Amalgams of finite inverse semigroups and deterministic context-free languages

Let S be the amalgamated free product of two finite inverse semigroups. We prove that the Schützenberger graph of an element of S with respect to a standard presentation of S is a context-free graph in the sense of Müller and Schupp (Theor. Comput. Sci. 37:51-75, 1985), showing that the language L recognized by the Schützenberger automaton is context-free. Moreover we construct the grammar generating L proving that L is a deterministic context-free language and we use this fact for solving some algorithmic problems.