EMBEDDING TOPOLOGICAL MEDIAN ALGEBRAS IN 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 there exists a two-dimensional metric continuum with a continuous median operator, for which there is no median-preserving embedding in a product of finitely many dendrons. Our method involves ideas and results 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 that m(a,b,b) = b for all a,beX. Such structures occur as median-stable subsets of distributive lattices; see [1]. If X is a topological space, then {X, m) is a topological median algebra provided that m is continuous. Weaker structures (where m is required to satisfy the absorption law only) have already been studied in [10,11] in relation to absolute retracts in topology. Median algebras have further connections with the theory of binary (Helly number 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]: a compact, connected median algebra of finite dimension is built up by a number of compact, connected, distributive lattices. It was shown in [15] that, if such a lattice is of dimension n, then it embeds as a sublattice in an n-cube. The question arose in [16] of whether a similar result holds for compact connected median algebras. As observed there, the cube should be replaced by a product of dendrons (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 three dendrons for such an embedding. A.M.S. (1980) subject classification: 05C10, 52A01, 54H99. Proc. London Math. Soc. (3) 58 (1989) 439-453.