  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.
