Free products of combinatorial strict inverse semigroups (Q1328994)

Each combinatorial strict inverse semigroup \(S\) is determined by (1) a partially ordered set \(X\) which in fact is the partially ordered set of the \(\mathcal J\)-classes of \(S\), (2) pairwise disjoint sets \(I_ \alpha\) indexed by the elements of \(X\) which in fact form the collection of \(\mathcal D\)- (equivalently: \(\mathcal J\)-) related idempotents and (3) structure mappings \(f_{\alpha,\beta}: I_ \alpha \to I_ \beta\) for \(\alpha \geq \beta\) satisfying certain compatibility conditions. The multiplication on \(S\) can be described in terms of the parameters \(X\), \(I_ \alpha\), \(f_{\alpha,\beta}\). Conversely, the system \((X;I_ \alpha,f_{\alpha,\beta})\) can be characterized abstractly so that it defines a uniquely determined combinatorial strict inverse semigroup. In this paper, the constituting parameters \(X\), \(I_ \alpha\), \(f_{\alpha,\beta}\) of the combinatorial strict inverse free product \(S\) of a collection of combinatorial strict inverse semigroups \(S_ i\) are described in terms of the parameters of the semigroups \(S_ i\). As an application it is shown that the word problem for such a free product in general is not decidable.