The formal description familiar from group theory can be carried out in those monoid extensions whose congruence classes have representatives generating them under one-sided action of the kernel. This is first developed systematically, without further restrictive assumption. The original description of Rédei ‘9’ falls out when the kernel acts faithfully on these representatives. The “quasi-decompositions” and “triple sums” of Schmidt ‘11’ are seen to be the split extensions in the commutative case; and their generalization by Krishnan ‘5’ those in an (unnecessarily) symmetrized non-commutative one. (Split extensions under faithful action, alias “semidirect products”, also appear in ‘8, p. 186’ and more specially, with the kernel a group, in ‘10, p. 191’.) The “left coset” and “ H -coextensions” of Grillet and Leech ‘3, 7’ do not quite fit into the formulation but a slight modification covers them as well, while freeing them from restriction to a group “kernel”; thus liberated the construction applies, again back in the split commutative case, to yield Schmidt's “semigroup of semigroups”. p]An appendix extends recent work of Köhler to develop a Kaloujnine–Krasner theorem for these extensions.