The Boone–Higman programme for semigroups: embedding finitely presented semigroups into finitely presented congruence-free, bisimple, and regular monoids

A 2013 note of Victor Maltcev (J. Algebra 389, 1–5) initiated the semigroup analogue of the Boone–Higman programme, proving that every finite semigroup embeds in a finitely presented congruence-free monoid and posing two explicit embedding questions — whether every finitely presented semigroup embeds in a finitely presented bisimple monoid (Question 1.2), and whether every finitely presented semigroup embeds in a finitely presented regular monoid (Question 1.3) — and announcing the broader Boone–Higman-style programme whose natural target question is whether every finitely generated semigroup with soluble word problem embeds in a finitely presented congruence-free semigroup (the note proves only the finite case as a first step toward it). No component has been resolved or refuted since. The known embedding theorems stop at finitely generated containers (Byleen) or at finite embedded semigroups (Maltcev), and a 2023 survey of the Boone–Higman conjecture explicitly records the semigroup version as open and 'much more difficult'.

Background

A semigroup is a set with an associative binary operation, and a monoid is a semigroup with an identity element. A semigroup or monoid is finitely presented if it can be described by finitely many generators subject to finitely many defining relations. Its word problem asks, given two words over the generators, whether they represent the same element; the word problem is soluble (decidable) when an algorithm decides this for all pairs. A congruence on a semigroup is an equivalence relation compatible with multiplication in both arguments, i.e. the kernel of a homomorphism; a semigroup is congruence-free if its only congruences are equality and the universal relation, which makes it the natural semigroup-theoretic analogue of a simple group, since it admits no nontrivial quotient constructions. Green's relation $\mathcal{D}$ is the equivalence generated by mutual divisibility ($a$ can be written using $b$ as a factor and vice versa, allowing one-sided factorizations on each side); a monoid is bisimple if it has exactly one $\mathcal{D}$-class (which forces it to be simple, i.e. to have no proper two-sided ideals). A monoid is regular if every element $x$ admits an element $y$ with $xyx=x$; every bisimple monoid is regular, but not conversely. In the Adian–Rabin tradition, a property $P$ of finitely presented semigroups is called a Markov property if it is invariant under isomorphism, some finitely presented semigroup has $P$, and some finitely presented semigroup embeds into no finitely presented semigroup with $P$; such properties are algorithmically unrecognizable, which is why embedding questions of the kind below are tied to decision problems.

Boone and Higman (1974) proved that a finitely generated group has soluble word problem if and only if it embeds in a simple subgroup of a finitely presented group, and raised the conjecture, still open, that every finitely generated group with soluble word problem embeds in a finitely presented simple group. On the semigroup side, Byleen (1984) proved that every countable semigroup embeds in a 2-generated bisimple monoid, and (1990) that every countable semigroup embeds in a 2-generated congruence-free semigroup; Evans (1978) proved the analogous general-algebra statement that an algebra has soluble word problem if and only if it embeds in a finitely generated simple algebra. In all of these results the container is merely finitely generated, not finitely presented — and that gap is the crux of the Boone–Higman programme, since finite presentability of the container is what transfers decidability structure in the group theorem. For groups the programme remains active: the hyperbolic groups were recently shown to satisfy the conjecture (Belk, Bleak, Matucci, Zaremsky, Duke Math. J. 2026), alongside partial results for self-similar groups, $\mathrm{Aut}(F_n)$, mapping class groups, and groups acting on locally finite trees; the general group conjecture is open.

Maltcev (2013) began the semigroup version of this programme, taking congruence-free semigroups as the counterpart of simple groups, and proved the first case as the note's main theorem: every finite semigroup embeds in a finitely presented congruence-free monoid. The note then poses two embedding questions — whether every finitely presented semigroup embeds in a finitely presented bisimple monoid, and whether every finitely presented semigroup embeds in a finitely presented regular monoid — remarks that these ask whether bisimplicity and regularity are Markov properties for finitely presented semigroups, and records the author's belief that the two questions are connected to the Boone–Higman programme for semigroups. Companion papers of the same period (Maltcev 2014; Al-Kharousi, Cain, Maltcev, Umar 2015; Cain and Maltcev 2015) constructed further example families of finitely presented congruence-free monoids — bisimple $\mathcal{H}$-trivial ones, infinite 0-simple and simple ones, and a simple non-bisimple one — thereby enriching the supply of candidate containers without addressing the embedding questions themselves.

Problem Statement

Determine which of the following embedding statements from the semigroup Boone–Higman programme initiated in Maltcev (2013) are true:

1. (Boone–Higman for semigroups) Does every finitely generated semigroup with soluble word problem embed in a finitely presented congruence-free semigroup? (The case where the embedded semigroup is finite is the note's theorem; the statement for arbitrary finitely generated semigroups with soluble word problem is the open target, formulated from the programme the note announces rather than from a numbered question in it.)

2. (Question 1.2 of the source) Does every finitely presented semigroup embed in a finitely presented bisimple monoid?

3. (Question 1.3 of the source) Does every finitely presented semigroup embed in a finitely presented regular monoid?

These three components share one scientific objective, the semigroup-theoretic Boone–Higman programme: as the source note frames it, components 2 and 3 ask whether bisimplicity and regularity are Markov properties of finitely presented semigroups, and the note records the author's belief that the two questions are connected to the Boone–Higman programme, though it does not state that belief as a precise conjecture. The three questions are posed jointly, and none may be answered by narrowing to a restricted class of source semigroups, a fixed container, or a special case.

References

1. Victor Maltcev, Finite semigroups embed in finitely presented congruence-free monoids, Journal of Algebra 389 (2013) 1–5. DOI: 10.1016/j.jalgebra.2013.05.015, arXiv:1301.5336, https://arxiv.org/abs/1301.5336
2. W. W. Boone, G. Higman, An algebraic characterization of groups with soluble word problem, Journal of the Australian Mathematical Society 18 (1974) 41–53. DOI: 10.1017/S1446788700019108, https://doi.org/10.1017/S1446788700019108
3. Karl Byleen, Embedding any countable semigroup in a 2-generated bisimple monoid, Glasgow Mathematical Journal 25 (1984) 153–161. DOI: 10.1017/S0017089500005565, https://doi.org/10.1017/S0017089500005565
4. 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
5. Trevor Evans, An algebra has a solvable word problem if and only if it is embeddable in a finitely generated simple algebra, Algebra Universalis 8 (1978) 197–204. DOI: 10.1007/BF02485389, https://doi.org/10.1007/BF02485389
6. Victor Maltcev, A countable series of bisimple $\mathcal{H}$-trivial finitely presented congruence-free monoids, Semigroup Forum 88 (2014) 279–285. DOI: 10.1007/s00233-013-9508-5, arXiv:1301.5343, https://arxiv.org/abs/1301.5343
7. Fatma Al-Kharousi, Alan J. Cain, Victor Maltcev, Abdullahi Umar, A countable family of finitely presented infinite congruence-free monoids, Acta Scientiarum Mathematicarum (Szeged) 81 (2015) 437–445. DOI: 10.14232/actasm-013-028-z, arXiv:1304.4687, https://arxiv.org/abs/1304.4687
8. Alan J. Cain, Victor Maltcev, A simple non-bisimple congruence-free finitely presented monoid, Semigroup Forum 90 (2015) 184–188. DOI: 10.1007/s00233-014-9607-y, arXiv:1307.6580, https://arxiv.org/abs/1307.6580
9. James Belk, Collin Bleak, Francesco Matucci, Matthew C. B. Zaremsky, Progress around the Boone–Higman Conjecture (2023). arXiv:2306.16356, https://arxiv.org/abs/2306.16356
10. James Belk, Collin Bleak, Francesco Matucci, Matthew C. B. Zaremsky, Hyperbolic groups satisfy the Boone–Higman conjecture, Duke Mathematical Journal 175(9) (2026). DOI: 10.1215/00127094-2025-0055, arXiv:2309.06224, https://arxiv.org/abs/2309.06224