Pretzel monoids

TL;DR

This paper introduces pretzel monoids, a new class of left adequate monoids constructed from birooted graphs, which serve as free idempotent-pure expansions of right cancellative monoids, linking algebraic and geometric perspectives.

Contribution

It defines pretzel monoids as a novel class of left adequate monoids, providing a geometric model and establishing their relation to right cancellative monoids and existing expansions.

Findings

Pretzel monoids are monoids of birooted graphs with a natural operation.

They are the free idempotent-pure expansions of right cancellative monoids.

The construction generalizes the geometric model of free left adequate monoids.

Abstract

We introduce an interesting class of left adequate monoids which we call pretzel monoids. These, on the one hand, are monoids of birooted graphs with respect to a natural `glue-and-fold' operation, and on the other hand, are shown to be defined in the category of left adequate monoids by a natural class of presentations. They are also shown to be the free idempotent-pure expansions of right cancellative monoids, making them, in some sense, the left adequate analogues of Margolis-Meakin expansions for inverse monoids. The construction recovers the second author's geometric model of free left adequate monoids when the right cancellative monoid is free.