9 papers
Order convergence in ordered vector spaces: compactness, atoms, pervasiveness, and ice cream spaces
Eugene Bilokopytov, Mark Roelands, Samuel Tiersma +3
We study order convergence in ordered vector spaces that are not necessarily lattices. In this setting, we consider two generalizations of order convergence on vector lattices, a d…
The Dedekind completion of an Archimedean ordered vector space as a reflector
Antonio Avilés, Eugene Bilokopytov
We consider the category of Archimedean ordered vector spaces with linear maps which preserve all existing suprema, and its full subcategories , $\m…
Normed lattices majorizing in their norm completions
Eugene Bilokopytov, Viktor Bohdanskyi
This note is a follow-up to \cite{bt}. We focus on conditions under which a normed lattice is majorizing in its norm completion. We show that \cite[Question 8.17]{bt} -- namely…
Relative uniform completion of a vector lattice
Eugene Bilokopytov, Vladimir G. Troitsky
In the paper, we revisit several approaches to the concept of uniform completion of a vector lattice . We show that many of these approaches yield the same res…
Countability conditions in locally solid convergence spaces
Eugene Bilokopytov, Viktor Bohdanskyi, Jan Harm van der Walt
We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which re…
Locally solid convergence structures
E. Bilokopytov, J. Conradie, V. G. Troitsky +1
While there is a well developed theory of locally solid topologies, many important convergences in vector lattice theory are not topological. Yet they share many properties with lo…
Atomicity of Boolean algebras and vector lattices in terms of order convergence
Antonio Avilés, Eugene Bilokopytov, Vladimir G. Troitsky
We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an A…
Uniformly closed sublattices of finite codimension
Eugene Bilokopytov, Vladimir G. Troitsky
The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (o…
Order and uo-convergence in spaces of continuous functions
Eugene Bilokopytov, Vladimir G. Troitsky
We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions , , , and , extending results of [vdW18…