Primitive groups, road closures, and idempotent generation

We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if this semigroup is idempotent-generated for all possible choices of $a$, then $G$ is the symmetric or alternating group of degree $n$, with three exceptions (having $n=5$ or $n=6$). Our purpose here is to prove stronger results where we assume that $\langle G,a\rangle\setminus G$ is idempotent-generated for all maps of fixed rank $k$. For $k\ge6$ and $n\ge2k+1$, we reach the same conclusion, that $G$ is symmetric or alternating. These results are proved using a stronger version of the \emph{$k$-universal transversal property} previously considered by the authors. In the case $k=2$, we show that idempotent generation of the semigroup for all choices of $a$ is equivalent to a condition on the permutation group $G$, stronger than primitivity, which we call the \emph{road closure condition}. We cannot determine all the primitive groups with this property, but we give a conjecture about their classification, and a body of evidence (both theoretical and computational) in support of the conjecture. The paper ends with some problems.