Local minimalities in paratopological groups

In this paper, we mainly discuss the cardinal invariants on some class of paratopological groups. For each i∈{0,1,2,3,3.5}i∈{0,1,2,3,3.5}, we define the class of locally TiTi-minimal paratopological groups by the conditions that, for a TiTi paratopological group (G,τ)(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 TiTi-semigroup topology on G which is strictly coarser than τ  . We mainly prove that (1) each UFSS and TiTi-paratopological Abelian group (G,τ)(G,τ) is locally TiTi-minimal; (2) if (G,τ)(G,τ) is a regular locally T1T1-minimal Abelian paratopological group then χ(G)=πχ(G)χ(G)=πχ(G); (3) if (G,τ)(G,τ) is an Abelian locally T3T3-minimal paratopological group then we have w(G)=nw(G)w(G)=nw(G). Moreover, we also discuss some relations of locally TiTi-minimal paratopological groups and some properties of subgroups of TiTi-minimal paratopological groups. Some questions are posed.