Local minimalities in paratopological groups

Abstract

In this paper, we mainly discuss the cardinal invariants on some class of paratopological groups. For each i∈{0,1,2,3,3.5}, we define the class of locally Ti-minimal paratopological groups by the conditions that, for a Ti paratopological group (G,τ), there exists a τ-neighborhood U of the neutral element such that U fails to be a neighborhood of the neutral element in any Ti-semigroup topology on G which is strictly coarser than τ. We mainly prove that (1) each UFSS and Ti-paratopological Abelian group (G,τ) is locally Ti-minimal; (2) if (G,τ) is a regular locally T1-minimal Abelian paratopological group then χ(G)=πχ(G); (3) if (G,τ) is an Abelian locally T3-minimal paratopological group then we have w(G)=nw(G). Moreover, we also discuss some relations of locally Ti-minimal paratopological groups and some properties of subgroups of Ti-minimal paratopological groups. Some questions are posed.