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Cocyles on cancellative semigroups. (English) Zbl 0860.39037
Let \(M\) denote an abelian monoid with \(O\) as identity element, and let \(G\) be an abelian group. A function \(F:M^2 \to G\) is called a cocycle if for all \(x,y,z\in M\), \(F(x,y)+ F(x+y,z) = F(x,y+z) + F(y,z)\). A cocycle is symmetric if, in addition, \(F(x,y) = F(y,x)\) for all \(x,y \in M\). For each function \(f\) from \(M\) to \(G\) one define \(\widehat f: M^2 \to G\) by \(\widehat f(x,y): =f(x) + f(y)- f(x+y)\). A symmetric cocycle \(F\) to \(G\) is called a coboundary if there is a function from \(M\) to \(G\) such that \(F= \widehat f\) on \(M^2\). The main result is given in the following Theorem.
If \(M\) is a cancellative abelian monoid and \(G\) is a divisible abelian group, then every symmetric cocycle on \(M\) to \(G\) is a coboundary.
In the last section a result about arbitrary cocycles is proved with an additional hypothesis.

39B52 Functional equations for functions with more general domains and/or ranges