Problem 6.1 of Dékány–Szendrei–Szittyai: classifying the varieties of completely simple semigroups for which every E-solid locally inverse extension embeds into a λ-semidirect product

For which varieties $\mathcal{V}$ of completely simple semigroups is it true that whenever an E-solid locally inverse semigroup $S$ carries an inverse semigroup congruence $\rho$ whose idempotent $\rho$-classes belong to $\mathcal{V}$, the extension $(S,\rho)$ embeds into a λ-semidirect product extension of a member of $\mathcal{V}$ by the inverse semigroup $S/\rho$? This is Problem 6.1 of Dékány–Szendrei–Szittyai (2018/2019), the stated generalization of their structure theorem for E-solid locally inverse semigroups. The answer is known to be affirmative for the variety of all completely simple semigroups (their main theorem), for the variety of rectangular bands (yielding the characterization of generalized inverse semigroups), and for every variety of groups (settled by the group-variety theorem of Billhardt–Szittyai recalled below); the classification of the remaining varieties is open.

Background

A semigroup is regular if every element $a$ has an inverse, i.e. an element $x$ with $axa=a$ and $xax=x$; it is an inverse semigroup if every element has a unique inverse. A locally inverse semigroup is a regular semigroup in which, for every idempotent $e$, the local submonoid $eSe$ is an inverse semigroup; the class contains, among others, all inverse semigroups and all completely simple semigroups. A completely simple semigroup is a semigroup with no proper two-sided ideals that also possesses minimal left and right ideals; by the Rees theorem such semigroups are exactly the Rees matrix semigroups $\mathcal{M}(G;I,\Lambda;P)$ over a group $G$, and a variety of them is a class closed under homomorphic images, regular (equivalently, completely simple) subsemigroups, and arbitrary direct products, equivalently a class defined by identities. A rectangular band is a semigroup of idempotents satisfying $xyz=xz$. A regular semigroup is E-solid if its core $C(S)$, the subsemigroup generated by all idempotents, is completely regular (a union of groups); equivalently, it is an extension of a completely regular semigroup by an inverse semigroup, and by a theorem of Yamada and Hall this is also equivalent to the idempotent classes of its least inverse semigroup congruence being completely simple subsemigroups.

A congruence $\rho$ on a regular semigroup $S$ is an inverse semigroup congruence if the quotient $S/\rho$ is an inverse semigroup; the classes of $\rho$ containing idempotents are then subsemigroups of $S$, and the pair $(S,\rho)$ is called an extension by the inverse semigroup $S/\rho$. Such a pair $(S,\rho)$ embeds into another extension $(S',\rho')$ if there is an injective homomorphism $\varphi\colon S\to S'$ with $\rho=\ker(\pi_{\rho'}\circ\varphi)$, where $\pi_{\rho'}$ is the natural quotient map. A λ-semidirect product of a semigroup $K$ by an inverse semigroup $T$ (Billhardt; Kuřil–Szendrei) is a common generalization of the semidirect product in which $T$ no longer acts globally on $K$ but by maps subject to conditions tied to the natural partial order of $T$, the product living on a subset of $K\times T$; it generalizes the role wreath products play for groups in the classical Kaloužnin–Krasner embedding theorem for group extensions.

Billhardt and Szittyai (2000) proved the group-variety analogue: for any variety $\mathcal{U}$ of groups, an idempotent-separating extension of an inverse semigroup whose classes belong to $\mathcal{U}$ embeds into a λ-semidirect product extension of a member of $\mathcal{U}$. Dékány, Szendrei and Szittyai (2018, published 2019) proved the corresponding theorem at the top of the hierarchy: if $S$ is an E-solid locally inverse semigroup and $\rho$ an inverse semigroup congruence whose idempotent classes are completely simple, then $(S,\rho)$ embeds into a λ-semidirect product of a completely simple semigroup by $S/\rho$, which yields the structure theorem that the E-solid locally inverse semigroups are, up to isomorphism, exactly the regular subsemigroups of λ-semidirect products of completely simple semigroups by inverse semigroups. They also observed that for the variety of rectangular bands the analogous statement holds with a much simpler argument, re-deriving the classical fact that the generalized inverse semigroups — the orthodox locally inverse semigroups, i.e. those whose idempotents form a normal band — are exactly the regular subsemigroups of λ-semidirect products of a rectangular band by an inverse semigroup. The paper then poses, as its stated direction for future work, the classification problem for the remaining varieties.

Problem Statement

Determine exactly the varieties $\mathcal{V}$ of completely simple semigroups with the following property: whenever $S$ is an E-solid locally inverse semigroup and $\rho$ is an inverse semigroup congruence on $S$ such that every idempotent $\rho$-class (viewed as a subsemigroup of $S$) belongs to $\mathcal{V}$, the extension $(S,\rho)$ is embeddable — in the extension sense defined in the background, i.e. via an injective homomorphism $\varphi$ with $\rho=\ker(\pi_{\rho'}\circ\varphi)$ — into a λ-semidirect product extension of a member of $\mathcal{V}$ by the inverse semigroup $S/\rho$. This is Problem 6.1 of Dékány–Szendrei–Szittyai (Acta Sci. Math. 85 (2019), 381–411; Problem 4.4.1 of the thesis version). Known affirmative instances: $\mathcal{V}$ = all completely simple semigroups (the paper's main theorem), $\mathcal{V}$ = rectangular bands (Proposition 6.2), and $\mathcal{V}$ = any variety of groups (settled by the group-variety theorem of Billhardt–Szittyai); the requested classification covers all remaining varieties. For an individual remaining variety $\mathcal{V}$, a proof that the embedding construction adapts to $\mathcal{V}$ establishes that $\mathcal{V}$ has the property, and an explicit extension with idempotent $\rho$-classes in $\mathcal{V}$ witnessing failure establishes that it does not.

References

1. Tamás Dékány, Mária B. Szendrei, István Szittyai, E-solid locally inverse semigroups, Acta Scientiarum Mathematicarum 85 (2019), 381–411. DOI: 10.14232/actasm-018-311-5, https://doi.org/10.14232/actasm-018-311-5 (preprint arXiv:1807.01265, https://arxiv.org/abs/1807.01265)
2. Mária B. Szendrei, Structure theory of regular semigroups, Semigroup Forum 100 (2020), 119–140. DOI: 10.1007/s00233-019-10055-8, https://doi.org/10.1007/s00233-019-10055-8 (published online November 2019; preprint arXiv:1909.05280, https://arxiv.org/abs/1909.05280)
3. Jiří Kaďourek, A wreath product decomposition of E-solid locally inverse semigroups, Communications in Algebra 53 (2024), no. 2, 826–841. DOI: 10.1080/00927872.2024.2390638, https://doi.org/10.1080/00927872.2024.2390638
4. Tamás Dékány, E-solid locally inverse semigroups as extensions, PhD thesis, University of Szeged, 2019. DOI: 10.14232/phd.9998, https://doi.org/10.14232/phd.9998
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