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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.

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