Pretzel Monoids

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.