  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.
