International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 803-811
doi:10.1155/S0161171297001099
Comultiplication on monoids
Martin Arkowitz1
and Mauricio Gutierrez2
1Mathematics Department, Dartmouth College, Hanover 03755, NH, USA
2Mathematics Department, Tufts University, Medford 02155, MA, USA
Abstract
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.