Axiomatization and undecidability results for metrizable betweenness relations

Abstract

Abstract. Let d be a metric on a nonempty set A . The ternary betweenness relation T¡¡ induced by d on A is defined by Td{x,y,z) o d{x,y) + d{y, z) = d(x, z) for x, y, z e A. Allowing the range of d to vary over some "reasonable" ordered additive algebraic structures (not just the real numbers), we will prove that the class JK of all metrizable ternary structures, i.e., the class of all structures (A, Tj), where d is some metric on A, is an elementary class which can be axiomatized by a set of universal Horn sentences. Further, using an algorithm of linear programming, we will show that the first-order theory of Jt is recursively axiomatizable and its universal part is decidable. On the other hand, the theory of J( is not finitely axiomatizable and the theory of finite members of JK is hereditarily undecidable.In his fundamental work Grundlagen der Geometrie [H 1899] David Hubert introduced the ternary relation of betweenness which he used in formulating the second group of his axioms, namely the axioms of order. (In fact " y is between x and z " in Hilbert's formulation corresponds to " y lies between x and z and x ^ y ^ z" in ours.) Later on Alfred Tarski [T 1959] used the betweenness relation and the quaternary equidistance or congruence relation (i.e., "the distance of x, y and z, u is the same") to obtain a complete and decidable axiomatization of elementary geometry. Since the 1920s the notion of betweenness became of interest in its own right, as it naturally occurred in diverse mathematical structures, like linearly and partially ordered sets and lattices (see [B 1967[B ], [Bl 1953, metric spaces (see [M 1928[M ], [Bl 1953, linear spaces over ordered fields and normed linear spaces (see [Sm 1943], [Bl 1953]), etc. However, because of its intuitively clear geometrical meaning acquired in some "nice", though by far not all, metric spaces, the metric betweenness, defined as above, still has kept its central position. So far the paper [Mo 1977] by M. Moszyñska seems to be the only contribution to the study of the first-order theory of betweenness and equidistance relations in metric spaces.

Geometric Optimization of the Evaluation of Finite Element Matrices

“…The theory of ordered ternary relations has been used to provide a notion of betweenness for some time [15]. Recent applications have also appeared in [17], which is based on a technical report [16] that provides a historical survey of the use of ordered ternary relations as a type of geometry, as does [15]. In [15], ternary relations associated with a metric d are studied.…”

Metrizable and $\mathbb {R}$-metrizable betweenness spaces

“…As proved by R. Mendris and P. Zlatoš in [1], the class M of all metrizable betweenness spaces, i.e., the class of all first-order structures of the form (A, T d ), where d is a metric on A, is a universal elementary one. On the other hand, being not closed under elementary extensions, the class M 0 of all R-metrizable betweenness spaces is not elementary.…”

Pretrees and the shadow topology

“…The interval spaces satisfying (A 1 2 ), (A4), and (A5) were studied independently (see, e.g., [37, I, §4] and [19,24,11]). As in [37], such interval spaces will be called geometric.…”

“…The class E F (M) (in contrast to the pretrees) is not finitely based (it cannot be defined by a finite list of forbidden subspaces) and does not coincide with E k (M) for any k ∈ N (see [24], the proof 8 of assertion (v) of the main theorem). Let E F (M) be the class formed by the interval spaces all of whose finite subspaces are metrizable, and let E k (M) be the class of interval spaces all of whose subspaces of cardinality at most k are metrizable.…”

“…24 See Definition 2.11. (Indeed, if a point s ∈ T is not terminal, then there exist v, w ∈ T such that s ∈ v, w , whence by (A3) it follows that s lies in [a, v ∪ [a, w so that by (A4) we have either s < a v or s < a w, i.e., s is not a maximal element.)…”