Banaschewski's theorem for \(S\)-posets: regular injectivity and completeness.
From the authors' abstract with some additions: In this paper we study the notion of injectivity in the category \(\mathbf{Pos}\text{-}\mathbf S\) of \(S\)-posets for a pomonoid \(S\). First we see that, although there is no non-trivial injective \(S\)-poset with respect to monomorphisms, \(\mathbf{Pos}\text{-}\mathbf S\) has enough (regular) injectives with respect to regular monomorphisms (sub \(S\)-posets). Then, recalling Banaschewski's theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for \(S\)-posets. We see for example, that regular injective \(S\)-posets are exactly the retracts of cofree \(S\)-posets over complete posets. The authors get some homological classification of pomonoids and pogroups like the following. All complete \(S\)-posets, over a pomonoid \(S\) where the identity is maximal, are regular injective iff \(S\) is a pogroup. If the identity is a top, then this is the case iff \(S=1\). Recall that regular monomorphisms are equalizers which in the present case are exactly order-embeddings.
scientific article; zbMATH DE number 5710597
scientific article