On regular torsionless $S$-posets

This paper shall be concerned with the notion of regular torsionless in the category of $S$-posets. Besides elementary basic properties of regular torsionless $S$-posets, we consider cyclic regular torsionless $S$-posets and also study when regular torsionless property is preserved under coproducts. Then we characterize pomonoids over which all free or projective $S$-posets are regular torsionless. Finally, we present conditions on $S$ which follow if all regular torsionless $S$-posets are principally weakly po-flat, weakly po-flat, strongly flat, or projective.