On dominions of certain ample monoids

Abstract

A semigroup S is called left ample if it can be embedded in the symmetric inverse semigroup I_X of partial bijections of a non-empty set X such that the image of S is closed under the unary operation α → αα⁻¹, where α⁻¹ is the inverse of α in I_X. Right ample semigroups are defined dually. A semigroup is called ample if it is both left and right ample. A monoid is (left, right) ample if it is (left, right) ample as a semigroup. We observe that the dominion of an ample subsemigroup of I_X coincides with the inverse subsemigroup of I_X generated by it. We then determine the dominions of certain submonoids of I_n, the symmetric inverse semigroup over a finite chain 1<2<⋯<n.