In this paper, we first study monomorphisms and regular monomorphisms and show that monomorphisms in PosAct-S are exactly one-one morphisms and regular monomorphisms in PosAct-S are exactly order embeddings. Then recalling the fact that the category Pos does not have any non-trivial (non-singleton) injective object with respect to monomorphisms, we see that PosAct-S has no non-trivial injective object, too. Then we study regular injectivity, that is, injectivity with respect to regular monomorphisms.
