Trees, lattices, order, and betweenness
Marlow Sholander · Proceedings of the American Mathematical Society · 1952
In this paper we consider postulates expressed in terms of "segments," "medians," and "betweenness."Characterizations are obtained for trees, lattices, and partially ordered sets.In general a characterization is given by a system of three postulates.These systems fall in pairs; systems of a pair have two postulates in common.An algebra which has both lattices and trees as special cases is given in the final section. 1. Segments.Consider a set S of elements a, b, c, • • ■ such that to each pair a, b, of elements in 5 there corresponds a unique subset of 5 denoted by (a, b) and called the segment from a to b.By assumption, these segments have as properties: (S) To each set of three elements a, b, and c, there corresponds an element d such that (a, b)f~\(b, c) = (¿>, d).(T) (a, b)C(a, c) implies (a, ¿>)Pi(&, c) = {&}.The segments of Duthie [l]2 are segments in this sense (see §3).Paths in a tree are also segments in this sense (see §2).Setting a = b=c in (T) we have(1.1) {a} = (a, a).From this, and from (T) with b=c, we have(1.2) be(a,b).(1.3) ae(a,b).Proof.From (S), we may choose d so that (a, a)C\(a, b) = (a, d).By (1.1) and (1.2), d(£(a, d)C.(a, a) = {a}.Hence d = a and a£(a, d) C(a, b).(1.4) (a, b) = (b,a).Proof.From (S), we may choose d so that (a, b)f\(b, a) = (&, d).By (1.2) and (1.3), aE(b, d).By (T), (b, d)C\{d, a)={d}.Hence a£_{d}, a = d, and (b, a) = {b, d)Q{a, b).By symmetry, (a, b)C(b, a).(1.5) b G (a, c) if and only if (a, b) C (a, c).