10 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 $T_n$ be the full transformation monoid on an $n$-element set, and let $U_{k,l}$ be the monoid defined above. For a language $L$ over an alphabet of size $2$, write…

For $n e 7$ be an integer, choose coprime integers $k,l e 2$ with $k+l=n$, and let $U{k,l}$ be the transformation monoid defined by the two-cycle permutation construction above. **…

Tightness conjecture. The matching lower bounds in these three instances are tight.

Let $D$ be a digraph and let $\langle D\rangle$ be the semigroup generated by the arc transformations of $D$. A maximal subgroup of $\langle D\rangle$ is a subgroup associated with…

The minimum-length conjecture. For every odd $n\geq 3$, every $r\in[n-1]$, and every $T\in\mathrm{Tour}_n$,

The extremal length conjecture. For every $n\geq 3$, $r\in[n-1]$, and $T\in\mathrm{Tour}_n$,

Synchronization conjecture. Every primitive permutation group synchronizes every non-uniform map.

Let $\Omega$ be a finite set and let $G$ be a primitive permutation group on $\Omega$. A transformation $f$ of $\Omega$ has a non-uniform kernel when the inverse-image partition of…

Let $\Omega$ be a finite set, let $G$ be a primitive permutation group on $\Omega$, and let $f:\Omega\to\Omega$ be a transformation whose kernel partition has parts that are not al…