International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 803-811

Comultiplication on monoids

Martin Arkowitz1 and Mauricio Gutierrez2

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 S is a homomorphism m:SSS (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.