An S-system A is injective if any S-homomorphism \(\theta\) : \(N\to A\), where N is an S-subsystem of an S-system M, can be extended to an S- homomorphism \(\phi\) : \(M\to A\). By imposing conditions on M and N we obtain definitions of various forms of weak injectivity, namely, absolutely pure, \(\alpha\)-injective (for any cardinal \(\alpha\) greater than 1) and coflat. These notions include the familiar concepts of weakly injective and weakly f-injective. In contrast to the corresponding case for modules over a ring, it has been shown that injectivity in S-systems is not equivalent to the Baer criterion. This condition is, however, equivalent to weak injectivity. We generalise this latter result by showing that for any cardinal \(\alpha >1\) an S-system is \(\alpha\)- injective if and only if it satisfies the \(\alpha\)-Baer criterion.
We prove that these notions are all related to what may be called ’purity’ properties, that is, the solubility in an S-system of certain restricted systems of equations. To complete the results of this type we show that an S-system A is injective if and only if any consistent system of equations, with constants from A, has a solution in A. If all S- systems are injective then S is said to be completely right injective. Internal characterisations of such monoids have been developed independently by several authors. We deal with the corresponding problems of classifying those monoids over which all S-systems are absolutely pure, \(\alpha\)-injective, or coflat.