LAUSR

Abstract For a given paratopological group ( H , τ ) , the quasitopological group Q 2 ( H ) with a weaker topology than τ were investigated and some problems were posed by I. Sanchez and M. Tkachenko. In this note, we give partial answers to two of the problems. Some cardinal inequalities are obtained: (1) if H is a quasitopological group, then i b ( H ) ≤ e ( H ) ; (2) if H is a paratopological group, then i b ( Q 3 ( H ) ) ≤ w l ( H ) ; (3) if H is a saturated semitopological group, then i b ( H ) ≤ s ( H ) ; (4) if H is a semitopological group, then i b ( Q 3 ( H ) ) ≤ c ( H ) .