Related papers: On separability properties in direct products of s…

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…

We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals…

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of…

We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.

One approach to monitoring a dynamic system relies on decomposition of the system into weakly interacting subsystems. An earlier paper introduced a notion of weak interaction called separability, and showed that it leads to exact…

Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and let $N$ be its free submodule. We consider the problem when a given element of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of the…

We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of…

Recently, the authors gave some conditions under which a direct product of finitely many groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. In this paper, we extend…

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the…

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…

We find necessary and sufficient conditions for the finite separability of finitely generated commutative rings. Namely, we prove that every such ring is a finite extension of its torsion ideal $I_k$ where $k$ is square-free, and $I_k$ is a…

Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.

Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We…

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in…

The direct product $\mathbb{N}\times\mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise non-isomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite…

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are…

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact…

We prove that finitely presented residually free groups are subgroup conjugacy separable. Furthermore, if they are of type $FP_\infty$, then they are also subgroup conjugacy distinguished. Using a connection between conjugacy separability…

We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include…