Right Cancellative and Left Ample Monoids: Quasivarieties and Proper Covers,

The aim of this paper is to study certain quasivarieties of left ample monoids. Left ample monoids are monoids of partial one–one mappings of sets closed under the operation α αα − 1. The idempotents of a left ample monoid form a semilattice and have a strong influence on the structure of the monoid; however, a left ample monoid need not be inverse. Every left ample monoid has a maximum right cancellative image and a proper cover which is also left ample. The structure of proper left ample monoids is well understood. Let be a class of right cancellative monoids. A left ample monoid has a proper cover over if it has a proper cover with maximum right cancellative image in . We show that if is a quasivariety determined within right cancellative monoids by equations, then the left ample monoids having a proper cover over form a quasivariety. We achieve our aim using the technique of graph expansions to construct proper left ample monoids from presentations of right cancellative monoids.