On Regularity of Acts

In this article we give a characterization of monoids for which torsion freeness, ((principal) weak, strong) flatness, equalizer flatness or Condition (E) of finitely generated and (mono) cyclic acts and Condition (P) of finitely generated and cyclic acts implies regularity. A characterization of monoids for which all (finitely generated, (mono) cyclic acts are regular will be given too. We also show that monoids for which all regular right acts are WPF, WKF, PWKF, TKF or satisfy Conditions (P), (WP) and (PWP) are the same as those for which all regular right acts are projective or strongly flat. Monoids S with E(S)?C(S) or those for which every element of E(S)\{1} is right zero will be characterized, when all (finitely generated, cyclic) right S-acts satisfying Conditions (PWP), (WP), (PE) or (P) are regular. Simple monoids for which all (finitely generated, cyclic) right acts with property U (U is a property of acts over monoids implied by Condition (P)) are regular will be characterized too.