Explicit semiband embedding for semigroups whose regular D-classes are aperiodic semigroups

Every semigroup embeds into a semiband (an idempotent-generated semigroup) by classical constructions, but no known explicit embedding preserves the property that all regular Green D-classes are aperiodic semigroups — the defining property of the pseudovariety DA of finite semigroup theory. Almeida and Moura (2011) proved non-constructively, using profinite techniques, that DA is generated as a pseudovariety by its idempotent-generated members, yet their argument yields no explicit semigroup construction. Oliveira (2014) therefore asked for an explicit semiband construction embedding any semigroup whose regular D-classes are aperiodic semigroups into a semiband with the same property, finite when the source semigroup is finite. This question has remained open since Oliveira (2014).

Background

An element $e$ of a semigroup $S$ is an idempotent if $e^2 = e$ ; the set of idempotents is denoted $E(S)$ , and $\langle E(S)\rangle$ is the subsemigroup they generate. A semigroup equal to the subsemigroup generated by its idempotents is called a semiband (or idempotent-generated semigroup); its depth is the least $n$ such that every element is a product of $n$ idempotents. Green's relations organize a semigroup by ideals: $x\,\mathcal{R}\,y$ if $x$ and $y$ generate the same principal right ideal ( $xS^1 = yS^1$ , where $S^1$ is $S$ with an identity adjoined if absent), $\mathcal{L}$ is the left-sided analogue, $\mathcal{H} = \mathcal{R}\cap\mathcal{L}$ , and $\mathcal{D}$ is their join; the equivalence classes of $\mathcal{D}$ are the D-classes. An element is regular if it has a quasi-inverse $y$ with $xyx = x$ , and a D-class is regular if it contains a regular element. A semigroup is aperiodic (H-trivial) if all its subgroups are trivial. The class of finite semigroups in which every regular D-class is an aperiodic semigroup is the well-studied pseudovariety DA (a pseudovariety being a class of finite semigroups closed under finite direct products, subsemigroups, and homomorphic images); in such a semigroup each regular D-class is in fact a rectangular band (a band satisfying $xyz = xz$ for all elements $x, y, z$ ), and DA plays a central role in finite semigroup theory and its connections to formal language theory.

A classical line of results, beginning with Howie's 1966 theorem that the singular (non-bijective) maps in the full transformation monoid $T_X$ of all self-maps of a set $X$ under composition form an idempotent-generated semigroup, shows that every semigroup embeds into a semiband, finitely when the source is finite and with controlled depth (Howie 1966; the series of constructions is surveyed in Oliveira 2014). However, none of these constructions preserves the DA property: each introduces regular D-classes that fail to be aperiodic semigroups. Almeida and Moura (2011) studied the operator sending a pseudovariety $\mathrm{V}$ to the pseudovariety $\mathrm{V}_{\mathrm{E}}$ generated by its idempotent-generated members, and proved $(\mathrm{DA})_{\mathrm{E}} = \mathrm{DA}$ : every finite semigroup in DA divides (is a homomorphic image of a subsemigroup of) an idempotent-generated member of DA. Their proof uses profinite machinery — free profinite semigroups (inverse limits of the free finite semigroups of each rank, carrying a compact zero-dimensional topology) and implicit operations on them (operations on profinite semigroups compatible with all continuous homomorphisms, such as $x \mapsto x^{\omega}$ , the limit of the powers of $x$ ) — and shows that the map $x_i \mapsto x_i^{\omega}y^{\omega}$ (with $y$ a new variable and $x^{\omega}$ the idempotent power) embeds the free profinite semigroup $\overline{\Omega}_n\mathrm{DA}$ into the subsemigroup generated by idempotents of $\overline{\Omega}_{n+1}\mathrm{DA}$ . Their paper explicitly notes that for DA this method yields only a division property, not an explicit construction. Oliveira (2014) contributed a new explicit embedding — for semigroups that are unions of their idempotent-generated principal left and right ideals (a class including all monoids and all regular semigroups) — into a semiband of depth 4 built inside a semidirect product of $S \times S$ with the two-element right-zero semigroup $R_2 = \{\sigma, \tau\}$ , in which $ab = b$ for all $a, b$ (a semidirect product combines the multiplication of one factor with an action of the other factor on it), of depth 2 when $S$ is regular; it preserves several properties of $S$ , but again not the aperiodicity of regular D-classes. Consequently, the explicit-construction question stated below was left open.

Problem Statement

Give an explicit construction which, for every semigroup $S$ in which each regular $\mathcal{D}$ -class is an aperiodic semigroup, produces a semiband $B(S)$ (an idempotent-generated semigroup) in which likewise each regular $\mathcal{D}$ -class is an aperiodic semigroup, together with an explicit embedding $S \hookrightarrow B(S)$ ; the construction must be uniform (a single explicit scheme or algorithm, not a case-by-case appeal to existence, that applies to every such $S$ ), and $B(S)$ must be finite whenever $S$ is finite. This is the question stated as open in Oliveira (2014): the pseudovariety identity $(\mathrm{DA})_{\mathrm{E}} = \mathrm{DA}$ of Almeida and Moura guarantees such semibands only through a non-constructive profinite argument, and every known explicit semiband embedding — including Oliveira's own semidirect-product construction — fails to preserve the aperiodicity of regular D-classes.

References

Luís Oliveira, On the question of embedding a semigroup into a semiband, Semigroup Forum 88 (2014), no. 3, 715–731. DOI: 10.1007/s00233-013-9564-x, https://doi.org/10.1007/s00233-013-9564-x (preprint: arXiv:1205.3454, under the title "On the question of embedding a semigroup into an idempotent generated one", https://arxiv.org/abs/1205.3454)
J. Almeida and A. Moura, Idempotent-generated semigroups and pseudovarieties, Proceedings of the Edinburgh Mathematical Society (2) 54 (2011), no. 3, 545–568. DOI: 10.1017/S0013091509001795, https://doi.org/10.1017/S0013091509001795
J. M. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup, Journal of the London Mathematical Society 41 (1966), 707–716. DOI: 10.1112/jlms/s1-41.1.707, https://doi.org/10.1112/jlms/s1-41.1.707
Gracinda M. S. Gomes and Nik Ruškuc, John Macintosh Howie: work and legacy, Semigroup Forum 89 (2014), no. 1, 2–19. DOI: 10.1007/s00233-014-9628-6, https://doi.org/10.1007/s00233-014-9628-6
A. Moura, E-local pseudovarieties, Semigroup Forum 85 (2012), no. 1, 169–181. DOI: 10.1007/s00233-012-9413-3, https://doi.org/10.1007/s00233-012-9413-3
Luís Oliveira, Regular semigroups weakly generated by idempotents, International Journal of Algebra and Computation 33 (2023), no. 5, 851–891. DOI: 10.1142/S0218196723500388, https://doi.org/10.1142/S0218196723500388 (preprint: arXiv:2112.11310, https://arxiv.org/abs/2112.11310)