Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is (omega)-balanced (totally omega-narrow) and the Hausdorff number of G is Countable. i.e., for every neighbourhood U of the neutral element e of G there exists a Countable family gamma of neighbourhoods of e such that boolean AND(V is an element of gamma) VV-1- subset of U. Similarly, we prove that a regular paratopological group G call be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is (omega)-balanced (totally omega-narrow) and the index of regularity of G is countable. As a by-product. we show that a regular totally omega-narrow paratopological group with Countable index of regularity is Tychonoff. (C) 2008 Elsevier B.V. All rights reserved.