On dense left ideal pure $S$-acts

In this paper, similar to the Lembek's idea concerning a generalization of the notion of purity associated with a radical in the category of R-modules, we give the notion of purity related to a Hoehnke radical, d.l.i.pure, in the category of S-acts and investigate this notion. We also show that absolutely d.l.i.pure S-acts are exactly the r-weakly injective S-acts, for every Hoehnke radical r, and we give several characterization of d.l.ipure S-acts. We then present a closure operator induced by an r-dense left ideal which is closely related to the notion of d.l.i.purity and give a characterization of d.l.i.pure  S-acts by this closure operator.