# Events

### Monday, Apr 13, 2015 - 3:30pm - Allen 411

Top Combo Fun seminar

The cocycle equation on commutative semigroups

Dr. Bruce Ebanks, Mathematics, Msstate

**Title:** The cocycle equation on commutative semigroups**Abstract:** Let (*S*, ⋅) be a commutative semigroup and (*G*, +) be an abelian group. Any solution *F* : *S* × *S* → *G* of the functional equation

*F*(*x*, *y*) + *F*(*xy*, *z*) = *F*(*x*, *yz*) + *F*(*y*, *z*), for all *x*,*y*,*z* ∈ *S*,

is called a __cocycle__, and the equation is called the __cocycle equation__. It is easy to check that any *F* of the form

*F*(*x*, *y*) := *f*(*x*) + *f*(*y*) - *f*(*xy*), *x*,*y* ∈ *S*,

is a cocycle. The question is whether all symmetric cocycles are of this form. The answer is "yes" if *S* is an abelian group (known for about 50 years), but
the general theory on commutative semigroups is far from complete. We introduce a framework
which allows a significant advance in the development of this theory.