Monatsh Math DOI 10.1007/s00605-016-0953-6 Completions of paratopological groups and bounded sets Manuel Sanchis 1 · Mikhail Tkachenko 2 Received: 14 April 2016 / Accepted: 8 July 2016 © Springer-Verlag Wien 2016 Abstract In this paper we consider two questions in the realm of paratopological groups: When does multiplication on a given Tychonoff paratopological group H admit an extension to continuous multiplication on the Dieudonné completion, μ H , of H in such a way that H turns into a dense subgroup of the paratopological group μ H ? and, if A and B are bounded subsets of paratopological groups G and H , respectively, is A × B bounded in G × H ? The motivation for these questions comes from the field of topological groups and they are receiving a special attention as an interesting source of open problems. Keywords Topological semigroup · (para)topological group · Dieudonné completion · Hewitt realcompactification · Bounded subset · Glicksberg’s theorem Mathematics Subject Classification 22A05 · 22A30 · 54C50 · 54D35 · 54C20 Communicated by A. Constantin. The first author was supported by Universitat Jaume I Grant P1-1B2014-35 and Generalitat Valenciana Grant AICO/2016/030. B Manuel Sanchis sanchis@uji.es Mikhail Tkachenko mich@xanum.uam.mx 1 Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I de Castelló, Castellón de la Plana, Spain 2 Departamento de Matemáticas, Universidad Autónoma Metropolitana (Iztapalapa campus), Mexico, DF, Mexico 123