The primitive normality of a class of weakly injective $S$-acts

The notion of weakly injective $S$-act can be regarded as a generalization of the notion of injective $S$-act. This article describes the finite monoids over which each weakly injective $S$-act has a primitively normal theory. Moreover, we show that the primitive normality of the class of all principally weakly injective $S$-acts is equivalent to $S$ being totally ordered.