18 problems

Logarithmic multipermutation-level conjecture. One has

Let $(X,r)$ be a square-free solution of the set-theoretic Yang–Baxter equation, and let $S(X,r)$ be its associated semigroup. For $x,y,x',y'\in X$, write $r(xy)=y'x'$. Conjectures…

Let $(X,r)$ be a multipermutation square-free solution of level $m$. Generalized twisted-union conjecture. Every multipermutation square-free solution of level $m$ is a generalized…

Let $(X,r)$ be a square-free, nondegenerate, involutive solution of the set-theoretic Yang–Baxter equation, with associated semigroup $S(X,r)$. Skew-polynomial ordering conjecture.…

Let $(X,r)$ be a square-free set-theoretic solution of the Yang–Baxter equation. Retractability conjecture. Every square-free solution of the Yang–Baxter equation is retractable in…

Let a category of bijective non-degenerate set-theoretic solutions to the Yang–Baxter equation be fixed, where a solution is a set equipped with a map satisfying the Yang–Baxter eq…

Expected implication. If $H$ has a $\Phi$-set theoretic basis, then $H$ has the positive basis property.

A set-theoretic solution of the Yang–Baxter equation is a solution considered up to isomorphism, and its size is the cardinality of its underlying set. Let $n$ denote the size of t…

Let $(X,r)$ be a non-degenerate involutive set-theoretic solution to the Yang–Baxter equation, with permutation group $G(X)$ acting on $X$. Assume that $(X,r)$ is indecomposable an…

Let $(X,r)$ be an involutive solution of size $n$, and let $g(n)$ denote the bound used in the surrounding text for solutions whose permutation group is abelian. Abelian permutatio…

Let $(X,r)$ be an involutive solution of size $n$, and define … Dehornoy's class bound conjecture. The Dehornoy's class is bounded by $a_n$, and the bound $a_n$ is minimal. The cla…

Let $(X,r)$ be an indecomposable involutive solution, and let the Dehornoy's class denote its associated class invariant. Dehornoy's class conjecture for indecomposable solutions.…

Let $(X,r)$ be an indecomposable involutive solution, and let $\mathcal{G}(X,r)$ denote its permutation group. A solution is uniconnected when its associated permutation action is…

Let $X$ be an indecomposable solution of the Yang–Baxter equation whose permutation group is isomorphic to the dihedral group $D_{2n}$. Dihedral size conjecture. Then … For an inde…

Let $n\in\mathbb{N}$, and consider indecomposable involutive solutions of the Yang–Baxter equation of multipermutation level $2$ and size $n$. Exponential upper-bound conjecture. T…

Yang–Baxter solution enumeration conjecture. There are

Gateva-Ivanova's conjecture. The structure monoid $M(X,r)$ is cancellative if and only if the solution $(X,r)$ is involutive.

Let $(X,r)$ be a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, and let $G(X,r)$ be its structure group, presented with generators $X$ and relations…