A comultiplication on a monoid S is a homomorphism m:S→S∗S (the free
product of S with itself) whose composition with each projection is the identity homomorphism.
We investigate how the existence of a comultiplication on S restricts the structure of S. We show
that a monoid which satisfies the inverse property and has a comultiplication is cancellative and
equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a
comultiplication if and only if S is the free product of a free monoid and a free group. We call
these monoids semi-free and we study different comultiplications on them.