Abstract
We characterize aperiodic relational morphisms as those that are injective on regular $\H$-classes. This result is applied to obtain simple proofs and generalizations of McAlister’s results on joins of aperiodic semigroups and groups. Also, we show that if $\pv H$ is a proper, non-trivial pseudovariety of groups, then \[\pv A\ast \pv H\subsetneq (\pv A\ast \pv G)\cap \ov {\pv H}.\] We provide coordinate-free formulations and proofs of Rhodes’s Presentation Lemma and generalizations. As an application, we give simpler proofs of Tilson’s theorem on the complexity of semigroups with at most $2$ non-zero $\J$-classes and Rhodes’s theorem that complexity is not local.
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Steinberg, B. On Aperiodic Relational Morphisms. Semigroup Forum 70, 1–43 (2005). https://doi.org/10.1007/s00233-004-0148-7
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DOI: https://doi.org/10.1007/s00233-004-0148-7