We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in products and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.

We prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

A group A is said to be endoprimal if its term functions are precisely the functions which permute with all endomorphisms of A. In this paper we describe endoprimal groups in the following three classes of abelian groups: torsion groups, torsionfree groups of rank at most 2, direct sums of a torsion group and a torsionfree group of rank 1.