Amalgams vs Yamamura's HNN-extensions of inverse semigroups

We investigate the connections between amalgams and Yamamura's HNN-extensions of inverse semigroups. In particular, we prove that amalgams of inverse semigroups with an identity adjoint are quotient semigroups of some special Yamamura's HNN-extensions. As a consequence, we show how to obtain the Schutzenberger graph of a word w with respect to the presentation of an amalgamated free product starting from the Schutzenberger graph of a suitable word w' with respect to the presentation of the associated HNN-extension by simply making V-quotients and deletions of edges.