On some kinds of ω-balancedness in semitopological groups

In this article, we introduce two notions which are called M-omega-balancedness and C-omega-balancedness, respectively. We get the following conclusions. A regular semitopological group G is topologically isomorphic to a subgroup of the product of a family of Moore semitopological groups if and only if G is M-omega-balanced and Ir(G) <= omega. A T(1 )semitopological group G is topologically isomorphic to a subgroup of the product of a family of T-1 semitopological groups with a point-countable base if and only if G is C-w-balanced and Sm(G) <= omega. If G is a Hausdorff countably compact semitopological group with Hs(G) <= omega, then the M-omega-balancedness (C-omega-balancedness) of G implies that G is a topological group. Let G be an omega-balanced paratopological group and let e be the identity of G. We show that if for each U is an element of N(e) there exist omega-good set V is an element of N(e) with V subset of U and A subset of G such that boolean OR{g(V boolean AND V-1) : g is an element of A} = G and {g(V boolean AND V-1)V : g is an element of A} is star-countable, then G is completely omega-balanced, where N(e) is the family of open neighborhoods of e in G. (C) 2020 Elsevier B.V. All rights reserved.