C-compact and r-pseudocompact subsets of paratopological groups

Our study of C-compactness, r  -pseudocompactness, and close notions is motivated by the fact that an arbitrary product ∏i∈IBi∏i∈IBi of C  -compact subsets BiBi of respective topological groups GiGi is C  -compact in the product group ∏i∈IGi∏i∈IGi, and the same conclusion remains valid for products of r-pseudocompact subsets of topological groups. In fact, it is known that the two notions of boundedness coincide for subsets of topological groups (but they are quite different for subsets of Tychonoff spaces).Our aim here is to extend the aforementioned results to paratopological groups. We find several wide classes of paratopological groups in which the C-compact and r-pseudocompact subsets coincide (these include totally ω-narrow paratopological groups, commutative paratopological groups with countable Hausdorff number, precompact or Lindelöf paratopological groups). Similarly, we present several classes of paratopological groups in which C-compact and/or r-pseudocompact subsets remain to be productive, as in topological groups.