Multilinear equations in amalgams of finite inverse semigroups

Let S = S_1 ∗_U S_2 = Inv<X;R> be the free amalgamated product of the finite inverse semigroups S_1 , S_2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations w_L ≡ w_R with w_L,w_R ∈ (X ∪ X^{−1} ∪ Ξ ∪ Ξ^{−1})^+ such that each x ∈ Ξ labels at most one edge in the Schutzenberger automaton of either w_L or w_R relative to the presentation <X ∪Ξ|R>. We prove that the satisfiability problem for such equations is decidable using a normal form of the words w_L, w_R and the fact that the language recognized by the Schutzenberger automaton of any word in (X ∪ X^{−1})^+ relative to the presentation <X|R> is context-free.