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 × SG of the functional equation

F(x, y) + F(xy, z) = F(x, yz) + F(y, z),    for all    x,y,zS,

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,yS,

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.

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