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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.

MSC:

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

Citations:

Zbl 0532.20032
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References:

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