Abstract
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.
Highlights
1.1 Transformation and phase spaceLet Γ be a digraph, namely a pair consisting of its vertex set V(Γ) and arc set E(Γ) ⊆ V(Γ) × V(Γ)
The main purpose of this note is to propose a study of the possible regularity in the strong/weak shape of a semigroup acting on a valuated poset
The key lemma is Lemma 3.2, which gives us some information of the strong/weak shapes of a poset under some semigroup action, provided the semigroup consists of hereditary endomorphisms and that some linear map associated with the poset is injective
Summary
Let Γ be a digraph, namely a pair consisting of its vertex set V(Γ) and arc set E(Γ) ⊆ V(Γ) × V(Γ). The composition of maps provides an associative product on the set ΩΩ and turns it into a monoid, namely a semigroup with a multiplicative unit We call this monoid the full transformation monoid on Ω and denote it by T(Ω). If the transformation semigroup S is generated by a set G ⊆ ΩΩ, namely S consists of products of elements of G of positive length, we call (S, G) a deterministic automaton on Ω [67, §1]. For any set Ω, a subset of T(Ω) forms a permutation group on Ω whenever it is a transformation semigroup and each element has an inverse in it, namely it is a set of bijective transformations of Ω and is closed under compositions and taking inverses. Moving from group actions to semigroup actions is just to consider general deterministic automata instead of reversible ones
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