On the decidability of the word problem for amalgamated free products of inverse semigroups. (Q2481330)

The paper is investigating the decidability of the word problem of the amalgamated free product \(S_1*_US_2\) of an inverse semigroup amalgam \([S_1,S_2;U]\). The authors extend the methods of \textit{A. Cherubini, J. Meakin} and \textit{B. Piochi} [Semigroup Forum 54, No. 2, 199-220 (1997; Zbl 0872.20049)], based on \textit{P. Bennett}'s results [J.~Algebra 198, No. 2, 499-537 (1997; Zbl 0890.20042], and establish a list of five conditions on an inverse semigroup amalgam that guarantee the decidability of the word problem. The proof is technically rather involved. At the end of the paper an example is presented of two (isomorphic) inverse semigroups \(S_1\), \(S_2\) given by an idempotent pure presentation, with a common free monogenic inverse subsemigroup \(U\) and show that their theorem applies, whence \(S_1*_US_2\) has decidable word problem. It is remarked that the semigroups \(S_i\) have infinite \(\mathcal R\)-classes (the latter has long been considered to be an obstacle for the decidability of the word problem).