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 } , we define the class of locally T i -minimal paratopological groups by the conditions that, for a T i 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 T i -semigroup topology on G which is strictly coarser than τ. We mainly prove that (1) each UFSS and T i -paratopological Abelian group ( G , τ ) is locally T i -minimal; (2) if ( G , τ ) is a regular locally T 1 -minimal Abelian paratopological group then χ ( G ) = π χ ( G ) ; (3) if ( G , τ ) is an Abelian locally T 3 -minimal paratopological group then we have w ( G ) = n w ( G ) . Moreover, we also discuss some relations of locally T i -minimal paratopological groups and some properties of subgroups of T i -minimal paratopological groups. Some questions are posed.