Sasane, Amol
Published in
Topological Algebra and its Applications

The set 𝒜 := δ0+ 𝒟+′, obtained by attaching the identity δ0 to the set 𝒟+′ of all distributions on with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equivalentl...

Banakh, Taras Bardyla, Serhii Ravsky, Alex
Published in
Topological Algebra and its Applications

We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergen...

Banakh, Taras Bardyla, Serhii Guran, Igor Gutik, Oleg Ravsky, Alex
Published in
Topological Algebra and its Applications

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special c...

dal Verme, Giulia Weigel, Thomas
Published in
Topological Algebra and its Applications

In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C*-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives ris...

Dardano, Ulderico Dikranjan, Dikran Salce, Luigi
Published in
Topological Algebra and its Applications

If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms th...

Bruno, Anna Giordano
Published in
Topological Algebra and its Applications

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

Castellano, I. Cook, G. Corob Kropholler, P.H.
Published in
Topological Algebra and its Applications

We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.

Castellano, Ilaria
Published in
Topological Algebra and its Applications

Let 𝕂 be a discrete field and (V, ϕ) a pair consisting of a locally linearly compact 𝕂-space V and a continuous endomorphism ϕ: V → V. We provide the formulae to compute the topological entropy ent* of the flow (V, ϕ) subject to either extension or restriction of scalars.

Virili, Simone
Published in
Topological Algebra and its Applications

Inspired by the work of Crawley-Boevey on additive functions in locally finitely presented Grothendieck categories, we describe a natural way to extend a given exact Sylvester rank function on the category of finitely presented left modules over a given ring R, to the category of all left R-modules.

Cánovas, Jose S.
Published in
Topological Algebra and its Applications

In this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.