A finiteness condition for \(\mathcal R\)-classes of amalgams of inverse semigroups (Q2755040)

Let \(S_1\) and \(S_2\) be two inverse semigroups and suppose \(U=S_1\cap S_2\) is a non-empty common subsemigroup such that, for any two idempotents \(g\) and \(h\) in~\(U\), \(J_g\geq J_h\) in~\(S_1\) if and only if \(J_g\geq J_h\) in~\(S_2\). Suppose further that the amalgam \([S_1,S_2;U]\) is lower bounded in the sense of \textit{P. Bennett} [J. Algebra 198, No. 2, 499-537 (1997; Zbl 0890.20042)]. Under these hypotheses, the paper establishes a necessary and sufficent condition for finiteness of \(\mathcal R\)-classes of the amalgamated free product \(S=S_1*_US_2\). This finiteness condition implies that the Schützenberger automata of words of~\(S\) are effectively constructible and therefore that the word problem for~\(S\) is solvable. A different necessary and sufficient condition for finiteness of \(\mathcal R\)-classes of~\(S\) was obtained earlier by \textit{P. Bennett} [Int. J. Algebra Comput. 7, No. 5, 577-604 (1997; Zbl 0892.20033)].