Top-heavy phenomena for transformations

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