10 problems

Let $G$ be a finitely generated group, and let $S$ be a finite generating set. Write $\mathrm{cp}_S(G)$ for the limiting commuting probability associated with the word-ball exhaust…

The commuting-degree realization conjecture. Every value in $\mathcal{CD}(n)$ is the commuting degree of some BCK-algebra of order $n$, for every $n\geq 3$.

Joseph's conjectures. One should have $\ell\in\mathbb{Q}$, $x_i\geq\ell$ for all but finitely many $i$, and $\ell\in S$. Equivalently, the second conjecture says that $S$ is well-o…

Let $\mathfrak R$ be the class of finite rings, and let $\mathfrak{R}_{sa}$ denote the class of finite semiassociative rings. Write $\mathfrak{S}_c(\mathfrak R)$ for the set of com…

Antolín, Martino and Ventura's meta-conjecture. For any “reasonable” sequence $M=(\mu_n)_{n=1}^\infty$ of probability measures on $G$, we have $\operatorname{dc}_M(G)>0$ if and onl…

Joseph's conjectures. The following assertions hold:

Let $G_{1}$ and $G_{2}$ be finite groups, and let $Pr^{2n}(G)$ denote the generalized commuting probability of $G$ at level $2n$. Write $Z(G)$ for the center of $G$. The conjecture…

Let $G$ range over finite groups, and write $\operatorname{Pr}(\mathcal{G})$ for the set of their commuting probabilities. Joseph's third conjecture. The set … is a closed subset o…

Let $G$ range over finite groups, and write $\operatorname{Pr}(\mathcal{G})$ for the set of their commuting probabilities. If $l$ is a limit point of this set, then Joseph's second…

Let $G$ range over finite groups, and write $\operatorname{Pr}(\mathcal{G})$ for the set of their commuting probabilities. A limit point of a subset $S\subseteq\mathbb{R}$ is a rea…