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Finite presentability of HNN extensions of inverse semigroups. (English) Zbl 1088.20035
HNN extensions of inverse semigroups, in which the associated inverse subsemigroups are order ideals of the base, are defined through a construction based on the isomorphism between the categories of inverse semigroups and inductive groupoids. It is proved that if \(S\) is a finitely generated inverse semigroup then its HNN extension over the inverse subsemigroup \(U\) is finitely generated if and only if \(U\) is finitely generated and \(\mathcal J\)-dominated, which means that \(U\) has a subset of idempotents that dominate \(U\) with respect to the \(\mathcal J\)-order.
If \(S\) is finitely presented, a necessary and sufficient condition, which also incorporates \(\mathcal J\)-domination, is found for the corresponding HNN extension to be finitely presented. Interestingly, unlike the case for groups, it is not necessary that \(U\) is finitely generated and examples highlight this and a range of other behaviours of these HNN extensions.

20M18 Inverse semigroups
20M05 Free semigroups, generators and relations, word problems
20M50 Connections of semigroups with homological algebra and category theory
Full Text: DOI
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