DSC semigroups: which groups are DSC, and is the DSC class closed under direct products?

A semigroup $S$ is called DSC ("diagonal subsemigroups are congruences") if every subsemigroup of the direct square $S\times S$ that contains the diagonal $\{(s,s):s\in S\}$ — equivalently, every reflexive relation on $S$ compatible with multiplication — is necessarily a congruence on $S$. Barber and Ruškuc (2025), who introduced this notion, proved that every DSC semigroup is simple; that within the periodic, stable, inverse and completely simple classes the only DSC semigroups are groups; and that non-group DSC semigroups exist, obtained from Byleen's congruence-free monoid construction (even regular and bisimple examples). Two determinate questions they explicitly posed remain open: (1) whether every DSC group is periodic — all periodic groups are DSC, the infinite cyclic group is not, and it is currently unknown whether any DSC group contains an element of infinite order; and (2) whether the class of DSC semigroups is closed under formation of direct products.

Background

A semigroup is a set equipped with an associative binary operation. A congruence $\rho$ on a semigroup $S$ is an equivalence relation that is compatible with the multiplication, i.e. $(x,y),(z,t)\in\rho$ implies $(xz,yt)\in\rho$; congruences are exactly the kernels of semigroup homomorphisms and are the objects with respect to which quotient semigroups are formed. Viewed as a subset of the direct square $S\times S$, a congruence is precisely a relation that (i) contains the diagonal $\Delta=\{(s,s):s\in S\}$ (reflexivity), (ii) is a subsemigroup of $S\times S$ under componentwise multiplication (compatibility), and (iii) is symmetric and transitive. For algebras possessing a Mal'cev term — a ternary term $m(x,y,z)$ with $m(x,y,y)=x=m(y,y,x)$, such as $xy^{-1}z$ in a group — conditions (i) and (ii) force (iii): a classical folklore fact is that any subgroup of $G\times G$ containing $\Delta$ is automatically a congruence of the group $G$. For semigroups this implication fails: in the two-element left zero semigroup ($ab=a$ for all $a,b$), the relation $\{(x,x),(x,y),(y,y)\}$ is a subsemigroup of $S\times S$ containing the diagonal, but it is not symmetric.

Barber and Ruškuc (2025) call a semigroup DSC if every diagonal subsemigroup of $S\times S$ — that is, every reflexive and compatible relation on $S$ — is a congruence, and they initiate the study of this class. (In their terminology a diagonal subsemigroup of $S\times S$ is a subsemigroup containing $\Delta$; the same idea for general algebras, where one additionally requires symmetry, is related to the classical notion of a tolerance relation.) Their main findings are: every DSC semigroup is simple, meaning it has no two-sided ideal other than itself; a semigroup that is completely simple (simple with a primitive idempotent, i.e. an idempotent minimal in the natural order $e\le f \Leftrightarrow ef=fe=e$), stable (a finiteness condition on Green's relations $\mathcal{R},\mathcal{L},\mathcal{J}$, which relate elements generating the same principal right, left or two-sided ideals), periodic (every element satisfies $s^m=s^n$ for some $m\ne n$), or inverse (every element has a unique semigroup inverse, where $t$ is an inverse of $s$ when $sts=s$ and $tst=t$) can only be DSC if it is a group; and the class of DSC semigroups is closed under quotients but not under subsemigroups. Constructions of simple semigroups such as Bruck–Reilly extensions and the generalized Baer–Levi semigroups $B(p,q)$ (injective self-maps of a set of cardinality $p$ whose image misses $q$ points) never yield DSC semigroups. Nevertheless, non-group DSC semigroups do exist: they arise from a monoid presentation $C_1(S;\alpha,\beta;P)$ introduced by Byleen (1990) to build congruence-free monoids, where $\alpha,\beta$ are faithful left and right actions of a monoid $S$ on sets $B$ and $A$ and $P$ is a $2$-transitive $A\times B$ matrix (every two distinct rows, and every two distinct columns, realize every pair of entries; such matrices over countable index sets exist by Quick–Ruškuc 2010). Such examples can moreover be regular (every element has an inverse) and bisimple (Green's relation $\mathcal{D}$, the join of $\mathcal{R}$ and $\mathcal{L}$, is universal).

A further wrinkle is that, despite the folklore fact about groups, not every group is DSC, because a group may contain subsemigroups that are not subgroups: in the infinite cyclic group $\mathbb{Z}$, the relation $\{(x,y): x\le y\}$ is a diagonal subsemigroup of $\mathbb{Z}\times\mathbb{Z}$ that is not symmetric. Every periodic group is DSC, since in a periodic group every diagonal subsemigroup is in fact a diagonal subgroup, to which the folklore fact applies. The paper's closing section therefore poses, among other directions, two open questions: a description of the DSC groups — in particular, whether every DSC group must be periodic, i.e. whether some group mixing infinite-order elements with the DSC property can exist — and whether a direct product of two DSC semigroups is again DSC, for which nothing is currently known in either direction. (The same authors also pose the much broader goals of a complete structure theory of all DSC semigroups — which they describe as out of reach at present — and a systematic study of which combinations of reflexivity, symmetry, transitivity and compatibility imply the remaining properties; these research programs are context for the present problem, not part of its completion boundary.)

Problem Statement

Determine the extent of the class of DSC semigroups — semigroups in which every reflexive and multiplication-compatible relation is a congruence — at the two points explicitly left open by Barber and Ruškuc (2025):

(1) Decide whether every DSC group is periodic (equivalently, whether any DSC group contains an element of infinite order). Every periodic group is DSC and the infinite cyclic group is not; a proof that every DSC group is periodic characterizes the DSC groups intrinsically as exactly the periodic groups, and a counterexample is an explicit DSC group containing an element of infinite order.

(2) Decide whether the class of DSC semigroups is closed under formation of direct products: either prove that for any two DSC semigroups $S$ and $T$ the direct product $S\times T$ is DSC, or exhibit two DSC semigroups whose direct product is not DSC, together with an explicit diagonal subsemigroup of the product that is not a congruence.

The broader programs mentioned in the same source (a complete structure theory of all DSC semigroups, and the general interdependence of reflexivity, symmetry, transitivity and compatibility) are context, not part of the required completion.

References

1. Callum Barber and Nik Ruškuc, Semigroup Congruences and Subsemigroups of the Direct Square, Bulletin of the Australian Mathematical Society, vol. 113, pp. 123–134 (published online July 2025), DOI 10.1017/S0004972725100166, https://doi.org/10.1017/S0004972725100166; preprint arXiv:2504.02428, https://arxiv.org/abs/2504.02428
2. Callum Barber and Nik Ruškuc, Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups, preprint arXiv:2602.17286 (February 2026), https://arxiv.org/abs/2602.17286
3. Karl Byleen, Embedding any countable semigroup in a 2-generated congruence-free semigroup, Semigroup Forum 41 (1990), 145–153, DOI 10.1007/BF02573385, https://doi.org/10.1007/BF02573385
4. John M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs New Series 12, Oxford University Press, 1995, DOI 10.1093/oso/9780198511946.001.0001, https://doi.org/10.1093/oso/9780198511946.001.0001
5. Martyn Quick and Nik Ruškuc, Growth of generating sets for direct powers of classical algebraic structures, Journal of the Australian Mathematical Society 89 (2010), 105–126, DOI 10.1017/S1446788710001473, https://doi.org/10.1017/S1446788710001473