18 problems

Let $T_n$ be the full transformation monoid on an $n$-element set. A monoid is synchronising if it contains a constant mapping. Cameron's conjecture. The probability that two rando…

Let $S \subseteq \mathbb{Q}^{n \times n}$ be a finite semigroup of rational matrices containing the zero matrix, generated by a set $S_0$. The mortality threshold of $S$ is the min…

Null-summand conjecture. There is some $m\ge 1$ such that $Z$ is a direct summand of $V^{\otimes m}$, and consequently

Let … be a Burnside semigroup, and for $s\in\mathcal{B}(m,n)$ let $S^{\mathscr{J}}_s$ be the finite semigroup formed by the elements $\mathscr{J}$-above $s$, together with zero…

Let $(S,A)$ be a finite semigroup with generating set $A$, and let $\mathsf{KR}(S,A)$ and $\mathsf{Mc}$ denote the Karnofsky–Rhodes and McCammond expansions, respectively. The expa…

Let $S$ be a finite semigroup with generating set $A$, let $(\mathsf{KR}(S,A),A)$ be its Karnofsky–Rhodes expansion, and let $K(\mathsf{KR}(S,A))$ be a Kleene expression for its no…

Let $Q_8$ be the quaternion group of order eight and $D_4$ the dihedral group of order eight. Write $\langle\!\langle S\rangle\!\rangle$ for the compact pseudovariety generated by…

Let ${\sf Sl}$ be the pseudovariety of finite semilattices, let ${\sf L}$ be the pseudovariety of finite locally trivial semigroups, and let ${\sf GLT}$ denote the pseudovariety of…

Let $S$ be a finite semigroup. A subsemigroup $T$ of $S$ contains the idempotents if it contains every idempotent of $S$, and it satisfies the closure condition that whenever…

Let $p>2$ be a prime and let $f(x)\in \mathbb{F}_p[x]$ be any nonconstant polynomial. Let $R=\mathbb{F}_p[x]/(f(x))$, let $\mathcal{S}_R$ be the multiplicative semigroup of $R$, an…

Let $p>2$ be a prime, let $f(x)\in \mathbb{F}_p[x]$ be a non-constant polynomial, and let $\mathcal{S}_{f(x)}^p$ denote the multiplicative semigroup of the quotient ring…

Let $\Gamma$ be a finite directed acyclic graph. Its universal dynamics monoid $D(\Gamma)$ is the smallest quotient of $\operatorname{\mathbf{HK}}_\Gamma$ through which all evaluat…

Kleitman–Rothschild–Spencer conjecture. Almost all finite semigroups are nilpotent of index at most three.

Erdős–Ginzburg–Ziv conjecture for semigroups. For any finite commutative semigroup $\mathcal{S}$,

Let $A_1,\dots,A_k\in\operatorname{Mat}_{n}(\mathbb{F})$, with $k$ arbitrary, and let the semigroup generated by $A_1,\dots,A_k$ be finite and contain the zero matrix $O$. By a pro…

Let $S$ be a finite $3$-nilpotent semigroup, meaning that every product of three elements of $S$ is zero. The effective dimension over a field is the least dimension of an effectiv…

Subsemigroup conjecture. If $S$ is not adequate, then $S$ contains a subsemigroup isomorphic to $M$.

Let $M$ be the finite monoid under consideration, let $X$ be its generating alphabet, let $R$ be the relevant set on which $Omega(X)$ acts, and let $P(M)$ denote the power set of…