Road systems and betweenness
Paul Bankston · Bulletin of Mathematical Sciences · 2013
A road system is a collection of subsets of a set—the roads—such that every singleton subset is a road in the system and every doubleton subset is contained in a road. The induced ternary (betweenness) relation is defined by saying that a point $$c$$ lies between points $$a$$ and $$b$$ if $$c$$ is an element of every road that contains both $$a$$ and $$b$$ . Traditionally, betweenness relations have arisen from a plethora of other structures on a given set, reflecting intuitions that range from the order-theoretic to the geometric and topological. In this paper we initiate a study of road systems as a simple mechanism by means of which a large majority of the classical interpretations of betweenness are induced in a uniform way.