Embedding theorems for HNN extensions of inverse semigroups

A variant of an HNN extension of an inverse semigroup introduced by Gilbert [N.D. Gilbert, HNN extensions of inverse semigroups and groupoids, J. Algebra 272 (2004) 27–45] is defined provided that associated subsemigroups are order ideals. We show this presentation still makes sense without the assumption on associated subsemigroups in the sense that it gives a semigroup deserving to be an HNN extension, and it is embedded into another variant using the automata theoretical technique based on combinatorial and geometrical properties of Schützenberger graphs.