International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 803-811
Comultiplication on monoids
1Mathematics Department, Dartmouth College, Hanover 03755, NH, USA
2Mathematics Department, Tufts University, Medford 02155, MA, USA
Received 14 April 1997
Copyright © 1997 Martin Arkowitz and Mauricio Gutierrez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A comultiplication on a monoid is a homomorphism (the free
product of with itself) whose composition with each projection is the identity homomorphism.
We investigate how the existence of a comultiplication on restricts the structure of . 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 which satisfies the inverse property admits a
comultiplication if and only if 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.