9 problems

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 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 $X$ 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…

Gateva-Ivanova's conjecture. Every finite nondegenerate involutive square-free set-theoretic solution $r:X\times X\to X\times X$ such that $|X|\geq2$ is a multipermutation solution…

Logarithmic multipermutation-level conjecture. One has

Retractability and multipermutation conjecture. Every finite square-free solution $(X,r)$ is retractable. Moreover, every finite square-free solution $(X,r)$ of finite order $n$ is…