Cardinal invariants in locally -minimal paratopological groups

In this paper, cardinal invariants in locally T i -minimal paratopological groups with i = 1 , 2 , 3 are studied. It mainly shows that: (1) If ( G , τ ) is a T 2 locally T 1 -minimal 2-oscillating paratopological group, then χ ( G ) = π χ ( G ) ⋅ inv ( G ) ; (2) Let ( G , τ ) be a locally T 1 -minimal paratopological group, then χ ( G ) = ψ ( G ) ⋅ inv ( G ) ; (3) If ( G , τ ) is a locally T 2 -minimal paratopological group, then χ ( G ) = ψ ( G ) ⋅ inv ( G ) ⋅ H s ( G ) ; (4) If ( G , τ ) is a locally T 3 -minimal paratopological group, then χ ( G ) = ψ ( G ) ⋅ inv ( G ) ⋅ Ir ( G ) . These results generalize the corresponding results in [9] and also give positive answers to two questions posed by F.C. Lin in [9].