Geodesic
Top-heavy phenomena for transformations
Ars Mathematica Contemporanea, Tome 22 (2022) no. 4, article no. 09, 26 p.

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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 ℓ-homogeneous provided it can send A to B for any two (not necessarily distinct) ℓ-subsets A and B of Ω. On the condition that k ≤ ℓ  k + ℓ ≤ |Ω|, we show that every ℓ-homogeneous transformation semigroup acting on Ω 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.
DOI : 10.26493/1855-3974.1753.52a
Keywords: Incidence operator, kernel space, rank, strong shape, valuated poset, weak shape 
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Yaokun Wu; Yinfeng Zhu. Top-heavy phenomena for transformations. Ars Mathematica Contemporanea, Tome 22 (2022) no. 4, article  no. 09, 26 p. doi : 10.26493/1855-3974.1753.52a. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1753.52a/

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