Fundamental bisimple embeddings for solid regular and locally inverse semigroups (the Uniform Bands problem)

Albert and Pastijn (Uniform Bands, 2014) proved that every orthodox semigroup embeds into a bisimple orthodox semigroup — fundamental unless the original is a rectangular group — whose band of idempotents generates the same band variety. They extended the surrounding machinery to solid regular semigroups (idempotents generating a completely regular semigroup) and locally inverse semigroups (all local submonoids inverse), proving the uniformization of the idempotent algebra and the fundamental-to-fundamental-bisimple step, but stated that their orthodox-case device, a strong composition (Płonka sum) with an inverse semigroup, needs to be modified without elaborating. The open problem is whether every solid regular, respectively locally inverse, semigroup embeds into a fundamental bisimple member of the same class preserving the variety generated by its idempotent algebra. The problem is open.

Background

A semigroup is a set with an associative multiplication; an element $e$ with $e^2=e$ is an idempotent, and a band is a semigroup consisting entirely of idempotents. A semigroup $S$ is regular if every $a\in S$ has an inverse $a'$ with $aa'a=a$ ; it is completely regular if it is a union of groups, and an inverse semigroup if every element has a unique inverse, equivalently if it is regular with commuting idempotents (which then form a semilattice, a commutative band). An orthodox semigroup is a regular semigroup whose idempotents form a subsemigroup, necessarily a band. Two wider classes of regular semigroups are central here. A regular semigroup is locally inverse if for every idempotent $e$ the local submonoid $eSe$ is an inverse semigroup. A regular semigroup is called solid regular (in older terminology, E-solid) if its idempotents generate a completely regular semigroup. Every orthodox semigroup is solid regular (its idempotents form a band, which is completely regular), and every inverse semigroup is both orthodox and locally inverse.

Nambooripad's structure theory attaches to every regular semigroup $S$ its regular biordered set $(E(S),\circ)$ : the set of idempotents with the partial operation $f\circ e=fe$ , defined exactly when $\{e,f\}\cap\{ef,fe\}\neq\emptyset$ . In the two classes above this partial operation extends to a total binary operation $\wedge$ on $E(S)$ : for solid regular $S$ , $f\wedge e=(fe)^0$ , the identity of the maximal subgroup containing $fe$ ; for locally inverse $S$ , $f\wedge e$ is the unique inverse of $ef$ lying in $fSe$ . The resulting algebra $(E(S),\wedge)$ is a regular solid idempotent algebra, respectively a pseudo-semilattice — in each case meaning an abstract algebra with one binary operation satisfying precisely the identities that hold in the corresponding class of concrete algebras $(E(S),\wedge)$ — and these classes of algebras form varieties, i.e. equational classes closed under homomorphic images, subalgebras, and products (Broeksteeg; Nambooripad). The variety generated by such an algebra is invariant under isomorphism and monotone under extension, so asking that an embedding preserve this variety is asking that the idempotent structure be enlarged as little as possible in equational terms.

A regular semigroup is bisimple if Green's relation $\mathcal{D}$ — the equivalence generated by mutual divisibility, i.e. $a=xby$ and $b=uav$ for some elements $u,v,x,y$ of the semigroup — is universal, i.e. there is a single $\mathcal{D}$ -class; bisimple semigroups play the role that 'one-layer' objects play in structure theory. It is fundamental if the greatest idempotent-separating congruence $\mu$ — the largest congruence of $S$ that restricts to the identity on $E(S)$ — is the identity; equivalently, $S$ has no nontrivial congruence invisible on idempotents. Every regular semigroup $S$ maps homomorphically into the Hall–Nambooripad semigroup $T_B$ of partial isomorphisms $e\wedge B\wedge e\to g\wedge B\wedge g$ of its idempotent algebra $B=(E(S),\wedge)$ modulo a natural congruence $\kappa$ , and $T_B$ is fundamental; the map is an embedding precisely when $S$ is fundamental, and $T_B$ is bisimple exactly when $B$ is uniform, meaning all the subalgebras $e\wedge B\wedge e$ are isomorphic.

Hall (1969–1971) proved that every regular semigroup whose idempotents form a subsemigroup embeds into a bisimple one with an isomorphic band of idempotents. Albert and Pastijn (2014) refined this to the level of varieties: their Theorem 2.3 embeds every band $B$ into a uniform band $B'$ generating the same band variety, and their Theorem 3.8 embeds every orthodox semigroup $S$ into a bisimple orthodox semigroup $S'$ with $E(S')$ generating the same band variety as $E(S)$ ; if $S$ is not a rectangular group (a direct product of a rectangular band and a group), $S'$ can moreover be chosen fundamental. The device behind the fundamental step is the strong composition, in the sense of Płonka sums: $S$ is composed with a fundamental inverse semigroup $I$ via the least inverse congruence $Y$ of $S$ and a Vagner–Preston embedding $S/Y\hookrightarrow I$ , yielding an intermediate fundamental orthodox semigroup $S_0=S\cup I$ with cross-products $si=(s\varphi)i$ , $is=i(s\varphi)$ computed in $I$ ; $S_0$ is then embedded into the bisimple fundamental semigroup $S'$ of Theorem 3.8. In their final section they carry the uniformization and the Hall-semigroup construction over to the two wider classes: Theorem 4.1 embeds every regular solid idempotent algebra (pseudo-semilattice) into a uniform one generating the same variety, and Proposition 4.2 embeds every fundamental solid regular (locally inverse) semigroup into a fundamental bisimple one preserving the variety of $(E(S),\wedge)$ . However, the strong-composition device has no known analogue in these classes, because the least inverse congruence and the inverse-semigroup target are special to the orthodox setting; they state that proving the analogue of Theorem 3.8 'needs' this device 'to be modified', and stop there.

Problem Statement

For each of the two classes of regular semigroups — solid regular semigroups and locally inverse semigroups — decide whether the analogue of Theorem 3.8 of Albert–Pastijn (2014) holds: given an arbitrary solid regular (respectively, locally inverse) semigroup $S$ , construct, or prove the impossibility of, an embedding of $S$ into a semigroup $S'$ of the same class such that (i) $S'$ is bisimple, (ii) $S'$ is fundamental, outside the analogue of the rectangular-group exceptional case of the orthodox theorem — the extent of this exceptional class in each of the two classes is itself to be determined and stated as part of the answer — and (iii) the idempotent algebras $(E(S),\wedge)$ and $(E(S'),\wedge)$ , a regular solid idempotent algebra respectively a pseudo-semilattice, generate the same variety. By Proposition 4.2 of the source, the essential open step is (the analogue of) embedding an arbitrary member of the class into a fundamental member of the class while preserving the variety generated by the idempotent algebra; the orthodox device — strong composition (Płonka sum) of $S$ with a fundamental inverse semigroup via the least inverse congruence — is known not to carry over directly; whether it can be adapted, or must be replaced, is open. The two-class question is settled for a class only when the embedding question is decided for every semigroup of that class (no restriction on size or generation); a resolution of one class leaves the other open.

References

Justin Albert and Francis Pastijn, "Uniform Bands", Proceedings of the Edinburgh Mathematical Society (Series 2) 57 (2014), no. 3, 575–587. DOI: 10.1017/S0013091513000709, https://doi.org/10.1017/S0013091513000709
P. A. Azeef Muhammed, Mikhail V. Volkov, and Karl Auinger, "Cross-connection structure of locally inverse semigroups", International Journal of Algebra and Computation 33 (2023), no. 1, 123–159. DOI: 10.1142/S0218196723500091, arXiv:1912.00214, https://doi.org/10.1142/S0218196723500091, https://arxiv.org/abs/1912.00214
Tamás Dékány, Mária B. Szendrei, and István Szittyai, "E-solid locally inverse semigroups", Acta Scientiarum Mathematicarum (Szeged) 85 (2019), no. 3–4, 381–411. DOI: 10.14232/actasm-018-311-5, arXiv:1807.01265, https://doi.org/10.14232/actasm-018-311-5, https://arxiv.org/abs/1807.01265
Luís Oliveira, "A combinatorial approach to the structure of locally inverse semigroups", Semigroup Forum 103 (2021), no. 2, 575–621. DOI: 10.1007/s00233-021-10203-z, arXiv:1903.10236, https://doi.org/10.1007/s00233-021-10203-z, https://arxiv.org/abs/1903.10236
Jiří Kad’ourek, "A wreath product decomposition of E-solid locally inverse semigroups", Communications in Algebra 53 (2025), no. 2, 826–841. DOI: 10.1080/00927872.2024.2390638, https://doi.org/10.1080/00927872.2024.2390638
Francis Pastijn and Justin Albert, "Free split bands", Semigroup Forum 90 (2015), no. 3, 753–762. DOI: 10.1007/s00233-014-9625-9, https://doi.org/10.1007/s00233-014-9625-9