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On the structure of semigroups whose regular elements are completely regular

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Abstract

We study the structure of strongly 2-chained semigroups, which can be defined alternatively as semigroups whose regular elements are completely regular. The main result is a semilattice decomposition of these semigroups in terms of ideal extensions of completely simple semigroups by poor semigroups and idempotent-free semigroups.

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Notes

  1. For idempotent-generated semigroups however, two idempotents are isomorphic if and only if they are related by a (possibly long) chain of associate idempotents. This follows from Fitz-Gerald results on regular products of idempotents [26].

  2. Incidentally, we can deduce from the previous equations that \(x^\#=ex^\#e\), so that, ultimately, \(x''=x^\#\).

  3. Actually, Tamura proved in 1966 [77] that the set of identities \({x^2 = x, xy = yx}\), which defines semilattices, is the only proper set of identities, T, that provides for any semigroup a T-decomposition into T-indecomposable subsemigroups. This major result is a compelling explanation for the prominent role that semilattice decompositions play in the structure theory of semigroups.

  4. Lallement’s lemma states that every idempotent congruence class of a regular semigroup contains an idempotent.

  5. If S is not idempotent-generated, then the result holds but without the surjectivity assumption. In this case, the image of \(IG\left( {\mathcal {E}}\right) \) by \(\psi \) is the subsemigroup \(S'=\langle E\rangle \) of S generated by its idempotents.

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Acknowledgements

This research was conducted as part of the Labex MME-DII project (ANR11-LBX-0023-01) within the FP2M federation (CNRS FR 2036).

We extend our deepest gratitude to the referee for their meticulous review, insightful feedback, and valuable suggestions. Their contributions have greatly improved the quality of this article.

We would also like to warmly thank I. Dolinka and J. East for graciously sharing their egg-boxes pictures (Figs. 5 and 6).

The material presented in this article was originally shared at the thirty-second NBSAN meeting held in York in June 2022. We would like to express our heartfelt thanks to the organizer, V. Gould, for kindly inviting us to participate in this event.

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    Xavier Mary

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Mary, X. On the structure of semigroups whose regular elements are completely regular. Semigroup Forum 107, 692–717 (2023). https://doi.org/10.1007/s00233-023-10394-7

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