Top-heavy phenomena for transformations

Let S be a transformation semigroup acting on a set Ω. The action of S on Ω can be naturally extended to be an action on all subsets of Ω. We say that S is `-homogeneous provided it can send A to B for any two (not necessarily distinct) `-subsets A and B of Ω. On the condition that k ≤ ` < k + ` ≤ |Ω|, we show that every `-homogeneous transformation semigroup acting on Ω must be k-homogeneous. We report other variants of this result for Boolean lattices and projective geometries. In general, any semigroup action on a poset gives rise to an automaton and we associate some sequences of integers with the phase space of this automaton. When the poset is a geometric lattice, we propose to study various possible regularity properties of these sequences, especially the so-called top-heavy property.