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We prove an instance of the so-called Addition Theorem for the algebraic entropy of actions of cancellative right amenable monoids $S$ on discrete abelian groups $A$ by endomorphisms, under the hypothesis that $S$ is locally monotileable (that is, $S$ admits a right F\o lner sequence $(F_n)_{n\in\mathbb N}$ such that $F_n$ is a monotile of $F_{n+1}$ for every $n\in\mathbb N$). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by Weiss and developed further by other authors recently.

We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. 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.

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; they 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 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.

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 that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.

We study the topological entropy h ( f ) of continuous endomorphisms f of compact-like groups. More specifically, we consider the e-spectrum E t o p ( K ) for a compact-like group K (namely, the set of all values h ( f ) , when f runs over the set E n d ( K ) of all continuous endomorphisms of K). We pay particular attention to the class E ∞ of topological groups without continuous endomorphisms of infinite entropy (i.e., ∞ ∉ E t o p ( K ) ) as well as the subclass E 0 of E ∞ consisting of those groups K with E t o p ( K ) = { 0 } . It turns out that the properties of the e-spectrum and these two classes are very closely related to the topological dimension. We show, among others, that a compact connected group K with finite-dimensional commutator subgroup belongs to E ∞ if and only if dim ⁡ K ∞ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ω-bounded (in particular, compact). Examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results (e.g., compact connected semi-simple groups K with dim ⁡ K = ∞ and K ∈ E 0 , or countably compact connected abelian groups with the same property). Since the class E ∞ is not stable under taking closed subgroups or quotients, we study also the largest subclasses S ( E ∞ ) and Q ( E ∞ ) , respectively, of E ∞ , having these stability properties. We provide a complete description of these two classes in the case of compact groups, that are either abelian or connected. The counterpart for S ( E 0 ) and Q ( E 0 ) is done as well.

This is a survey on the contribution of W. Wistar Comfort to the theory of topological groups through the looking glass of “dense subgroups of topological groups”. It turns out that under this umbrella one can find an amazing variety of results, as well as problems posed by Comfort that gave a significant input in the progress in the field of topological groups in the last five decades.

Given a compact abelian group G, we study the poset G δ ( G ) of G δ -dense subgroups of G and the subgroup δ - den ( G ) : = ⋂ G δ ( G ) , named G δ -density nucleus of G. It turns out that for every compact abelian group G: (i) the subgroup δ - den ( G ) is compact metrizable and coincides with the intersection of just two members D 0 , D 1 ∈ G δ ( G ) ; (ii) if δ - den ( G ) = { 0 } , then there exists some independent family F in G δ ( G ) (i.e., with 〈 ⋃ F 〉 = ⨁ i ∈ I D i ) such that: (ii1) | F | = r ( G ) , the free rank of G, if mG is not metrizable for any positive m ∈ N ; otherwise, (ii2) when G is (necessarily) bounded torsion, | F | coincides with the least leading Ulm-Kaplansky invariant of G. (iii) in option (ii1) the members of F can be chosen to be free subgroups of G precisely when G admits a free dense subgroup (equivalently, when r ( G ) ≥ d ( G ) ); F can have (the maximum possible) size | G | if and only if r ( G ) = | G | ; (iv) in option (ii2) the family F can have size | G | if and only if all leading Ulm-Kaplansky invariants of G coincide with | G | . Resuming, the compact abelian groups G with trivial G δ -density nucleus δ - den ( G ) are, roughly speaking, “maximally fragmented” into G δ -dense subgroups. More precisely, the existence of a pair of subgroups D 0 , D 1 ∈ G δ ( G ) with D 0 ∩ D 1 = { 0 } yields the existence of a large independent family F = { D i ∈ I } ⊆ G δ ( G ) with I is as big as possible (i.e., as big as the algebraic structure of G permits in terms of some immediate necessary restraints involving relevant algebraic cardinal invariants of the group, in the spirit of those in (ii)–(iv)).

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

We study the locally compact abelian groups in the class $\mathfrak E_{, Comment: 16 pages

Every group G carries an intrinsically defined (by means of solution sets of one-variable equations) topology Z G , named Zariski topology. It is related to another topology M G having as closed sets all unconditionally closed sets of G, named Markov topology, after Markov who implicitly introduced both topologies in dealing with a series of problems related to group topologies. The aim of this survey is to enlighten the utility of these topologies in resolving Markov problems, as well as other challenging problems in the area of topological groups, mainly related to topologization via group topologies with certain properties. We show that these topologies shelter under the same umbrella as distant issues as abelian groups and highly non-abelian ones, as permutation groups and homeomorphism groups.

A subgroup H of the circle group $$\mathbb T$$ is said to be characterized by a sequence $$\mathbf {u}= (u_n)_{n\in \mathbb N}$$ of integers if $$H=\{x\in \mathbb T: u_nx\rightarrow 0\}$$ . The characterized subgroups of $$\mathbb T$$ are known also under the name topologically $$\mathbf {u}$$ -torsion subgroups. This survey paper is dedicated to the characterized subgroups of $$\mathbb T$$ : we recall their main properties and collect most of the basic results from the wide bibliography, following, when possible, the historical line, and trying to show the deep roots of this topic in several areas of Mathematics. Due to this universality of the topic, many notions and results were found independently by various authors working unaware of each other, so our effort is also directed towards giving credit to all of them to the best of our knowledge. We provide also some background on the notions of characterized subgroup and topologically $$\mathbf {u}$$ -torsion subgroup in the general case of topological abelian groups, where they differ very substantially.

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = ( u n ) n ∈ N 0 in N such that ‖ u n x ‖ → 0 uniformly on A . In particular, A is contained in the subgroup t u ( T ) : = { x ∈ T : ‖ u n x ‖ → 0 } , which is the subgroup of T characterized by u . Using strictly increasing sequences u in N such that u n divides u n + 1 for every n ∈ N , we find in T a family of closed perfect D -sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D -sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

We study properties of the pseudocompact spaces X with a weak selection, and we dedicate a particular attention to the weak selection topologies on X. In case when X is also locally compact, we obtain a convenient decomposition of X into a finite union of clopen sets, which are either almost compact or connected with a remainder of size two in their Stone–Cech compactification.

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

Let G be a locally essential subgroup of a locally compact abelian group K. Then: (i) t ( G ) = χ ( G ) = χ ( K ) , where t ( G ) and χ ( G ) are the tightness and the character of G, respectively; (ii) if G is radial, then K must be metrizable; (iii) if G is non-discrete, then G contains a super-sequence S converging to 0 such that | S | = χ ( G ) = χ ( K ) ; in particular, G has non-trivial convergent sequences. Items (i)–(iii) hold when G is a dense locally minimal subgroup of K. It follows that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal κ, we construct a Frechet-Urysohn minimal nilpotent group G of nilpotency class 2 and character κ such that the connected component of G is an open normal ω-bounded subgroup of G (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.

Research articles: Global Krull dimensions and global dual Krull dimension of valuation rings The Luroth theorem for some complete valued fields AB-5 for module and ring extensions Ko of semiartinian unit-regular rings Idempotents of the class semigroup of a Prufer domain of finite character Greatest common divisors generalized Subgroups of the Baer-Specker group with prescribed endomorphism ring and large dual Isomorphic automorphism groups of torsion-free p-acid modules A theorem on extensions of bimodules Communicative rings with acc on irreducible ideals Self linearly compact rings and dualities Extensions of compact Abelian groups by discrete ones and their duality theory II A note on quasi-duality endomorphism, rings of modules whose cardinality is cofinal to Omega Endomorphism rings and automorphism groups of separable torsionfree modules over valuation domains Torsionless modules and rings with finite essential socle Building noetherian domains inside an ideal-adic completion G-minimal topological groups An adjointness relation for finite partition lattices The variety of topological groups generated by the class of all Banach spaces On thin generating sets in topological groups Abelian topological groups without irreducible Banach representations Cyclic modules over end (Vp) whose endomorphism ring is an ultrapower of D Automorphisms that determine an Abelian p-group Dualities and pure-semisimple rings Suitable sets in products of topological groups and in groups equipped with the Bohr topology On a duality with less than usual reflexive abstract modules The Roelcke compactification of unitary groups Tilting in module categories. (Part contents).

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Ausenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Diaz Nieto and Martin Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory., 28 pages

In this paper we define some ballean structure on the power set of a group and, in particular, we study the subballean with support the lattice of all its subgroups. If $G$ is a group, we denote by $L(G)$ the family of all subgroups of $G$. For two groups $G$ and $H$, we relate their algebraic structure via the ballean structure of $L(G)$ and $L(H)$., ballean, coarse structure, coarse map, asymorphism, asymptotic dimension, balleas defined by ideals, hyperballeans

We discuss the growth function of a finitely generated cascade and its connection to the growth function of its related semi-direct product (Conjecture 1.9). The results is applied for simpler proo...

A topological group G is called locally minimal if there exists a neighbourhood V of 1 in G such that if H is a Hausdorff group and f : G → H is a continuous isomorphism such that f ( V ) is a neighbourhood of 1 in H, then f is open. This paper is focused on the study of quotients of locally minimal groups. A topological group G is called locally q-minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f : G → H is a continuous surjective homomorphism such that f ( V ) is a neighbourhood of 1 in H, then f is open. We find a close connection between locally q-minimality and divisibility, by showing that a dense subgroup of R n is locally q-minimal if and only if it is divisible. A description of the locally q-minimal subgroups of the n-dimensional torus T n is also given. Two weaker versions of local q-minimality are proposed – a topological group G is: (a) locally t-minimal, if all Hausdorff quotients of G are locally minimal; (b) locally q ⁎ -minimal, if there exists a neighbourhood V of 1 in G such that whenever H is a Hausdorff group and f : G → H is a continuous surjective homomorphism such that f ( V ) is a neighbourhood of 1 in H and ker ⁡ f ⊆ V , then f is open. While local q-minimality and local t-minimality are preserved by taking quotients, local q ⁎ -minimality fails to have this property (so does not coincide with local t-minimality), yet it has a relevant advantage. By means of an appropriate notion of local t-density, one can show that a dense subgroup H of a Hausdorff group G is locally q ⁎ -minimal if and only if G is locally q ⁎ -minimal and H is locally t-dense in G. Similar criterion for local q-minimality is not available (examples are given to show that a topological group with a dense locally q-minimal subgroup need not be locally q-minimal). The interrelations of these three versions of local minimality, as well as their prominence properties are studied in detail.

We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the underlying space and $T:X\to X$ a transformation. We see entropies as functions $h:\mathfrak X\to \mathbb R_+$, associating to each flow $(X,T)$ of a category $\mathfrak X$ either a non negative real or $\infty$. We introduce the notion of semigroup entropy $h_\mathfrak S:\mathfrak S\to\mathbb R_+$, which is a numerical invariant attached to endomorphisms of the category $\mathfrak S$ of normed semigroups. Then, for a functor $F:\mathfrak X\to\mathfrak S$ from any specific category $\mathfrak X$ to $\mathfrak S$, we define the functorial entropy $h_F:\mathfrak X\to\mathbb R_+$ as the composition $h_{\mathfrak S}\circ F$. Clearly, $h_F$ inherits many of the properties of $h_\mathfrak S$, depending also on the properties of $F$. Such general scheme permits to obtain relevant known entropies as functorial entropies $h_F$, for appropriate categories $\mathfrak X$ and functors $F$, and to establish the properties shared by them. 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 formalize the notion of Bridge Theorem between two entropies $h_i:\mathfrak X_i\to \mathbb R_+$, $i=1,2$, with respect to a functor $\varepsilon:\mathfrak X_1\to\mathfrak X_2$. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem, which allows us to put under the same umbrella various relations between pairs of entropies.