The cocycle equation on periodic semigroups

Abstract

The cocycle functional equation \(F(x,y) + F(xy, z) = F(x,yz) + F(y,z)\) has a long and rich history, with important roles in contexts from homological algebra to polyhedral algebra to information theory. This paper deals with the problem of finding explicit forms of symmetric cocycles on periodic semigroups. A semigroup S is periodic if each of its elements has finite order, that is if the cyclic subsemigroup \(\langle a \rangle = \{ a^k \mid k = 1,2,3,\ldots \}\) generated by each element a of S is finite. For several classes of abelian semigroups, including idempotent semigroups, cancellative semigroups, and certain types of topological semigroups, it is known that every symmetric cocycle F is a coboundary (in other words, a Cauchy difference), that is \(F(x,y) = f(x) + f(y) - f(xy).\) We show that in fact every cocycle has this form on periodic semigroups.