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 <= n. We say that G has the k-existential transversal property, or k-et, if there exists a k-subset A (of the domain Omega) whose orbit under G contains transversals for all k-partitions P of Omega. 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 Omega.Our first task in this paper is to investigate the k-et property and to decide which groups satisfy it. For example, it is known that for k < 6 there are several families of k-transitive groups, but for k > 6 the only ones are alternating or symmetric groups; here we show that in the k-et context the threshold is 8, that is, for 8 <= k <= n/2, the only transitive groups with k-et are the symmetric and alternating groups; this is best possible since the Mathieu group M24 (degree 24) has 7-et. We determine all groups with k-et for 4 <= k <= n/2, up to some unresolved cases for k = 4, 5, and describe the property for k = 2, 3 in permutation group language. These considerations essentially answer Problem 5 proposed in the paper on k-ut referred to above; we also slightly improve the classification of groups possessing the k-ut property.In that earlier paper, the results were applied to semigroups, in particular, to the question of when the semigroup < G,t > 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 (G,0 is regular for all maps t with image A. This turns out to be much 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, the regularity question for k < n/2 is solved except for one sporadic group. The paper ends with a number of problems on combinatorics, permutation groups and transformation semigroups, and their linear analogues.