Amenable Orders Associated with Inverse Transversals

If S is a regular semigroup with an inverse transversal S° = {x°; x S} then an order ≤ on S is said to be amenable with respect to S° if• (1) ≤ is compatible with the multiplication of ;• (2) on the idempotents, ≤ coincides with the natural order ≤;• (3) ≤ ° ≤ °, ° ≤°. 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° is a quasi-ideal). We give a complete description of all amenable orders on S and characterise the natural order ≤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°, whence every amenable order on S° extends to a unique amenable order on S.