Let S be a monoid. An S-system A is absolutely pure if every finite consistent system of equations, with constants from A, has a solution in A. A monoid S is coherent if every finitely generated S-subsystem of every finitely presented cyclic S-system is finitely presented. A right S-system A is called weakly injective (\(\alpha\)-injective) if for every right ideal I of S (with a generating set of fewer than \(\alpha\) elements) and for every homomorphism \(\theta\) : \(I\to A\) there exists an extension \(\psi\) : \(S\to A\). The author proves that the class of all \(\alpha\)-injective S-systems is axiomatizable iff for every natural number n less than \(\alpha\), the kernel of every homomorphism from the free S-system with n generators to S is finitely generated. The class of weakly injective S-systems is axiomatizable iff every right ideal of S is finitely presented; the class of injective S-systems is axiomatizable iff S is coherent and all absolutely pure S-systems are injective. An S-system A is \(\alpha\)-algebraically closed if every finite consistent system of equations with constants from A, in less that \(\alpha\) variables, has a solution in A. Monoids are described, for which the class of \(\alpha\)-algebraically closed S-systems is axiomatizable \((1<\alpha \leq \aleph_ 0)\). Particularly, the class of 2-algebraically closed S-systems is axiomatizable iff S is coherent.