Locally Inverse Semigroups with Inverse Transversals

Abstract:

      Let $S$ be a locally inverse semigroup with an inverse transversal $S^{\circ}$. In this paper, we construct an amenable partial order on $S$ by an $R$-cone. Conversely, every amenable partial order on $S$ can be constructed in this way. We give some properties of a locally inverse semigroup with a Clifford transversal. In particular, if $S$ is a locally inverse semigroup with a Clifford transversal, then there is an order-preserving bijection from the set of all amenable partial orders on $S$ to the set of all $R$-cones of $S$.