String Numbers of Abelian Groups

Bruno, Anna Giordano; Virili, Simone

doi:10.1142/s0219498812501617

Cited by 3 publications

(2 citation statements)

References 19 publications

(75 reference statements)

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“…Then we discuss the contravariant set-theoretic entropy h * from [36]. These entropies h and h * are related to invariants for selfmaps of sets (i.e., the string number and the antistring number, see [3,45,58,65]).…”

Section: Set-theoretic Entropymentioning

confidence: 99%

Entropy on normed semigroups (towards a unifying approach to entropy)

Dikranjan

Bruno

2019

Dissertationes Math.

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We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, various forms of topological entropy, several notions of algebraic entropy, and two forms set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs (X, T ), where X is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and T : X → X is a transformation of X (e.g., a measure preserving transformation, a continuous selfmap, a group homomorphism, a selfmap). We see entropies as functions h : X → R + , associating to each flow (X, φ) of a category X either a non negative real number or ∞.First, we introduce the notion of semigroup entropy h S : S → R + , which is a numerical invariant attached to endomorphisms of the category S of normed semigroups. Then, for a functor F : X → S from any specific category X to S, we define the functorial entropy h F :Clearly, this entropy h F inherits many of the properties of h S , depending also on the properties of the functor F . Motivated by this aspect, we study in detail the properties of h S . Such general scheme, using elementary category theory, permits to obtain many relevant known entropies as functorial entropies h F , for appropriately chosen categories X and functors F : X → S. For example, most of the above mentioned measure entropy, topological entropies, algebraic entropies and set-theoretic entropies are functorial entropies. Furthermore, we exploit our scheme to elaborate a common approach to establish the properties shared by those entropies that we find as functorial entropies. In this way we point out their common nature.Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we first formalize the notion of Bridge Theorem between two entropies h 1 : X 1 → R + and h 2 : X 2 → R + with respect to a functor ε : X 1 → X 2 , taking inspiration from the theorem relating the topological and the algebraic entropy by means of Pontryagin duality. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem. It allows us to put under the same umbrella various relations between pairs of entropies (e.g., the above mentioned connection of the topological and the algebraic entropy and their relation with the set-theoretic entropy).

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Section: Set-theoretic Entropymentioning

confidence: 99%

Entropy on normed semigroups (towards a unifying approach to entropy)

Dikranjan

Bruno

2019

Dissertationes Math.

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“…In connection with the topological entropy, the algebraic entropy h alg of group endomorphisms was introduced somewhat later (see [2,22,37,46,47,52]), and the adjoint algebraic entropy h * alg more recently (see [24,40]). Moreover, the set-theoretic entropy h set of selfmaps of a set provided with no further structure was defined in [3] (see also [28,33,38]), and used for computing the topological entropy of generalized shifts. For the details about the origin of all these entropies as well as the connections between them see the surveys [19,31].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic entropy for generalized quasimetric semilattices

Castellano¹,

Dikranjan²,

Freni³

et al. 2020

Preprint

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We introduce the notion of intrinsic semilattice entropy h in the category L qm of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories X and functors F : X → L qm , we find specific known entropies h X on X as intrinsic functorial entropies, that is, as h X = h • F . These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.

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Discrete dynamical systems in group theory

Dikranjan

Bruno

2013

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String Numbers of Abelian Groups

Cited by 3 publications

References 19 publications

Entropy on normed semigroups (towards a unifying approach to entropy)

Entropy on normed semigroups (towards a unifying approach to entropy)

Intrinsic entropy for generalized quasimetric semilattices

Discrete dynamical systems in group theory

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