  It is shown that the category of \emph{semi-biproducts} of monoids is
equivalent to the category of \emph{pseudo-actions}. A semi-biproduct of
monoids is a new notion, obtained through generalizing a biproduct of
commutative monoids. By dropping commutativity and requiring some of the
homomorphisms in the biproduct diagram to be merely identity-preserving maps,
we obtain a semi-biproduct. A pseudo-action is a new notion as well. It
consists of three ingredients: a pre-action, a factor system and a correction
system. In the category of groups all correction systems are trivial. This is
perhaps the reason why this notion, to the author's best knowledge, has never
been considered before.
