Embedding Topological Median Algebras in Products of Dendrons

EMBEDDING TOPOLOGICAL MEDIAN ALGEBRASIN PRODUCTS OF DENDRONS

H.-J. BANDELT and M. VAN DE VEL

[Received 13 November 1986—Revised 22 September 1987]

ABSTRACT

Dendrons and their products admit a natural, continuous median operator. We prove that thereexists a two-dimensional metric continuum with a continuous median operator, for which there is nomedian-preserving embedding in a product of finitely many dendrons. Our method involves ideas andresults concerning graph colouring and abstract convexity. The main result answers a question in [16]negatively, and is sharply contrasting with a result of Stralka [15] on embeddings of compact lattices.

0. Introduction

A median algebra is a set X together with a ternary operation

m: X^-^X

which, among other things, is symmetric and 'absorbing' in the sense thatm(a,b,b) = b for all a,beX. Such structures occur as median-stable subsets ofdistributive lattices; see [1]. If X is a topological space, then {X, m) is atopological median algebra provided that m is continuous. Weaker structures(where m is required to satisfy the absorption law only) have already been studiedin [10,11] in relation to absolute retracts in topology.

Median algebras have further connections with the theory of binary (Hellynumber 2) convexity [1, 7, 18], and with the theory of median graphs [1, 13, 14].Some more information will be given in § 1 below.

For the sake of motivation, let us loosely formulate a main result from [19]: acompact, connected median algebra of finite dimension is built up by a number ofcompact, connected, distributive lattices. It was shown in [15] that, if such alattice is of dimension n, then it embeds as a sublattice in an n-cube. Thequestion arose in [16] of whether a similar result holds for compact connectedmedian algebras. As observed there, the cube should be replaced by a product ofdendrons (which are the compact, connected median algebras of dimension 1). In[20] an example of dimension 2 was given (as suggested in Fig. 1) requiring threedendrons for such an embedding.

A.M.S. (1980) subject classification: 05C10, 52A01, 54H99.Proc. London Math. Soc. (3) 58 (1989) 439-453.

440 H.-J. BANDELT AND M. VAN DE VEL

Note that this space—a subalgebra of the 3-cube—is built up by five distributivelattices—squares actually. The point-labels will be explained later on.

For some time it seemed that this was the worst thing that could happen indimension 2. We now present a construction, leading to a 2-dimensional metricmedian continuum, which cannot be embedded in a product of finitely manydendrons. Surprisingly, the basis of this construction lies in considerationsconcerning vertex-colourings of finite graphs. Let us outline the main steps.

(1) For each graph F we construct a median graph G (called the simplex graphof F) such that the chromatic number of F equals the number of trees, minimallyneeded for a product in which G embeds. Also, G has no (graph) n-cubes forn > 2 if F is triangle-free. By a classical result in graph theory (see [21, p. 82 ffj),there exist triangle-free graphs of any chromatic number.

(2) As explained in [17], each median graph G can be 'realized' as a sort ofpolyhedron \G\ with a natural median structure by replacing each graph n-cubeby an affine n-cube. Then the polyhedra, obtained from (1), are 2-dimensional.After relating median graph embeddings in products of trees with embeddings ofmedian polyhedra in products of dendrons, and after constructing some inverselimit of median polyhedra, we arrive at the example announced earlier.

Section 1 below includes some basic information on the various structuresmentioned above. Sections 2 and 3 deal with Step (1) of our construction, and § 4deals with Step (2).

1. Preliminaries

Except for a small amount of topology (where we follow the terminology of [6])this paper exposes the reader to three closely related structures: median graphs,median algebras, and a class of binary convex structures. They really representdifferent viewpoints on one and the same object, but they all contributeessentially to the subsequent techniques and results.

1.1. Median algebrasA median (operator) on a set X is a function

m: X3^>X

with the following properties:(Ml) m(o(a), o(b), o(c)) = m(a, b, c) for each permutation o of (a, b, c)

(symmetry law);(M2) m{a, b, b) = b (absorption law);(M3) m(m(a, b, c), u, v) = m(a, m(b, u, v), m(c, u, v)).

The pair (X, m) is then called a median algebra. A subset Y of X with m(Y3) c Yis called median-stable. A function / : X-+X' between median algebras (X, m)and (X', m') is median-preserving (MP) provided

m\f(a),f(b)J(c))=f{m(a,b,c)).

With the obvious meaning of isomorphism, an embedding/: X-+X' of medianalgebras is an MP function giving an isomorphism between X and f(X).

EMBEDDING TOPOLOGICAL MEDIAN ALGEBRAS 4 4 1

For a general reference, see [1, 7].

1.2. GraphsAll graphs are assumed to be reflexive (i.e. with a loop at each point) and to

have no multiple edges. Let F be a graph. We say that Fis connected provided foreach x, y e F there is a sequence

ao = x,alt ...,an=y

of points in F {vertices) such that each pair (a,-, ai+l) forms an edge. This sequenceis a path joining x to y, and the number n is its length. The distance between twopoints x, y in a connected graph is defined as follows:

d(x, y) is the least possible length of a path joining x to y.

An induced subgraph is a subset with the induced edge-relation, and it is anisometric subgraph if its intrinsic distance function equals the subspace metric.With the obvious meaning of edge-preserving (EP) function and of isomorphismbetween two graphs, an embedding f: Fi—*F2of graphs is a function inducing anisomorphism between Fx and the induced subgraph f(F1).

A (vertex) colouring with n colours of a graph F is a partition {Flf..., Fn} of Finto subsets F{ (the ith colour) no two distinct points of which are joined by anedge. The chromatic number of F is the least number n such that F can becoloured with n colours. Clearly, if F includes an n-point simplex (a finite set inwhich all pairs form an edge), then the chromatic number of F is at least n. Wesometimes require a graph to be triangle-free, meaning that it does not contain a3-point simplex (triangle).

A median graph is a connected graph G such that for each triple of verticestfi> a2> a3 there is a unique vertex x eG with

Vi #y in {1, 2, 3}: d(ah x) + d(x, a,-) = d(ah a,).

Note that x lies simultaneously on a path of minimal length (a geodesic) from a, tofl; for each i±j. Median graphs do not include an n -point simplex for n5=3.Instead, they typically have induced subgraphs isomorphic to an n-cube (a graphof type {0, 1}", where two points form an edge if and only if they differ in at mostone coordinate); see [13, 14, 17].

1.3. Convex structuresLet <# be a collection of subsets of X closed under intersections and under

unions of updirected families (in particular, 0 , X e c€). Then Vo is a convexity onX, and (X, VD) is a convex structure. The members of *€ are referred to as convexsets. The convex hull co(A) of A c X is the smallest convex set including A. Asubset H of X is a half-space provided both H and X\H are convex. The subsets0 and A" are trivial examples of half-spaces. A convex structure satisfies theseparation property

St if all singletons are convex, andS2 if distinct points are in complementary half-spaces.

A collection Sf of sets is said to generate the convexity % of the set X provided^ c 1 ? and <# is the smallest convexity on X with this property. If (X, <€) is a

4 4 2 H.-J. BANDELT AND M. VAN DE VEL

convex structure and if Y^X, then the relative convexity of Y is the family{Cozy| Ce<£};see[8].

Let/: X->X' be a function between convex structures (X, <#), (X', <£'). Then/ is convexity-preserving (CP) provided f~\C) e % for each C" e <#'. With theobvious meaning of isomorphism, f is an embedding if it yields an isomorphismbetween X and the set f(X) with the relative convexity.

Let X be a convex structure (we omit reference to the convexity). A collection$? of half-spaces of X forms a direction provided each two half-spaces Hu H2e9€are compatible in the sense that either H1} H2 are comparable (that is, Hx c //2 or#2£H\)> o r #i> #2 a r e disjoint, or Hlt H2 are supplementary (that is, HX\JH2 =X); compare with [5, 2]. The reader should think of the ordinary convexity inEuclidean space to link this notion of direction with the intuitive idea. In a moregeneral form the notion appeared in [20]. We say that X is covered with ndirections provided there exist directions of half-spaces 9CU..., dfCn such that^ U . . . \jkn generates the convexity of X. Note that, if Jif is a direction ofhalf-spaces, then so is

X\J{X\H\

A convex structure is called binary provided each finite collection of pairwiseintersecting convex sets has a common point. For a general study of binary S2

convex structures, see [18].

1.4. Relations between the various structures(1) If A' is a median graph, then define m(a, b, c) as the unique point which is

simultaneously on a geodesic from a to b, b to c, and c to a, respectively. Theresulting pair (X, m) is a median algebra; see [1].

(2) Let {X, m) be a median algebra. Call a se tCcZconvex if

m(C xCxX)cC.

This results in a binary S2 convex structure; see [1]. The axiom (M3) plays anessential role in proving that

co{a, b} = {x\ m(a,b,x) = x}.

(3) Let A!" be a finite binary S2 convex structure. Then two points a, b eX aresaid to form an edge provided

co{a, b} = {a, b}.

The resulting graph is median; see [13, 17].(4) Let A!" be a binary S2 convex structure. For a, b, c e X, the set

co{a, b) n co{b, c} n co{c, a}

has exactly one point, which is denoted by m{a, b, c). The resulting pair (X, m) isa median algebra; see [1,18].

For a finite set X, the transitions described in (1), (2), (3) lead to a 1-1correspondence between the three kinds of structures. For an arbitrary set X, thetransitions (2) and (4) yield a 1-1 correspondence between median algebras andbinary S2 convexities on X. As an illustration of a composite transition, we notethat the convex sets of a median graph are exactly the geodesically convexsubsets.

EMBEDDING TOPOLOGICAL MEDIAN ALGEBRAS 443

As to the various morphisms introduced above, one should clearly distinguishbetween EP and MP, viz.: an EP function between median graphs need not beMP, and vice versa. The interested reader may verify the (non-trivial) fact that amedian-preserving surjection between median graphs must be edge-preserving.On the other hand, the MP functions correspond exactly to the CP functions; see[18]. The resulting notions of embedding correspond for median algebras andbinary S2 convex structures, but one must again be careful with embeddings ofmedian graphs.

1.5. Topological median algebrasWe now consider topological spaces X with a continuous median m: X3-*X.

One will not find any information in the literature under the heading 'topologicalmedian algebras'. But the subject has been studied in a disguised form as binaryconvex structures on a topological space, subject to the following two conditions:

(i) compactness of convex intervals co{a, b};(ii) existence of continuous CP functions into the unit interval (with the natural

convexity) separating pairs of disjoint convex closed sets.Condition (ii) is known as normality.

The following reformulations of results in [18, 19] establish the intendedrelationship.

(1) Let X be a compact space with a binary convexity satisfying (i) and (ii)above. Then the median of X is continuous.

(2) Let A!" be a topological median algebra with a compact, connected, andfinite-dimensional underlying space. Then the convexity of X satisfies (i) and (ii)(the topologically-minded reader may appreciate the additional result of [19] thatall of the usual topological dimension functions, and even the cohomologicaldimension function, coincide on compact median algebras).

In the sequel, we will freely 'translate' results on 'normal binary convexity'from the literature in accordance with the above quoted facts.

1.6. Dendrons, productsA dendron D is a compact connected space in which every two distinct points

get separated by the removal of some third point. The connected sets on Dconstitute a normal binary convexity, and hence D becomes a topological medianalgebra. It is known that there can be no other continuous median on D;see [9,12].

We will often have to deal with products of dendrons. In general, if (Xx, mx)and (X2, m2) are median algebras, then Xx x X2 is equipped with the median m,defined by

m((au a2), (bx, b2), (cx, c2)) = (mx(ax, bx, cx), m2(a2, b2, c2)).

The corresponding convexity is just the product convexity (see [8] for the latter)consisting of all sets of type Cx x C2, with C , c l ; convex.

For the sake of completeness, we describe a related concept of a Cartesianproduct of two graphs Fx, F2: two vertices of Fx X F2 form an edge if they differ inat most one coordinate, and if the ith coordinates form an edge in Fh for / = 1, 2.If Fx, F2 are median, then so is Fx X F2, and its derived median or convexity are ofthe above type.


2. The simplex graph of a general graph

2.1. Simplex graphsLet F be an arbitrary graph. Then F will denote the set of all simplices of F

(i.e. finite complete subgraphs of F). Two simplices of F form an edge if they arecomparable and differ in at most one point. The resulting graph F is called thesimplex graph of F. The first part of the next proposition is also mentioned in [4].

2.2. PROPOSITION. Let F be a graph. Then its simplex graph F is median, and thelargest possible dimension of a cube in F equals the largest number of points in asimplex of F (clique number of F).

Proof. Let 2F denote the collection of all finite subsets of F, with an edgerelation as above (for finite F, 2F is a copy of a graph cube). A path joiningS1)S2e2F can be regarded as a sequence of elementary operations, namely,adding or subtracting a point. For instance, one can first subtract the points ofS\\Si (yielding SiHS^ and then add the points of S NSj (yielding S^. Clearly,this is a geodesic between Sly S2, and d(Sx, S2) equals the number of points in thesymmetric difference oi S1,S2. Moreover, if S1} S^eF, then the above geodesicremains in F, showing that F is an isometric subgraph of 2F and is connected.

The median operator of 2F is given by the formula

m(su b, s,) = (Sx n u (s2 n s3) u (s3 n SO-

Note that, for Slt S2, S3eF, this yields another simplex of F. Hence F ismedian-stable in 2F, from which the first part of the theorem easily follows.

As to the second part, let 5 be a simplex of F. Then 2s is a subcube of F ofdimension #5. On the other hand, a subcube of F (hence of 2F) must have alargest member S. Clearly, the cube is included in 2s c F, and its dimension is atmost #5.

We use the following ad hoc notation. If 5 is a simplex of F, then

H(S)={T\ TeF,S^T}.

For 5 = {v} we simply write H(v). It is easily seen that H{S) is a geodesicallyconvex subset of F. We note from the above proof that, if Slt S2 e F, then

S2) = #((<S1\S2)U(S2\S1)).

It easily follows that any member S of a geodesic from 5X to 52 must satisfy

Si D Sz c S c Sl U S2.

2.3. LEMMA. Let F be a graph. Then the non-trivial half-spaces of the simplexgraph F are exactly the sets of type H(v) for v € F, and their complements.

Proof Let v e F and let Su 52 e F\H(v). Then v $ Sx U 52 and by the remarkabove, no S on a geodesic from Sx to 5 will contain the vertex v. It follows thatF\H(v) is geodesically convex.

Conversely, let H c F be a non-trivial half-space. Passing to its complement ifnecessary, we may assume that the empty simplex is not a member of H. For


Si, S2 e H we already know of a geodesic from St to 52 passing through St D S , soSiDSieH, whence S1nS2^0, by assumption. Then H, being downdirected,has a smallest member 50. One easily verifies that, for any simplex 5,

5eco{{u}| veS}.

As H is a half-space, we cannot have {t/} £ / / for each v eS0. So {u0} eH forsome u0e50, and then 50 = {u0}. This shows that H^H(v0). Consider anySeF\H. UvoeS, then

{vo}eco{0, S}gzF\H,

a contradiction. Hence S $ H(v0) and so H(v0) c H.

2.4. LEMMA. Let v¥^w be vertices of a graph F. Then H{v) and H(w) arecompatible if and only if v, w do not form an edge in F.

Proof. We have

{v} e H(v)\H(w), {w} e H(w)\H(v), 0 * H(v) U H(w).

Hence, H(v) and H(w) are compatible if and only if they are disjoint, that is, ifand only if v, w do not form an edge.

From Lemmas 2.3 and 2.4 we conclude:

2.5. THEOREM. For an arbitrary graph F the following are equivalent(1 =£«<«>);

(1) the chromatic number of F is at most n;(2) the simplex graph F can be covered with at most n directions.

To give an example we display the simplex graph of the complement of the7-cycle C7 in Fig. 2. By the complement F of a graph F one means the graph on

FIG. 2


the same vertex-set, in which two vertices are joined by an edge if and only ifthey are not adjacent in the original F. Observe that Fig. 1 is also a picture of agraph, which happens to be C5, the simplex graph of the 5-cycle C5. Its verticeshave been labelled in cyclic order: 1 to 5.

Examples of simplex graphs come up in the context of taxonomic models; see[3, 4]. For instance, the systems of n-trees and phylogenetic trees on a given setof objects can be regarded as simplex graphs.

3. Embeddings in Cartesian products of trees

We first recall that a graph is a tree if it is connected and has no cycles (i.e.there is no finite injective sequence of type a0, alf..., an (n> 1) such that eachpair (ait aI+1) as well as (an, a0) forms an edge). Such graphs arise from partiallyordered sets (X, =£) as follows: suppose that each order interval {y\ x^y^z} isfinite and each upper set {y\ x ^ v} is totally ordered {x, z e X). Two points a, bform an edge if they are comparable, and if no other point of X is between a, b.The resulting graph, if connected, is a tree.

We note that trees are median graphs with no n -cubes for n > 1; see [13].

3.1. CONSTRUCTION. Let X be a convex structure, and let $? be a direction ofhalf-spaces in X. Fix a point of reference p e X, and put

We say that %€is discrete if for each x eXand //0 e $?the number of half-spacesH in % with x e H c Ho is finite.

3.2. LEMMA. If Vt is discrete, then the partially ordered set (T(%€), c ) givesa tree.

Proof. Let HeT(%) and consider the upper set

Let Hlt H2eU{H). If Hx or H2 equals X, then they are comparable. IfHltH2±X, thenp $Hr UH2. As H*0, we also have HxnH2i= 0 . Since Hlt H2

are compatible, the only remaining possibility is that they are comparable.

3.3. CONSTRUCTION. Let X and $? be as above (with $? discrete). We constructa function

as follows:

JZ(X) is the smallest HeT(W) with xeH (xeX).

3.4. LEMMA. With the above notation, JZ is a well-defined function which is CP.If X is a median graph, then JZ is also EP.

Proof that K is well-defined. Certainly xeXeTffl). If xeHlfH2, whereHx,H2e T{2C), with HX^H2, and are both minimal with respect to this property,


then Hi, H2 are non-disjoint and non-comparable. Hence they are supplemen-tary. Then one of them contains the point of reference, p, and it must equal X, acontradiction.

Proof that n is CP. It suffices to show, for each pair of complementaryhalf-spaces of the tree T(3t), that one of them inverts to a half-space of X (thereason for this is that the half-spaces of a tree—or of any median algebra—generate the convexity; see [18]). So let T c T{%€) be a half-space. Without lossof generality, assume that 0^T¥^T(^C). Passing to its complement if necessary,we may also assume that X $ T. Being a convex set in (T(ffl), c ) the set T has alargest element Ho. Since T(ffl)\Tis also convex and contains X, it easily followsthat

Therefore, Jt~l(T) = Ho.Proof that n is EP in the case where X is a median graph. Let x1¥

zx2eX forman edge and assume without loss of generality that JZ{XX) =£ n(x2). Then xx and x2

are not in the same half-spaces of X constituting T(ffl), say: xx e Hx, x2 £ Hx forsome Hx eT(ffl). Now, two neighbours in X can be separated by only one pair ofcomplementary half-spaces, whence jt(xl) = Hl. Let H2eT{dfC) be such thatx2eH2. lfxi$H2, then by the above uniqueness remark we have H2 = G\Hi.But then p eH2, whence H2 = X, a contradiction. So xxeH2, and by theconstruction of n(xx), we find that Hx o H2. It follows that the smallest H2 econtaining x2 must be a neighbour of Hx in

This leads to an important intermediate result:

3.5. THEOREM. For a median graph G the following are equivalent:(1) there is an EP and CP embedding of G in a Cartesian product of n trees;(2) G can be covered with n directions.

Proof of (1) => (2). Suppose that G is a median induced subgraph of nr=i Th

where Tx,..., Tn are trees. As observed in § 1, the geodesic convexity of Il"=i Tt isjust the product convexity. The half-spaces of the latter are necessarily of typejtYx(H), where

is the ith projection and / / c 7] is a half-space. Now, a tree can be covered withone direction (all half-spaces are compatible; see, for instance, [5, 12]). HenceIl?=i T( can be covered with the n directions

% = {nr\H)\ H<=Tta half-space} (i = 1, ..., n).

Their traces on G yield n directions covering G.

Proof of (2)=>(1). Suppose G is covered with n directions VtX) ..., %6n ofhalf-spaces. Without loss of generality, we may assume that these families arepairwise disjoint. For each i, we construct a tree 7] = T ( ^ ) and a function

//: G-+Tt,

which is both edge- and convexity-preserving, as in Lemma 3.4. The product


function „

1=1

is again CP. We show that it is also EP. Let xi^y be neighbours in G. Thenfi(x),fi(y) form an edge in 7J, for i = 1,..., n. But, as x and y are separated by aunique half-space (which is in S£j;U... U %€n since this collection generates theconvexity) we obtain a unique index i for which f(x) =£/()>) (disjointness of thed/Ct), establishing our claim.

We next show that / is injective: suppose /(xi) = t = f(x2) for xx =£ x2. Then theconvex set / - 1 ( 0 includes a geodesic from xx to x2. Two neighbours on it are thenidentified by /, which is impossible by a previous argument.

An injective CP function between binary S2 convex structures is automaticallyan embedding of convex structures; see [18]. If t, t' e/(G) ('=£*') are neighboursin Il?=i Th then co{t, t'} = {t, t'} in the product convexity. The same is true forthe relative convexity, whence t, t' form an edge in/(G). We conclude that/(G)is an induced subgraph of the Cartesian product.

From Theorems 2.5 and 3.5 we obtain directly:

3.6. COROLLARY. For a graph F the following are equivalent:(1) the chromatic number of F is at most n;(2) there is an EP and CP embedding of the simplex graph F in a Cartesian

product of at most n trees.

It is time now to quote a well-known result in graph theory (cf. [21]): for eachnatural number n there is a finite graph Fn such that

(i) Fn is triangle-free,(ii) the chromatic number of Fn equals n.

By (i) and Proposition 2.2, the simplex graph Fn has no fc-cubes for k>2,whereas by (ii) and Corollary 3.6, it requires n trees for an embedding.

For instance, the graph indicated by Fig. 1 is a C5, and the 5-cycle C5 haschromatic number 3. Hence, the median graph C5 does not fit into a product ofless than three trees. On the other hand, the median continuum that is reallymeant by Fig. 1 does not embed in a product of less than three dendrons.Apparently, there is a deeper relationship between the two results, which will beexplained in the next section.

Since Fig. 1 represents what was previously known to be the worst case, it is ofinterest to have an explicit example of a finite median graph fitting only in aCartesian product of four trees. In Fig. 3 we exhibit a fairly simple graph F withchromatic number 4. The second part of the picture indicates the simplex graphF. The previous sections give detailed instructions for how to build a minimal setof tree factors. We have started with the colouring

{0}, {1, 6} {3, 5, 8, 10}, {2, 4, 7, 9}

and the 'point of reference' p = 0 to obtain the factor trees Tu ...,TA indicated inthe third part of the picture. The corresponding coordinate projections can becomputed explicitly. So, we obtain an EP and CP embedding in {-1,0,1}6 aswell, and the picture of P attempts to give a perspective view on this spatial copy.

(a)

EMBEDDING TOPOLOGICAL MEDIAN ALGEBRAS

9

449

(b)

(c)

*10

4 *

FIG. 3

4. Geometric realization

We now describe how the discrete results of §§2 and 3 can be turned intotopological results. The main ingredient is a procedure of [17] called geometricrealization. We give a description below, and extend the procedure to functions.


4.1. The processLet G be a finite median graph. For each n-cube C c G w e consider an affine

n-cube \C\ of which the corner point graph is isomorphic to C. One should takecare that, if Clt C2c.G are cubes, then

n \c2\ = \c1nc2\.The topological space \G\ is defined by

|G| = U { | C | | C c G a c u b e } .Since G is connected as a graph, \G\ will be a continuum. It was shown in [17]that \G\ admits a unique continuous median with the following convexityproperties:

(i) for each cube C^G, the relative convexity on \C\ is the subcubeconvexity;

(ii) if D c G is convex, then \D\ c \G\ is convex.

What we need now is a way to extend CP functions to the realizations. To thisend, we first give a more precise description of the set \G\. Fix any EP and CPembedding of G in a cube Q = {0,1}". If C c G is a graph cube, then we let \C\denote the (standard) convex hull of the embedded copy of C in \Q\ = [0, 1]", andconstruct \G\ as above.

4.2. PROPOSITION. Let G be a median-stable induced subgraph of a cube Q.Then \G\is a median-stable subset of \Q\, and its relative convexity satisfies (i) and(ii) above. In particular, \G\ is the geometric realization of G.

Proof. We argue by induction. Assume the result to be valid for graphs withfewer than k vertices, and that #G = k>\. If G is a graph cube itself, then theresult is obvious. If G is not a graph cube, then, as in the proof of [17, Theorem5.3] there exist overlapping convex sets Glf G2 <= G with G1UG2 = G, such thateach graph cube of G is included in Gx or in G2. In particular, \G\ = \Gi\ U \G2\.Since G is median-stable in Q, each convex set of G is relatively convex: thereexist subcubes Qu Q2 of Q with

Then also

|G,| = \G\ n \Q,\,

and by the induction hypothesis, |G,| is median-stable in \Q{\ and its relativeconvexity satisfies (i), (ii).

Take x,y, z e\G\. If all three points are in the same |G,|, then m(x, y, z)e\Gi\c\G\ since |G,| is median-stable. So assume, for example, that x, y e \Gi\ andz e\G2\. By [17] we can consider canonical CP functions between (relatively)convex sets (the so-called nearest point projections)

p: IGal-HdnGil, p': \Q2\^\QinQ2\.

By [18, 2.12], p is the restriction of/?', whereas by [18, 2.10],

m{x, v, z) = m{x, y,p'(z)).


Now, p'(z) =p(z) e \GX n G2\, and m(x, y, p{z)) e \GX\ since the latter is median-stable. So m(x, y, z) e \G\.

It is clear that the relative convexity of \G\ in \Q\ satisfies (i). As to (ii),let C^G be convex. Then C = GHQ' for some cube Q ' c Q , whence \C\ =\G\ n \Q'\ is relatively convex in \G\. This establishes the proposition.

Now, let / : GX->G2 be an EP and CP function between finite median graphsGlf G2. We wish to 'realize' / as a continuous CP function |G1|->|G2|. First,assume/to be surjective (then the condition 'EP' is redundant, cf. 1.4). Consideran EP and CP embedding

such that the coordinate projections

* , : & " • {0,1}

map G2 onto {0,1}. Put % = {Hh Hi}, where

Then S€{ consists of two non-empty, complementary half-spaces of Gx. To this listof directions, we add all remaining couples of non-empty, complementaryhalf-spaces of Gx, say:

Wn+\ = {Hn+\> Hl,+i}, ..., 2€m = {Hm, H'm}.

The techniques of § 3 then yield an EP and CP embedding

where the ith factor {0,1} is a copy of the tree T(%£/). If, for i = 1, ..., n, we takethe 'reference point' p in Hiy say, then T{df€t) = {Hif G2} and we identify 0 withHh 1 with G2. In this way, the projection

g- Qi->Qi> g{tx,...,tn,tn+l,...,tm) = {tx,...,tn)

extends/: Gx—* G2. Now g extends to a corresponding projection of solid cubes,

\8\: IG1I-HG2I,which evidently maps \GX\ into \G2\. We define | / | to be the resulting restriction.Note that with this 'external' procedure, the verification of | / | being CP iscompletely trivial. The same goes for continuity.

If / is not surjective, then replace G2 by G'2=f(G\)- This is a median inducedsubgraph because / is CP and EP. Now / extends to a continuous CP function\GX\—>|G2|. By virtue of Proposition 4.2 there is a canonical CP embedding|G2|—»|G2|. The composition of the latter two functions gives the describedcontinuous CP function |/ |.

4.3. THEOREM. For a finite median graph G, the following are equivalent:(1) there is an EP and CP embedding of G in a Cartesian product of n trees;(2) there is a topological and CP embedding of\G\ in a product of n dendrons.

Proof of (1) => (2). Without loss of generality, we may assume that the factor


trees are finite (otherwise, the convex hull of the embedded G is a product set,with finite subtrees as factors). The geometric realization of a finite tree isobviously a dendron. Finally, the reader should convince himself that

-nisi1=1

for finite trees (or even median graphs) Tt.Proof of (2)^>(1). This requires more care. Suppose |G| is CP embedded as a

subspace of n?=i A> where each A is a dendron. Let

be the ith projection. Then

*if: |G|->D,

is median-preserving. Since G is median-stable in \G\ by Proposition 4.2 we findthat Tt = Jtif{G) c A is also median-stable. Hence this finite set has an intrinsicmedian graph structure, and its geodesic convexity is the relative convexity fromA- It is then clear that Tt has no graph fc-cubes for k>\, whence T( is a tree.

Now we have n CP functions/: G-* Tt yielding a CP embedding

G-+f[Tt.i = l

We turn it into a CP and EP embedding as follows. Each tree 7] is covered by onedirection of half-spaces, which we invert with / into a direction of half-spaces ^in G. Note that U?=i % generates the convexity of G since G is CP embedded inthe Cartesian product. Then proceed as in the proof of Theorem 3.5: turn the 3£into a set of pairwise disjoint directions $?,', with U"=i %!= U?=i %> constructtrees T[ = T(%',), and produce a CP and EP embedding G-*n?=1 T\.

4.4. MAIN THEOREM. There is a compact, metric, connected median algebra ofdimension 2, which has no topological and CP embedding in a product of finitelymany dendrons.

Proof. Let Fn, with n ^ 2 , be triangle-free finite graphs such that Fn haschromatic number n. Put

(disjoint sum; there are no edges between the points of different pieces). Notethat Gn - F'n then consists of two pieces F ^ and Fn, meeting in the centralvertex ' 0 ' only. Hence both parts are geodesically convex. In particular, cf. [17],there is a CP function (nearest-point projection)

which is also a retraction (that is, fn(x) = x for x e Gn_x c Gn). In particular, fn issurjective (hence EP). This yields an inverse sequence


the inverse limit of which is a metric continuum X. As each Fn is triangle-free, wefind that each \Gn\ is of dimension 2, and hence so is the limit X (see [6]). Allbonding maps \fn\ are continuous and CP (or MP), and it is then easy to constructa continuous median on X. Since \fn\ has a right inverse (viz., the inclusion\Gn-x\ c \Gn\) we can embed each \Gn\ in X as a (convex) subalgebra. It easilyfollows that X has no topological and CP embedding in a product of n dendronsfor any n.

References

1. H.-J. BANDELT and J. HEDLIKOVA, 'Median algebras', Discrete Math. 45 (1983) 1-30.2. J.-P. BARTHELEMY, 'From copair hypergraphs to median graphs with latent vertices', Discrete

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comparison and consensus of classifications', preprint.5. P. BUNEMAN, 'The recovery of trees from measures of dissimilarity', Mathematics in the

archaeological and historical sciences (eds F. R. Hodson, D. G. Kendell, P. Tautu, EdinburghUniversity Press, 1971), pp. 387-395.

6. R. ENGELKING, General topology (PWN-Polish Scientific Publishers, Warszawa, 1977).7. J. R. ISBELL, 'Median algebra', Trans. Amer. Math. Soc. 260 (1980) 319-362.8. R. E. JAMISON, 'A general theory of convexity', Dissertation, University of Washington, Seattle,

1974.9. J. VAN MILL and M. VAN DE VEL, 'Convexity preserving mappings in subbase convexity theory',

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Appl. 13 (1982) 59-68.12. J. VAN MILL and E. WATTEL, 'Dendrons', MC tract 142 (Mathematisch Centrum, Amsterdam,

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629-640.16. M. VAN DE VEL, 'On the rank of a topological convexity', Fund. Math. 119 (1983) 101-132.17. M. VAN DE VEL, 'Matching binary convexities', Topology Appl. 16 (1983) 207-235.18. M. VAN DE VEL, 'Binary convexities and distributive lattices', Proc. London Math. Soc. (3) 48

(1984) 1-33.19. M. VAN DE VEL, 'Dimension of binary convex structures', Proc. London Math. Soc. (3) 48 (1984)

34-54.20. M. VAN DE VEL, 'Metrizability of finite dimensional spaces with a binary convexity', Canad. J.

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1974).

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Embedding Topological Median Algebras in Products of Dendrons

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