Free left and right adequate semigroups

TL;DR

This paper explores the geometric structure of free left and right adequate semigroups and monoids, providing embeddings into the two-sided case and analyzing their structural properties using labelled directed trees.

Contribution

It introduces natural embeddings of free left and right adequate semigroups into the two-sided case and simplifies their geometric representation.

Findings

Embedded free left and right adequate semigroups into the two-sided case

Provided explicit descriptions of the tree classes for these semigroups

Established basic structural properties of free left and right adequate semigroups

Abstract

Recent research of the author has given an explicit geometric description of free (two-sided) adequate semigroups and monoids, as sets of labelled directed trees under a natural combinatorial multiplication. In this paper we show that there are natural embeddings of each free right adequate and free left adequate semigroup or monoid into the corresponding free adequate semigroup or monoid. The corresponding classes of trees are easily described and the resulting geometric representation of free left adequate and free right adequate semigroups is even easier to understand than that in the two-sided case. We use it to establish some basic structural properties of free left and right adequate semigroups and monoids.