On dominions of certain ample monoids

A semigroup S is called left ample if it can be embedded in the symmetric inverse semigroup IX 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 IX. 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 IX coincides with the inverse subsemigroup of IX generated by it. We then determine the dominions of certain submonoids of In, the symmetric inverse semigroup over a finite chain 1 < 2 < · · · < n.