The algebraic law of left self-distributivity (LD), x(yz)=(xy)(xz)x(yz)=(xy)(xz), connects set theory, knot theory, and braid groups through the monoid LD+ generated by expansion operators on terms. Patrick Dehornoy proved that the associated category of the same name is a left-Garside category projecting onto the positive braid category, and that if this category is regular — meaning the conjugation-like functor ϕ\phi preserves normality of length-two simple sequences, a property implied by the identity gcd⁡(ϕt(a),ϕt(b))=ϕt(gcd⁡(a,b))\gcd(\phi_t(a),\phi_t(b))=\phi_t(\gcd(a,b)) for all terms tt and simple elements a,ba,b — then the long-standing Embedding Conjecture for free LD-systems follows. The problem is whether this regularity holds (Conjecture 6.11 of the source, dated 2008/2009). It remains open: Dehornoy's own 2019 survey still calls the Embedding Conjecture the main open problem of the area.

Background

A left-self-distributive system (LD-system) is a set with a binary operation obeying the LD law x(yz)=(xy)(xz)x(yz)=(xy)(xz); the law is obeyed by conjugation in groups, by applications of rank-to-rank elementary embeddings in set theory, and by shifted conjugacy on braid groups, and a substantial theory of the free LD-system was developed in the 1990s and summarized in Dehornoy's monograph Braids and Self-Distributivity (Birkhäuser, 2000). Syntactically, one applies the LD law in the expanding direction to terms (binary rooted trees with variables at leaves), producing LD-expansions; an expansion at a node with address α\alpha (a finite 0/1-word locating the node) is encoded by an operator DαD_\alpha, and the operators DαD_\alpha generate a monoid LD+ that partially acts on terms by t∙at \bullet a. The category LD+\mathcal{LD}^+ has terms as objects and triples (t,a,t∙a)(t, a, t\bullet a) as morphisms; Dehornoy proved it is a left-Garside category: a one-sided analogue of a Garside category (a category-theoretic version of a cancellative monoid with least common multiples and distinguished 'fundamental' elements whose left-divisors generate everything), equipped with a left-Garside map t↦Δtt \mapsto \Delta_t whose left-divisors, called simple elements, admit greedy normal forms (a canonical decomposition obtained by iteratively extracting the greatest simple left-divisor), and with a functor ϕ\phi acting like conjugation by Δt\Delta_t, determined on elements by Δt ϕt(a)=a Δt∙a\Delta_t\,\phi_t(a) = a\,\Delta_{t\bullet a}. The category admits a surjective functor onto the classical Garside category of positive braids that preserves right-least-common-multiples (right-lcms, i.e. least common right-multiples of morphisms), so Garside's braid theory appears as a quotient of the self-distributivity theory. In a two-sided Garside category, the functor ϕ\phi automatically preserves normality of length-two simple sequences — the defining property called regularity — but LD+\mathcal{LD}^+ is strongly asymmetric (it is not a Garside category), so regularity must be proved separately, and this is exactly what is not known: Conjecture 6.11 of the source paper states that LD+\mathcal{LD}^+ is regular (numbered Conjecture 1 in Section 6.4 of the published version). Its interest comes from Theorem 6.2 of the same paper: regularity implies the Embedding Conjecture of Chapter IX of the 2000 monograph, in any of six equivalent forms — the monoid LD+ embeds in a group; LD+ admits right-cancellation; two naturally associated categories are isomorphic; the functor ϕ\phi is injective; for each term, its LD-expansions form an upper-semilattice; and the known relations between the DαD_\alpha generate all relations. By Propositions 3.19 and 6.13 of the paper, regularity follows from the single identity gcd⁡(ϕt(a),ϕt(b))=ϕt(gcd⁡(a,b))\gcd(\phi_t(a),\phi_t(b)) = \phi_t(\gcd(a,b)) for every term tt and all simple elements a,ba,b for which t∙at\bullet a and t∙bt\bullet b are defined, where gcd⁡\gcd is the greatest common left-divisor — a sufficient criterion whose converse the paper does not establish; the paper verifies small cases, notes the analogous identity for least common multiples is simply false, and proposes a program: use the unique coordinate decomposition of simple elements a=∏α>Dα(eα)a = \prod_\alpha^{>} D_\alpha^{(e_\alpha)} (Proposition VIII.5.15 of the monograph, over addresses linearly ordered by α>α0β>α1γ\alpha > \alpha 0 \beta > \alpha 1 \gamma) to derive explicit formulas for the coordinates of ϕt(a)\phi_t(a) and of gcd⁡(a,b)\gcd(a,b), and then verify the identity for all triples (t,a,b)(t,a,b). That program has never been completed.

Problem Statement

Prove or disprove Conjecture 6.11 of Dehornoy's 'Left-Garside categories, self-distributivity, and braids' (Conjecture 1 of Section 6.4 of the published version): the left-Garside category LD+\mathcal{LD}^+ associated with the left self-distributivity law is regular, i.e. the functor ϕ\phi preserves normality of length-two sequences of simple morphisms. The paper (Proposition 6.13, via Proposition 3.19) establishes as a sufficient criterion for regularity the identity: for every term tt and all simple elements a,ba, b of the monoid LD+\mathrm{LD}^+ for which t∙at \bullet a and t∙bt \bullet b are defined,

gcd⁡(ϕt(a), ϕt(b))=ϕt(gcd⁡(a,b)),\gcd\big(\phi_t(a),\ \phi_t(b)\big) = \phi_t\big(\gcd(a,b)\big),

where gcd⁡\gcd denotes the greatest common left-divisor in LD+\mathrm{LD}^+ and ϕt(a)\phi_t(a) is the unique element satisfying Δt ϕt(a)=a Δt∙a\Delta_t\,\phi_t(a) = a\,\Delta_{t\bullet a}. This implication is one-way: the paper proves that the identity entails regularity, but does not prove the converse. A positive answer to the conjecture — regularity of LD+\mathcal{LD}^+ established by any means — implies the Embedding Conjecture for free LD-systems in all of its equivalent forms (in particular, that the LD-expansions of every term form an upper-semilattice and that LD+\mathrm{LD}^+ embeds in a group). A negative answer must exhibit a failure of ϕ\phi to preserve normality for some length-two sequence of simple morphisms; an explicit triple (t,a,b)(t,a,b) violating the displayed identity would show only that this sufficient criterion fails — it would not by itself refute the regularity conjecture, and it would not refute the Embedding Conjecture either, since Theorem 6.2 is a one-way implication, but it would close this proof route.

References

  1. Patrick Dehornoy, Left-Garside categories, self-distributivity, and braids, Annales Mathématiques Blaise Pascal 16(2) (2009) 189-244, DOI 10.5802/ambp.263, https://ambp.centre-mersenne.org/articles/10.5802/ambp.263/ (preprint arXiv:0810.4698, https://arxiv.org/abs/0810.4698)
  2. Patrick Dehornoy, Braids and Self-Distributivity, Progress in Mathematics 192, Birkhäuser Basel (2000), DOI 10.1007/978-3-0348-8442-6, https://doi.org/10.1007/978-3-0348-8442-6
  3. Patrick Dehornoy, Some aspects of the SD-world, Contemporary Mathematics 721 (2019) 69-96, DOI 10.1090/conm/721/14500, https://arxiv.org/abs/1711.09792
  4. Patrick Dehornoy, François Digne, Eddy Godelle, Daan Krammer, Jean Michel, Foundations of Garside Theory, EMS Tracts in Mathematics 22, European Mathematical Society (2015), DOI 10.4171/139, https://arxiv.org/abs/1309.0796 (Chapter XI is titled Left self-distributivity)
  5. Andrew D. Brooke-Taylor, Scott Cramer, Sheila K. Miller Edwards, A free two-generated left distributive algebra of elementary embeddings, arXiv:2508.02244 (2025), https://arxiv.org/abs/2508.02244
  6. Renrui Qi, Notes on Laver Tables, arXiv:2501.06733 (v1 2025, v5 revised July 2026), https://arxiv.org/abs/2501.06733