Right cancellative and left ample monoids: Quasivarieties and proper covers

The aim of this paper is to study certain quasivarieties of left ample mono ids. Left ample monoids are monoids of partial one-one mappings of sets clo sed under the operation alpha bar right arrow alpha alpha(-1). The idempote nts of a left ample monoid form a semilattice and have a strong influence o n the structure BE the monoid; however, a left ample monoid need not be inv erse. 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 m onoids is well understood. Let Sr be a class of right cancellative monoids. A left ample monoid has a proper cover over V if it has a proper cover wit h maximum right cancellative image in V. We show that if Sr is a quasivarie ty determined within right cancellative monoids by equations, then the left ample monoids having a proper cover over V form a quasivariety. We achieve our aim using the technique of graph expansions to construct proper left a mple monoids from presentations of right cancellative monoids. (C) 2000 Aca demic Press.