Top-heavy phenomena for transformations

english

2022

volume 22, issue 4

incidence operator   kernel space   rank   strong shape   valuated poset

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

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

02.10.2023