Geometry of Autometrized Lattice Ordered Monoids

In this paper, we study the geometry of AL-monoids by introducing the concept of metric betweeness and its properties t1, t2, B-linearity, D-linearity, lattice betweeness, B-linearity, and Dlinearity, segments and equilateral triangles. It is proved that there do not exist equilateral triangles in AL-monoids. It is also proved that any AL-monoid is ptolemaic.