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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 539573, 10 pages

http://dx.doi.org/10.1155/2013/539573

## Endpoints in -Quasimetric Spaces: Part II

Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa

Received 26 May 2013; Accepted 15 July 2013

Academic Editor: Salvador Hernandez

Copyright © 2013 Collins Amburo Agyingi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We continue our work on endpoints and startpoints in -quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valued -quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and the -hyperconvex hull of its natural -quasimetric space.

#### 1. Introduction

During his investigations on the hyperconvex (or injective) hull of a metric space Isbell [1] introduced the concept of an endpoint of a metric space and proved among other things that the hyperconvex hull of a compact metric space is equal to the hyperconvex hull of the subspace consisting of its endpoints (cf. also [2, 3]). A theory for -quasimetric spaces similar to the one for metric spaces due to Isbell can be developed (see, e.g., [4, 5]). In particular each -quasimetric space has a -hyperconvex (or injective) hull. For instance, it turns out that the hyperconvex hull of a metric space is isometric to the largest metric subspace containing the canonical copy of in the -hyperconvex hull of (see [5, Theorem 6]).

In [6] the authors defined the concept of an endpoint in an arbitrary -quasimetric space. In the quasimetric context it turned out to be natural to consider also the dual concept of an endpoint, which we called a startpoint.

Improving on a result from [6] in this note we will show that for any join compact -quasimetric space the set of the endpoints (resp., startpoints) of is equal to the set of the endpoints (resp., startpoints) of its -hyperconvex hull .

We also specialize some of our earlier results in [6] to two-valued -quasimetric spaces. It is well known that they are, essentially, the partially ordered sets and that in the category of partially ordered sets the injective hull coincides with the Dedekind-MacNeille completion (see, e.g., [5, 7, 8]). We will observe that in the case of a complete lattice our startpoints (resp., endpoints) turn out to be exactly the completely join-irreducible (resp., the completely meet-irreducible) elements. We also discuss for a partially ordered set in some detail the connection between its Dedekind-MacNeille completion and the -hyperconvex hull of its natural -quasimetric space. Our results can, for instance, be used to analyze the similarity between the following result due to Isbell [1]: *A compact injective metric space ** has a smallest closed subset ** such that the hyperconvex hull of ** is equal to *;
and the following well-known result from order theory (see, e.g., [9, Theorem 7.42]): *A lattice ** with no infinite chains is order isomorphic to the Dedekind-MacNeille completion of the partially ordered set *,
* where ** denotes the set of join-irreducible elements of ** and ** denotes the set of meet-irreducible elements of *. * Furthermore ** is the smallest subset of ** which is both join- and meet-dense in *.

#### 2. Preliminaries

In this section we first recall some of the basic definitions from asymmetric topology needed to read this paper. Then we recall some fundamental facts of the theory of the -hyperconvex hull of a -quasimetric space.

*Definition 1. *Let be a set and a function. (Here denotes the set of the nonnegative reals.) Then is *quasipseudometric* on if(a) whenever , and(b) whenever .

We will say that is a *-quasimetric space* provided that also satisfies the following condition: for each , implies that .

Let be quasipseudometric on a set . Then defined by whenever is also quasipseudometric, called the *conjugate quasipseudometric* of . Observe that if is a -quasimetric on , then is a metric on .

Given and a nonnegative real number we set . Note that this set is -closed, where is the topology having the balls with and as basic (open) sets.

A map between quasipseudometric spaces and is called *isometric* provided that whenever . Each isometric map with a -quasimetric domain is a one-to-one map.

Furthermore a map between quasipseudometric spaces and is called *nonexpansive* provided that whenever .

Given two real numbers and we will write for , which we will also denote by . Note that with defines the standard -quasimetric on the set of the reals.

Given a -quasimetric space , we recall that the *specialization (partial) order ** of * is defined as follows: for each , set if and only if .

For further basic concepts used from the theory of asymmetric topology we refer the reader to [10–12].

Many facts about hyperconvexity in metric spaces can be found in [13–15]. Connections between that theory and order theory are explored in [7, 16]. Throughout we will assume familiarity of the reader with the results of [6].

We next recall some facts mainly from [4] belonging to the theory of the -hyperconvex hull of a -quasimetric space (see also [5, 8, 17–19] for some related investigations).

Let be a -quasimetric space. We will say that a function pair on , where is *ample* provided that whenever .

Let denote the set of all ample function pairs on . (In such situations we may also write in cases where is not obvious.) For each we set

Then is an extended (if we replace in the definition of a quasipseudometric by we obtain the definition of an *extended* quasipseudometric. Of course, the triangle inequality for extended quasipseudometrics is interpreted in the self-explanatory way. (Indeed some authors prefer to work with extended quasipseudometrics throughout; see, e.g., [8]). For the purpose of this paper however real-valued -quasimetrics seem to be more appropriate; cf. Section 7) -quasimetric on .

We will call a function pair * minimal *on (among the ample function pairs on ) if it is ample and whenever is ample on , and for each we have and (for any function pairs and satisfying this relation we will write ); then . It is well known that Zorn’s Lemma implies that below each ample function pair there is a minimal ample pair (cf., e.g., [2, 20]). By we will denote the set of all minimal ample pairs on equipped with the restriction of to , which we will also denote by . Recall that is a (real-valued) -quasimetric on [4, Remark 6].

Furthermore if and only if the following equations are satisfied: whenever (cf. [21, Remark 2]). In particular note that such pairs are ample on .

Obviously the second component of a minimal ample pair on satisfies the following equation : whenever .

Given any real-valued function , satisfying , we can set whenever .

One readily checks that is an ample function pair on .

Furthermore, of course, whenever . Hence is a minimal ample pair on , and thus .

Hence characterizes exactly those functions that are second component of minimal ample pairs on (cf., e.g., [5, 8] concerning the underlying Isbell conjugation adjunction; see also [22]). Of course, an analogous result holds for the first component of minimal ample pairs on .

It is known (see [4, Lemma 3]) that implies that and whenever .

Moreover whenever (cf., [4, Lemma 7]).

For each we can define the minimal function pair whenever on . The map defined by whenever defines an isometric embedding of into (see [4, Lemma 1]).

Then is called the *-hyperconvex hull* of . A -quasimetric space is said to be -*hyperconvex* if implies that there is an such that (cf. [4, Corollary 4]). An intrinsic characterization of -hyperconvexity of a -quasimetric space can, for instance, be found in [4, Definition 2]: a -quasimetric space is -hyperconvex if and only if, given and families of nonnegative reals and such that whenever , we have that (see [4, Remark 2]).

Note also that and whenever and [4, Lemma 8].

The following important result (see [4, Remark 7]) is best understood as a kind of density of in . For any , we have that

Our first example shows that in that formula the step of taking the supremum with cannot be avoided in general.

*Example 2. *Let be such that , and let . Set
whenever . It is known that can be identified with the -hyperconvex hull of the subspace of (see [6, Example 4]).

Furthermore

#### 3. The Concept of Collinearity in Quasipseudometric Spaces

The following definition was given in [6] (cf. [3]). Let be a quasipseudometric space.(1)A finite sequence in is called *collinear* in provided that implies that .(2)An element is called an *endpoint* of provided that there exists an element in such that and such that for any collinearity of in implies that . We will say that *witnesses* that is an endpoint.(3)An element is called a *startpoint of * if it is an endpoint of .

Let us finally recall that a quasipseudometric space is called *join compact* provided that is compact.

The next result says intuitively that the points in the remainder of the -hyperconvex hull of a join compact -quasimetric space lie between the points of .

Proposition 3. *Let be a join compact -quasimetric space, and let be the canonical isometric embedding of into its -hyperconvex hull . Consider any . Then there are a startpoint and an endpoint in such that is collinear in .*

*Proof. *As stated above, for we will identify with .

Fix . Since is a -space, we have that or . We consider only the first case. The second one is analogous. By [6, Lemma 2] there is such that is collinear in .

Then . Therefore by [6, Corollary 3] there are a startpoint and an endpoint in such that is collinear in .

It follows that = , and hence is collinear in .

Lemma 4. *Let be a -quasimetric space. If is an endpoint with witness and is collinear in , then is also a witness that is an endpoint in . Similarly, if is a startpoint with witness and is collinear in , then is also a witness that is a startpoint in .*

*Proof. *Assume that is an endpoint with witness and that is collinear in . Note that . Suppose that for some is collinear in . Then inequalities imply by subtracting that (cf., [6, Lemma 1]); hence is collinear in , and thus by our assumption. Therefore witnesses that is an endpoint of . The dual result is proved analogously.

*Example 5. *Let be a join compact -quasimetric space with such that .

According to the proofs of [6, Proposition 3, Corollary 3] there exist a startpoint in with witness and an endpoint in with witness such that is collinear in . By Lemma 4 it follows that both witnesses that is an endpoint and witnesses that is a startpoint in

We next prove the result mentioned in Section 1.

Proposition 6. *Let be a join compact -quasimetric space. Then has exactly the same endpoints and startpoints as .*

*Proof. *In [6, Proposition 4] it was shown that each endpoint of is an endpoint of . In fact, it was proved that if is an endpoint in with witness , then witnesses that is an endpoint in .

Here we will show that each endpoint of is an endpoint of .

Suppose that witnesses that is an endpoint in . Then . Thus by [20, Proposition 5] there are sequences such that the increasing sequence converges to with respect to the usual topology on . (Note that here and in the following, for any , we will identify with

By join compactness of indeed we have that there is a subsequence of and such that and . Note that, for instance, for any .

Taking limits, therefore , and thus . Consequently is collinear in , since . Hence , and the statement follows.

We conclude that , since witnesses that is an endpoint in . Indeed then , and is collinear in . By Lemma 4 therefore is a witness belonging to that is an endpoint in . It follows that witnesses that is an endpoint in , because is a subspace of .

The assertion about startpoints is proved analogously.

#### 4. -Quasimetrics Induced by Partial Orders

Let be a partially ordered set and . In the following, we set and .

Given a partially ordered set , we equip it with the -quasimetric given by setting, for all if and otherwise. Indeed is a -ultraquasimetric, but we will not use this fact in this paper (see, e.g., [23]).

We will call the *natural **-quasimetric* of . Note that a map is nonexpansive if and only if is monotonically increasing. (Of course, here also denotes the restriction of to .)

Observe that, if is the natural -quasimetric induced by a partial order , then is the natural -quasimetric induced by the partial order .

As we noted in the Preliminaries section, with the help of the specialization order we can equip each -quasimetric space with a partial order. In [6, Proposition 1] we considered the following more sophisticated method.

Fix . For , set if is collinear in . Then is a partial order on according to [6, Proposition 1]. Our next result shows that, if the -quasimetric space originates from a partial order , then can be readily described.

*Example 7. *Let be a partially ordered set, and let be its natural -quasimetric. Furthermore let . Then agrees with if the two relations are restricted to or to . Furthermore restricted to is empty, and restricted to agrees with the complement of restricted to . Hence can be described as follows. For , we have if Case 1 is and [(i) or (ii) or if Case 2 is and . Note that obviously is a linear order if is a linear order.

Let us consider two specific examples of this construction.

*Example 8. *(a) Let be a partially ordered set with the smallest element . Then .

(b) Let be a partially ordered set with the largest element . Then is obtained on by removing the top element from and adding it as a new smallest element to .

Let be a -quasimetric space. Given we set that is collinear in (cf. [2, (4.2)]). We have that and is obviously -closed.

In the special case of -quasimetrics originating from partial orders the set can be described as follows.

*Example 9. *Let be a partially ordered set with natural -quasimetric , and let . Then is equal to the interval if and equal to if .

*Example 10. *Let be a partially ordered set and its natural -quasimetric. Suppose that is collinear in and . Then the sequence is equal to .

*Proof. *Of course . Furthermore with is impossible, since is collinear in . Thus we have that the sequence is alternating in .

We are next going to illustrate the concepts of a startpoint (resp., endpoint) in the case of -quasimetrics induced by partial orders. Let us first note that [6, Proposition 3] is useless for an infinite partially ordered set equipped with its natural -quasimetric , since is the discrete metric, and therefore is compact if and only if is finite.

Lemma 11. *Let be a partially ordered set, its natural -quasimetric, and . Then is a startpoint of witnessed by if and only if is a minimal element in .*

*Proof. *Suppose that is a startpoint of witnessed by . Then , and thus . Furthermore for all implies that . Hence and imply that . Thus is minimal in .

Assume that is minimal in . Then . Suppose that, for some is collinear in . Then ; thus and . Therefore , since is minimal in . Hence is a startpoint in witnessed by .

Corollary 12. *Let be a partially ordered set, its natural -quasimetric, and . Then is an endpoint in witnessed by if and only if is a maximal element in *

We next illustrate Lemma 11 and its corollary by two simple examples.

*Example 13. *Let be a set having at least two points and equipped with the discrete order . Then the -quasimetric induced by on is the discrete metric. Note that each point of is an endpoint and a startpoint in , witnessed by any other point.

*Example 14. *Let be a complete lattice, and let be compact (see [24, Definition .] for the definition), but such that . Since , there is such that . By Zorn’s Lemma, let be a maximal chain in containing . Set . Then , since is compact. Furthermore , since is a maximal chain. Thus is maximal in . Consequently is an endpoint witnessed by in , where denotes the natural -quasimetric of .

The following definition can essentially be found in [24, Definition ]. Let be a partially ordered set. An element is called *completely **-irreducible* if either is maximal in but different from the largest element or the set has a least element, which will be denoted by . Dually one defines *completely **-irreducible* elements in .

Proposition 15. *(a) Each completely -irreducible element in a partially ordered set is an endpoint in , where denotes the natural -quasimetric of .**(b) Let be a complete lattice equipped with its natural -quasimetric and (cf. [24, Remark ]). If is an endpoint of , then is completely -irreducible. *

*Proof. *(a) Suppose first that is maximal but not the largest element of . Then there is such that . Therefore is maximal in , and is an endpoint by Corollary 12.

Suppose now that has a least element . It will suffice by Corollary 1 to show that is maximal in . So let with . Then and consequently . Hence and is maximal in . Therefore is an endpoint in .

(b) For the convenience of the reader we include a proof, which follows [24, page 126]. Let witness that is an endpoint in . If is maximal, then is completely -irreducible, as it cannot be the largest element in , because .

If is not maximal in , then , since is maximal in by Corollary 12. Hence exists and , as . Thus has a least element. We have shown that is completely -irreducible in either case.

Corollary 16. *(a) Each completely -irreducible element in a partially ordered set is a startpoint in , where denotes the natural -quasimetric of .**(b) Let be a complete lattice equipped with its natural -quasimetric and (cf., [24, Remark ]). If is a startpoint of , then is completely -irreducible. *

Recall that an element in a complete lattice is called *completely join-irreducible* if, for each subset of implies that . Completely meet-irreducible elements are defined dually (see [9, Definition 10.26]).

Of course, in a complete lattice the completely -irreducible elements are exactly the completely join-irreducible elements and the completely -irreducible elements are exactly the completely meet-irreducible elements.

Corollary 17. *Let be a complete lattice and its natural -quasimetric. Then is a startpoint in if and only if is completely join-irreducible. Similarly, is an endpoint in if and only if is completely meet-irreducible in . *

*Example 18. *In a partially ordered set that is not complete and is equipped with its natural -quasimetric , an endpoint of need not be completely -irreducible. As an example consider the partially ordered set from [9, page 169] (see Figure 1).

Our characterizations (Lemma 11 and Corollary 12) immediately yield the set of startpoints and the set of endpoints . In particular is not completely -irreducible, although it is an endpoint.

As usual, an element in a complete lattice will be called *completely join-prime* if for some subset of means that there is such that . The concept of a *completely meet-prime element* is defined dually. It is easy to see that each completely join-prime element is completely join-irreducible and each completely meet-prime element is completely meet-irreducible (see [9, Definition 10.26]).

*Example 19. *Let be a complete lattice and its natural -quasimetric. Furthermore let be completely join-prime. It follows that is completely meet-prime. Furthermore one proves that . Indeed is a completely prime pair in the sense of [9, page 246]. We conclude that is an endpoint with witness in , since is obviously maximal in , and is a startpoint with witness in , since is minimal in .

We recall that a subset of a partially ordered set is called *join-dense* in provided that for each there exists such that (see [9, page 53]). Dually one defines the concept of a *meet-dense* subset of a partially ordered set .

It is known (see [24, Remark ]) that each meet-dense subset of contains all completely -irreducible elements. Similarly each join-dense subset of contains all completely -irreducible elements.

Proposition 20. *Let be a partially ordered set and its natural -quasimetric.*(a)*If ** is a join-dense subset of *,
* then all startpoints of ** belong to *. * Dually, if ** is a meet-dense subset in *,
* then all endpoints of ** belong to *. (b)*If ** is join- and meet-dense in *,
* then all startpoints (resp., endpoints) of ** are startpoints (resp., endpoints) of *.

*Proof. *(a) Suppose that is a startpoint of . Then our characterization (see Lemma 11) of a startpoint gives such that is a minimal element of .

Since , by join density of in , there must be such that and . Hence by the minimality property of we obtain . Therefore belongs to . The dual result is proved analogously.

(b) We continue the proof of part (a) dealing with startpoints. By the additional assumption of meet density of in , in the proof of part (a) we can find such that and . Then is minimal in . We conclude that witnesses that is a startpoint in , because . The result on endpoints is proved similarly.

#### 5. Examples

In this section we will discuss various examples in the light of the results in Section 4. Some of the details of the arguments are left to the reader.

*Example 21. *Let be a partially ordered set with the smallest element and the largest element , where , equipped with its natural -quasimetric . Given , collinearity of in obviously implies that . However we cannot conclude that is an endpoint of with witness , since .

Indeed we have the following result.

Proposition 22. *Let be a partially ordered set equipped with its natural -quasimetric . A smallest element of cannot be a startpoint of . Similarly a largest element of is never an endpoint of .*

*Proof. *Since , thus whenever ; there cannot be an element in witnessing that is a startpoint in . The dual statement is proved similarly.

Let be a linearly ordered set, and let be such that , but there does not exist an element such that . As usual, the pair is called a *jump* in . Note that for partially ordered sets one also says that*covers * in this case; see, for instance, [9, page 11].

Proposition 23. *Let be a linearly ordered set equipped with its natural -quasimetric . The first elements of jumps in are exactly the endpoints of . The second elements of jumps in are exactly the startpoints of .*

*Proof. *Suppose that is a jump in . Then is minimal in . Hence is a startpoint of by Lemma 11.

In order to prove the converse suppose that is a startpoint of . Then there is such that is minimal in . Since is a linear order, we have that and is a jump. Similarly one proves the stated result on endpoints.

Corollary 24. *In , equipped with its usual linear order and natural -quasimetric, is an endpoint but not a startpoint, while is a startpoint but not an endpoint.**In the set of integers equipped with the usual linear order and its natural -quasimetric, each point is an endpoint and a startpoint.**The closed unit interval of the set of rational numbers equipped with its usual linear order and the natural -quasimetric induced by that order does not have any endpoints nor any startpoints.*

Corollary 25. *Let be a partially ordered set equipped with its natural -quasimetric . If, for some consists of two elements, then is a startpoint in . Similarly, if, for some consists of two elements, then is an endpoint in .*

*Proof. *Suppose that with . Obviously is a minimal element of . The result follows from Lemma 11. The dual result is proved analogously.

Corollary 26. *Each atom in a partially ordered set (see [9, page 113] for the definition) with the smallest element and equipped with its natural -quasimetric is a startpoint of . Similarly each coatom in a partially ordered set with a largest element and equipped with its natural -quasimetric is an endpoint of *

*Example 27. *For a set with at least one element consider the complete lattice equipped with its natural -quasimetric , where is the power set of . Then the startpoints of are exactly the singletons. The endpoints of are exactly the complements of the singletons.

In fact, for each , the startpoint witnesses that is an endpoint, and the endpoint witnesses that is a startpoint. Observe that for each is a completely prime pair.

*Example 28. *Let be the usual topology on the set of the reals equipped with set-theoretic inclusion as a partial order, and let be its natural -quasimetric. Then there are no startpoints, and exactly the complements of singletons are the endpoints in .

Let us give a proof of this statement just using the basic definitions. Suppose that is a startpoint with witness . In particular . Let . Then we find such that . It follows that is collinear in —a contradiction. We conclude that there are no startpoints in .

On the other hand fix . Then collinearity of with implies that . Thus witnesses that is an endpoint in .

Assume that is an endpoint in . Then there is a witness such that . Let . Then is collinear in . By our assumption . Hence exactly the complements of singletons are the endpoints in .

Given a nonempty subset of a set , by we denote the filter generated by the filter base on .

Corollary 29. *Let be the set of filters (partially ordered under set-theoretic inclusion) on an infinite set and equipped with its natural -quasimetric . Then the set of endpoints of consists of all the ultrafilters on , and the set of startpoints of consists of all the filters with .*

*Proof. *It is well known that each filter on is the intersection of ultrafilters on and that the maximal elements in are the ultrafilters. Hence the endpoints in are exactly the ultrafilters on (see Propositions 15 and 20).

Furthermore has a smallest element, namely, . Since the set of with nonempty is join-dense in , the startpoints can only be of the form with proper by Propositions 20 and 22. We know by Corollary 26 that, for all is a startpoint in . We finally show that there are no other startpoints in .

Let be such that contains at least two points. Take any such that . We consider two cases.*Case 1*. . Then choose . It follows that is not minimal in , since and also . Indeed if , then there is such that ; hence and then —a contradiction.*Case 2*. There is . By our assumption about we can choose . Then and also , because , since . It follows that is not minimal in .

Thus we are done, since no exists that could witness that is a startpoint of

#### 6. The Dedekind-MacNeille Completion versus the -Hyperconvex Hull

Some of the results mentioned in the previous sections may have reminded the reader of the theory of the Dedekind-MacNeille completion (cf. also [16]). Of course this is not accidental but can be explained categorically (see, e.g., [5, 8]).

For the following discussion we need some basic facts from the theory of the Dedekind-MacNeille completion of a partially ordered set (see, e.g., [9, page 166]).

Let be a partially ordered set, and let . Then we define the *set of upper bounds* of , that is, whenever and the *set of lower bounds* of , that is, whenever . Let . The partially ordered set is a complete lattice. It is known as the *Dedekind-MacNeille completion of *. Furthermore defined by is an order embedding such that is both join-dense and meet-dense in . This is indeed the characteristic property of the Dedekind-MacNeille completion (cf. [9, Theorem 7.41]).

Proposition 30. *Let be a partially ordered set and its natural -quasimetric. Furthermore let be the natural -quasimetric of . Then and have the same startpoints (resp., endpoints). *

*Proof. *For the proof we consider a subset of . By Proposition 20(b) each startpoint (resp., endpoint) of is a startpoint (resp., endpoint) of , since is both join-dense and meet-dense in . Suppose now that is a startpoint of with witness . Let be such that is collinear in . Thus , and, therefore, , and . Since , is join-dense in , there is such that and . Thus and is collinear in . Hence and therefore , too. Consequently is a startpoint with witness in . The dual result is proved analogously.

*Example 31. *We continue the discussion of Example 18 (see [9, page 169] and Figure 2).

Considering as a subset of , in the light of Proposition 30 and according to Corollary 17 in the complete lattice , the set of the startpoints of becomes the set of the (completely) join-irreducible elements of , and the set of the endpoints of becomes the set of the (completely) meet-irreducible elements of .

We will say that a partially ordered set with its natural -quasimetric is -*hyperconvex* if, for any and any families in satisfying that whenever , it follows that . Observe that is -hyperconvex provided that is -hyperconvex.

Proposition 32. *Let be a complete lattice and its natural -quasimetric on . Then is -hyperconvex. *

*Proof. *Let , and let and be families in such that whenever . Set and . Note that contains at most one element, since is a -space. We observe that implies that and ; thus , and hence .

Furthermore by our assumption, obviously we have that , since whenever and . Since is a complete lattice, exists.

Then
since if and whenever . Thus is -hyperconvex.

As an illustration it may be useful to include here the following simple example (see [4, Example 8]).

*Example 33. *Let be equipped with its usual order and with its natural -quasimetric . Then can be identified with under the obvious inclusion . Hence is not -hyperconvex, although is a complete lattice.

For the following result compare the discussion preceding [8, Lemma 2.5].

Proposition 34. *Let be a bounded -hyperconvex -quasimetric space and its specialization order. Then is a complete lattice. *

*Proof. *Suppose that is an upper bound of .

Let and . Set if , and otherwise. Furthermore let if , and otherwise.

Consider now arbitrary . Assume first that and . Then and .

Suppose now that or . Consequently . We have shown that for the hypothesis of the condition of -hyperconvexity is satisfied.

We conclude that there is

Consequently . (Of course, similarly one could show that exists.) We deduce that is a complete lattice.

*Example 35. *Recall that is a -hyperconvex -quasimetric space (see [4, Example 1]). The specialization order of that space is the standard order on ; hence is not a complete lattice. So boundedness cannot be omitted in Proposition 34.

*Remark 36. *Let be a partially ordered set and its natural -quasimetric. If is -hyperconvex, then is a complete lattice. This is a consequence of the proof of Proposition 34 by setting , since we have .

Indeed, more generally, our next result shows explicitly how, given a partially ordered set equipped with its natural -quasimetric , the -hyperconvex hull of contains the Dedekind-MacNeille completion of .

Lemma 37. *Let be a partially ordered set and its natural -quasimetric. Furthermore let be the set of all those minimal ample function pairs on that only attain the values and .**Consider an arbitrary pair of functions . Then the following conditions are equivalent.*(a)*. *(b)* and .*(c)* and whenever .*

*Proof. * given , consider
(see from Section 2).

Since the functions and attain only the values and , we see that these equations are equivalent to . Indeed, given , we have that if and only if, for any implies that , and if and only if, for any implies that if and only if .

Similarly one verifies that is equivalent to whenever .

Since , condition is satisfied.

by we conclude that . Furthermore the second part of is equivalent to , as we have just noted above. Hence condition is satisfied.

: observe that and together imply that . But the latter equality is equivalent to whenever , as we have observed above. Thus , and condition holds.

Proposition 38. *Let be a partially ordered set with its natural -quasimetric , and let be defined as in Lemma 37.**Then the map defined by is an order isomorphism between (equipped with the specialization order induced on by the -quasimetric of the -hyperconvex hull of ) and the Dedekind-MacNeille completion of . Furthermore for each .*

*Proof. *By Lemma 37 each set belongs to the Dedekind-MacNeille completion, since . Moreover for each , . Also for each , obviously . The specialization order induced on by the -quasimetric is defined by if and only if whenever . Hence we have for any ,
if and only if

(Of course this means exactly that with respect to the usual pointwise order on real-valued functions.) In particular is injective. Furthermore for with we define so that and whenever . Then according to Lemma 37 and , and hence is surjective.

We conclude that the set can be identified with the ground set of the Dedekind-MacNeille completion of , and is an order isomorphism satisfying

*Remark 39. *Given a partially ordered set equipped with its natural -quasimetric and its -hyperconvex hull , the subspace identified above with in is obviously characterized by the property that it is the largest subspace of containing and such that the -quasimetric restricted to attains only values in .

*Example 40. *Let be equipped with the discrete order . As we have observed above, the natural -quasimetric on is the discrete metric. Furthermore can be identified with the set equipped with the -quasimetric
whenever , where is identified with and is identified with (see [6, Example 4]). Of course the Dedekind-MacNeille completion of consists only of the four corner points of endowed with the induced specialization order on (see Example in [9, page 169]).

*Problem 1. *Given a -quasimetric space , compare the set of the endpoints of with the set of the endpoints of .

In this paper we established that these two sets are equal if is compact.

#### 7. Extended -Quasimetrics

Given a partially ordered set , in the last three sections we worked with the natural -quasimetric of . The choice of the value in the definition of the natural -quasimetric looked rather arbitrary. A more canonical approach can be given if one uses extended -quasimetrics.

*Remark 41. *Let be a partially ordered set. For each , we set if and otherwise (see, e.g., [8, Section 2.1]). Obviously, is an extended -quasimetric on .

We note however that, for any , we have , but (so that is not closed under the operation of addition +). Therefore we obtain distinct theories of collinearity by considering or , respectively. For our purposes the -quasimetric seems to lead to a more interesting theory. We finally show how our characterizations of endpoints and startpoints (see Lemma 11 and Corollary 12) have to be modified to be applied to our new setting.

Proposition 42. *Let be a partially ordered set, its natural extended -quasimetric, and . Then is an endpoint with witness in if and only if one has .**Furthermore is a startpoint with witness in if and only if one has .*

*Proof. *Suppose that is an endpoint with witness in . Therefore , and, for all implies that . Hence . Consider any . Thus , which by our assumption implies that . Therefore .

For the converse suppose that . Then . Let be such that . Then and . Consequently . Thus is an endpoint with witness in . The dual result is proved analogously.

*Example 43. *Let us consider the set of the integers equipped with its usual linear order, and let be its natural -quasimetric and, its natural extended -quasimetric respectively. Then does not have any endpoints nor any startpoints by Proposition 42, while in each point is an endpoint and a startpoint (see Corollary 24).

#### 8. Conclusion

We showed that for any join compact -quasimetric space the set of endpoints (resp., startpoints) of is equal to the set of endpoints (resp., startpoints) of its -hyperconvex hull . We also specialized some of our earlier results on endpoints contained in [6] to two-valued -quasimetric spaces. In particular we observed that in the case of a complete lattice and its natural -quasimetric the startpoints (resp., endpoints) of are exactly the completely join-irreducible (resp., the completely meet-irreducible) elements. For a partially ordered set we finally explored the connection between its Dedekind-MacNeille completion and the -hyperconvex hull of its natural -quasimetric space.

#### Acknowledgments

The authors would like to thank the National Research Foundation of South Africa for partial financial support. This research was also supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme. Finally the authors would like to thank the referee for his or her suggestions.

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