Baer criterion for injectivity of projection algebras (Q818949)

Let \(\mathbb N^{\infty}\) be the monoid of all natural numbers with the greatest element \(\infty\) with respect to the operation \(\min\) added. A projection algebra is an \(M\)-set for the monoid \(M=\mathbb N^{\infty}\), and projection morphisms between projection algebras \(A\) and \(B\) are mappings \(f:A@>>>B\) with \(f(na)=nf(a)\) for all \(n\in \mathbb N^{\infty}\) and \(a\in A\). The category \(\mathbf P\mathbf R \mathbf O\) of all projection algebras and projection morphisms has enough injectives, thus injectives and absolute retracts coincide in \(\mathbf P\mathbf R \mathbf O\). We say that a projection algebra is principally weakly injective if it is injective relative to embeddings of all principal ideals into \(\mathbb N^{\infty}\), and it is weakly injective if it is injective relative to embeddings of all ideals into \(\mathbb N^{\infty}\). It is proved that every projection algebra is principally weakly injective, and a projection algebra is injective if and only if it is weakly injective (Baer Criterion for Injectivity). It is shown that the Baer Criterion does not hold for \(M\)-sets for all monoids \(M\), but there exists a great class of monoids such that the Baer Criterion holds for \(M\)-sets for all monoids \(M\) from this class.