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Inverse monoids and rational Schreier subsets of the free group. (English) Zbl 0757.20017

A Schreier subset of the free group \(FG(X)\) is a subset \(L\) of \(FG(X)\) which contains the identity element and which contains every prefix of the reduced word of each element of \(L\). If \(M\) is an inverse monoid presented by the set \(X\) of generators and a set \(T=\{(e_ i,f_ i): i\in I\}\) of relations where \(e_ i,f_ i\) are idempotents of the free inverse monoid \(\text{FIM}(X)\), then the natural homomorphism \(\sigma\) of \(M\) onto its greatest group homomorphic image \(FG(X)\) embeds each \(\mathcal R\)-class \(R\) of \(M\) into \(FG(X)\) and \(R\sigma\) is a Schreier subset of \(FG(X)\). In this case \(R\sigma\) is said to be naturally associated with \(\text{FIM}(X)\). Such Schreier subsets \(L\) of \(FG(X)\) are characterized in the main theorem of this paper, in terms of a finite test tree for the subtree of the Cayley graph \(\Gamma(X)\) of \(FG(X)\) whose vertices are labeled by the elements of \(L\). If \(I\) is finite, then \(R\sigma\) is a rational subset of \(FG(X)\), but not every rational Schreier subset of \(FG(X)\) is obtained in this way. Every rational Schreier subset of \(FG(X)\), with \(X\) finite, whose associated subtree of \(\Gamma(X)\) contains only a finite number of negatively labeled edges, is naturally associated with \(\text{FIM}(X)\).

MSC:

20M05 Free semigroups, generators and relations, word problems
20M18 Inverse semigroups
20M35 Semigroups in automata theory, linguistics, etc.
20E05 Free nonabelian groups
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