Free products of inverse semigroups. II. (English) Zbl 0742.20057

[For part I see the first author, Trans. Am. Math. Soc. 282, 293-317 (1984; Zbl 0532.20032).]
The free product of inverse semigroups \(S\) and \(T\) denoted by \(S \hbox{inv} T\), is their coproduct in the category of inverse semigroups. Using the graph theoretical techniques developed by Stephen the authors describe \(S \hbox{inv} T\) in the case \(S\) and \(T\) are given by their presentations with sets of free generators and defining relations. Let \(X\) be a nonempty set and \(X^{-1}\) be a set of formal inverses of \(X\). For every word \(u\in (X\cup X^{-1})\), the so called Schützenberger graph \(S\Gamma(u)\) and the Schützenberger automaton \(A(u)\) are defined. Let \(S=\hbox{inv}(X\mid P)\) and \(T=\hbox{inv}(Y\mid Q)\) be disjoint presentations of inverse semigroups \(S\) and \(T\) resp. A procedure is described for obtaining the Schützenberger automaton \(A(w)\) from the linear automaton of \(w\in (X\cup X^{-1}\cup Y\cup Y^{-1})\) in a finite number of steps. As a result a set of canonical forms for \(S \hbox{inv} T\) is provided. As an application, it is obtained that the free product of two residually finite inverse semigroups is again residually finite.


20M05 Free semigroups, generators and relations, word problems
20M18 Inverse semigroups
20M35 Semigroups in automata theory, linguistics, etc.
05C90 Applications of graph theory


Zbl 0532.20032
Full Text: DOI


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