*]]■ Thus, i\Pi[D] is surjective,

(iv)^(v): Because of (iv) it is enough to show that t-\{ji(ZIPi[D] is order reflecting. But this follows as in the proof of Corollary 3,4,

(v)—>-(i): For the following conclusions let l'-=l{jielPi[D]- Let (dn)neN be a monotone net in D and let A := {pi(dn) \ i E I, n E \ j-Clearly, A E ld(U^jj?i[i)]). Let i E I. Using (v) we find an element d E Pi[D] such that {Pi(dn) | n E X |4 = ^[A]= Pi(A) = i{d) = dJ,. Hence, d = Pi(dni) for some rii E N. Let n > rii. Then Pi{drH) < Pi{dn) because (dn)neN is monotone. On the other hand, Pi{dn) < d = Pi{drH). Therefore, Pi{dn) = Pi(dni) for all n> rii. We conclude that (dn)n^N is Cauchy, □

Corollary 5.7 Let V = (D,<, (pi)i^j) be an (I,<)-pop such that Pi[A] has a greatest element for all A E ld(D) and all i E I. Then the pop completion

C(V) = (C(£>),<, (pi)iei) is isomorphic to the ideal completion ld(X>) if and only if D = (JieJpi[D]. In this case, (C(D),<) is an algebraic dcpo with K(C(D)) = D.

Proof. The assertions follow from Theorem 5,6, Proposition 5,3(2), and Proposition 4.2(4). □

Note that the previous corollary generalizes Theorem 3.16 in Majster-Cederbaum and Baier [12]. This theorem states for pointed posets (D, <) with a weight function ||,|| that the metric completion is isometric to the ideal completion if D = UneNo^""^] an<^ ^ '1S finite f°r all A e ld(D),

where gll'" is the projection induced by ||,|| (see [12], cf. also [10], Section 4.2). The metric under consideration is induced by the projections (see also [9], Theorem 2.6).

Finally, we have a look at the completions of (I, <)-pop's whose projections are image-finite. They are closely related to bifinite domains.

Theorem 5.8 Let V = (D, <, (pi)i^j) be an (I, <)-pop such that pi has finite range for all i e I. Then C(V) and J(V) are pop isomorphic. Moreover, C(V) is compact in its pop topology and (C(D),<) is a bifinite domain with K{Q{D)) = {jiapl[Q{D)\.

Proof. Let tj) : D —y C(D) be the canonical homomorphism. Sincepj\C(D)] = V>[pi[£>]] is finite for all i e /, the pop completion C(X>) is totally bounded in its pop uniformity (Prop. 2.7 in [9]) and thus compact. Moreover, (C(£>), <) is a bifinite domain by Proposition 3.1 and Theorem 3.2(7). As finite directed sets have a greatest element, we may apply Theorem 5.6 to obtain that C(X>) and J(X>) are pop isomorphic and K(C(D)) = (JieIpi[C(D)]. □

Theorem 5.8 and Corollary 5.7 yield:

Corollary 5.9 Let V = (D, <, (pi)i^i) be an (J, <)-pop such thatpi has finite range for all i e I. Then the pop completion C(V) is isomorphic to the ideal completion ld(P) if and only if D = {jieJPi[D]. In this case, (C(D),<) is a bifi,nite domain and K(C(D)) = D. □

Acknowledgement

The author wishes to thank A.K. Seda for the invitation to this conference.

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