Amalgamation for Inverse and Generalized Inverse Semigroups

Abstract

For any amalgam $(S, T; U)$ of inverse semigroups, it is shown that the natural partial order on $S{{\ast }_U}T$, the (inverse semigroup) free product of $S$ and $T$ amalgamating $U$, has a simple form on $S \cup T$. In particular, it follows that the semilattice of $S{{\ast }_U}T$ is a bundled semilattice of the corresponding semilattice amalgam $(E(S), E(T); E(U))$; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.