Supplemented acts over monoids and their properties

In this talk, we generalize the notion of supplement in modules to S-acts for a monois S. ##In contrast to the case of modules, here we show that supplements of S-acts always exist and uniquely characterize the supplement of a proper subact of an S-act. ##We introduce supplemented acts as acts whose proper subacts all have proper supplements and study some connections between the property of being supplemented and some other properties of acts. ##Among other results, it is proved that supplemented acts are exactly completely reducible ones