  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.
