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Maximal subgroups of amalgams of finite inverse semigroups. (English) Zbl 1332.20060

Let \(S=S_1*_US_2\) be an amalgamated free product of finite inverse semigroups \(S_1\) and \(S_2\). It is proved that a maximal subgroup \(H_e\), \(e\in E(S)\), is either isomorphic or a homomorphic image to the group \(H_g\subseteq S_i\), for some \(g\in E(S_i)\), \(i\in\{1,2\}\), depending on whether \(e\) is \(\mathcal D\)-related in \(S\) to some idempotent from \(S_1\) or \(S_2\), or is not. All these groups are effectively computable.
Schützenberger graphs (and automata) are used for describing amalgams of finite inverse semigroups: for an inverse semigroup \(S=\text{Inv}\langle X;T\rangle\simeq(X\cup X^{-1})^+/\tau\), the Schützenberger graph \(S\Gamma(X,T;w)\) for a word \(w\in (X\cup X^{-1})^+\) has the \(\mathcal R\)-class of \(w\tau\) in \(S\) for its set of vertices and all the triples \((s,x,t)\) with \(x\in X\cup X^{-1}\) and \(s\cdot x\tau=t\) for its set of edges. This graph can be considered as a presentation of an automaton, defining the vertex \(ww^{-1}\tau\) as the initial state and the vertex \(w\tau\) as the terminal state. Necessary and sufficient conditions for \(H_e\), \(e\in E(S)\), to be infinite are also found for the case when \(e\mathcal Df\) in \(S\) for some \(f\in E(U)\).

MSC:

20M18 Inverse semigroups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E08 Groups acting on trees
20M35 Semigroups in automata theory, linguistics, etc.
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C05 Trees
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References:

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