7 problems

Let $X$ be a set, and let $\mathcal{T}(X)$ denote the full transformation semigroup on $X$. For the notation $N^{X}_{x_1,\ldots,x_t}$ and the functions $\xi$ and $\alpha$, use the…

Shortest-identity conjecture for $T_5$. This identity is the shortest identity of $T_5$. The preceding theorem establishes that this identity holds, while the conjecture concerns i…

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. **…

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.