Corner’s Realization Theorems from the Viewpoint of Algebraic Entropy

Goldsmith, Brendan; Salce, Luigi

doi:10.1007/978-3-319-65874-2_12

Cited by 2 publications

(1 citation statement)

References 15 publications

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“…These were generalized to all abelian groups in [23]. Details and results can be found in [2,4,23,29,38,42,43], in [40,41] for the non-abelian case, in [66,67] for the algebraic entropy for modules; see also the surveys [26,28,33,44,45].…”

Section: Introductionmentioning

confidence: 99%

Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

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We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group.For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.In Example 2.28 we show that the naïve generalization of this lemma for semidirect products fails, while Theorem 2.27 gives the correct generalization.Recall that a semigroup S is called Ore semigroup (or right reversible semigroup as in [63]) if it satisfies the left Ore condition, that is, for every a, b ∈ S there exist f, g ∈ S such that f a = gb.It is a well-known fact (see [63, Proposition 1.23]) that a right amenable semigroup S satisfies the left Ore condition, and that the group generated by S is of the form S −1 S.We use the following consequence of this property.Corollary 2.10. Let S be a right amenable semigroup, and s 1 , . . . , s k ∈ S. Then there exist t, r 1 , . . . , r k ∈ S such that t = r i s i for every i ∈ {1, . . . , k}.We conclude with the following relation between amenable monoids and amenable groups.

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Section: Introductionmentioning

confidence: 99%

Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

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Hilbert polynomial of length functions

Fornasiero

2024

Annali di Matematica

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Let $$\lambda $$ λ be a general length function for modules over a Noetherian ring R. We use $$\lambda $$ λ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of $$\lambda $$ λ . We show that the leading term $$\mu $$ μ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for R[X]-modules. Similar to algebraic entropy, $$\mu $$ μ in general is not additive for exact sequences of R[X]-modules: we demonstrate how to adapt certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

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Corner’s Realization Theorems from the Viewpoint of Algebraic Entropy

Cited by 2 publications

References 15 publications

Algebraic entropy for amenable semigroup actions

Algebraic entropy for amenable semigroup actions

Hilbert polynomial of length functions

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