The existential transversal property: a generalization of homogeneity and its impact on semigroups

Let $G$ be a permutation group of degree $n$, and $k$ a positive integer with $k\le n$. We say that $G$ has the $k$-existential property, or $k$-et for short, if there exists a $k$-subset $A$ of the domain $\Omega$ such that, for any $k$-partition $\mathcal{P}$ of $\Omega$, there exists $g\in G$ mapping $A$ to a transversal (a section) for $\mathcal{P}$. This property is a substantial weakening of the $k$-universal transversal property, or $k$-ut, investigated by the first and third author, which required this condition to hold for all $k$-subsets $A$ of the domain. Our first task in this paper is to investigate the $k$-et property and to decide which groups satisfy it. For example, we show that, for $8\le k\le n/2$, the only groups with $k$-et are the symmetric and alternating groups; this is best possible since the Mathieu group $M_{24}$ has $7$-et. We determine all groups with $k$-et for $4\le k\le n/2$, up to some unresolved cases for $k=4,5$, and describe the property for $k=2,3$ in permutation group language. In the previous work, the results were applied to semigroups, in particular, to the question of when the semigroup $\langle G,t\rangle$ is regular, where $t$ is a map of rank $k$ (with $k<n/2$); this turned out to be equivalent to the $k$-ut property. The question investigated here is when there is a $k$-subset $A$ of the domain such that $\langle G, t\rangle$ is regular for all maps $t$ with image $A$. This turns out to be more delicate; the $k$-et property (with $A$ as witnessing set) is a necessary condition, and the combination of $k$-et and $(k-1)$-ut is sufficient, but the truth lies somewhere between. Given the knowledge that a group under consideration has the necessary condition of $k$-et, we solve the regularity question for $k\le n/2$ except for one sporadic group.