We show that the two-sided quasi-uniformity of a regular paratopological group G is quiet. The Doitchinov completion DG of G can be considered a paratopological group containing G as a doubly dense subgroup whenever G is Abelian. Furthermore the completion quasi-uniformity on DG is the two-sided quasi-uniformity of DG. These results generalize in an appropriate way important results about topological groups to regular paratopological groups. A counterexample dealing with the non-Abelian case is presented. Furthermore we give conditions, depending on quasi-uniform completeness properties, under which a paratopological group is a topological group.

This article has been published. H.-P. A. K\"unzi, S. Romaguera\ and\ O. V. Sipacheva, The Doitchinov completion of a regular paratopological group. Serdica Math. J. 24 (1998), no. 1, 73--88; MR 2000c:54019; Zbl 0965.54029