Injectivity of \(S\)-posets with respect to down closed regular monomorphisms. (Q906969)

Let \(S\) be a pomonoid. An embedding \(f\colon A\to B\) of \(S\)-pomonoids is called \textit{down closed} if \(f(A)\) is a down closed \(S\)-poset of \(B\). An \(S\)-poset \(A\) is called \textit{down closed regular injective} or \textit{dc-injective} if it is injective with respect to down closed embeddings. \(A\) is called \textit{poideal injective} if every \(S\)-poset map \(I\to A\) from a right poideal \(I\) of \(S\) can be extended to an \(S\)-poset map \(S\to A\). It is proved that \(A\) is dc-injective if and only if it is injective with respect to the down closed embeddings of the form \(B\to{\downarrow\negthickspace(cS)}\). All right \(S\)-posets are poideal dc-injective if and only if \(S\) is a poregular pomonoid all of whose right poideals are principal as monoid ideals. This type of conditions are also found for some other properties and classes of \(S\)-posets. For example, pomonoids \(S\) are described for which all coproducts or direct sums of poideal dc-injective \(S\)-posets are poideal dc-injective, all poideal dc-injective \(S\)-posets are regular injective, all principally poideal dc-injective \(S\)-posets are finitely generated poideal dc-injective.