  Cosetal extensions of monoids generalise extensions of groups, special
Schreier extensions of monoids and Leech's normal extensions of groups by
monoids. They share a number of properties with group extensions, including a
notion of Baer sum when the kernel is abelian. However, unlike group extensions
(with fixed kernel and cokernel) there may be nontrivial morphisms between
them. We explore the structure of the category of cosetal extensions and relate
it to an analogue of second cohomology groups. Finally, the order structure and
additive structures are combined to give an indexed family of inverse
semigroups of extensions. These in turn can be combined into an inverse
category.
