On classification of monoids by properties of cofree acts
In the category \(\mathbf{Act}\)-\(S\) of right \(S\)-acts \(|I|\)-cofree acts are of the form \(I^S\) with the usual action (from the right), \(I\) a set. The author proves, for example, the equivalence of the following conditions: (1) All principally weakly injective \(S\)-acts satisfy condition \((E)\), (2) All weakly injective \(S\)-acts satisfy condition \((E)\), (3) All injective \(S\)-acts satisfy condition \((E)\), (4) All cofree \(S\)-acts satisfy condition \((E)\), (5) Every right congruence \(\varrho(s,t)\) is the kernel of some left translation, and similar results where condition \((E)\) is replaced by condition \((P)\) or by ``strongly flat'', where for commutative monoids the assertion corresponding to (4) is equivalent to \(S\) being a group. If condition \((E)\) is replaced by ``projective'', then the assertion corresponding to (4) is equivalent to \(S=\{1\}\). In the last section implications of the form ``divisible'' \(\Longrightarrow (E)/(P)/\)``strongly flat'' are used to characterize \(S=\{1\}\) or \(S=\{1,0\}\)/\(S\) a group/\(S=\{1\}\) in this order.
scientific article; zbMATH DE number 1330653