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