Geometric algebra for sets with betweenness relations

Given a betweenness relation on a nonempty set E, a certain abelian group T=TE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}= {{\mathbb {T}}}_E$$\end{document} given in terms of generators and relations is investigated. This group controls the given betweenness relation in an algebraic form. That is, the group structure algebraically unfolds geometric relations, and in turn allows us to read off geometric properties from algebraic relations emerging from them. The most important examples for betweenness relations arise from ordered sets on the one side and from intervals in metric spaces on the other side. The structure of T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}$$\end{document} will be determined completely in case of totally ordered sets as well as for several classes of metric spaces.