The Geometry Monoid

Abstract

Studying free LD-systems amounts to studying LD-equivalence of terms. Now t= _LD t’ holds ift can be transformed into t’ by iteratively applying the left self-distributivity identity, which can be seen as applying some operator that specifies where and how (from left to right or from right to left) the identity is applied. Applying the identity several times amounts to composing the associated operators. So we obtain a monoid of (partial) operators acting on terms, so that the LD-equivalence class of a term is its orbit under the action. The aim of this chapter is to study the monoid G_LDinvolved in this action, which we call the geometry monoid of(LD)as it captures a number of geometrical relations involving left self-distributivity.