Amenable orders associated with inverse transversals

If S is a regular semigroup with an inverse transversal S degrees = {x degrees ;x is an element of S} then an order less than or equal to on S is said to be amenable with respect to S degrees if

(1) less than or equal to is compatible with the multiplication of S;

(2) on the idempotents, less than or equal to coincides with the natural order less than or equal to (n):

(3) x less than or equal to y double right arrow x degreesx less than or equal to (n) y degreesy, xx degrees less than or equal to (n) yy degrees.

This notion is in fact independent of the choice of inverse transversal. Here we consider the case where S is locally inverse (equivalently, where S-degrees is a quasi-ideal). We give a complete description of all amenable orders on S and characterise the natural order less than or equal to (n) as the smallest of these. We also establish a bijection from the set of amenable orders definable on S to the set of McAlister cones of S degrees, whence every amenable order on S degrees extends to a unique amenable order on S. (C) 2001 Academic Press