Quasisums and generalized associativity

Summary.A quasisum is a real-valued function Q, defined on a rectangle, that has the form $$ Q(x,y) = \gamma (\alpha (x) + \beta (y)) $$ where α, β and γ are continuous and strictly monotonic functions defined on some intervals of positive length. In this paper we prove that if a two place function Q defined on the rectangle R is a local quasisum (that is, for each point of R there exists a rectangle, open in R and containing the point, on which Q is a quasisum) then it is a quasisum on the entire R. Applying this result we give a detailed and self-contained proof for the theorem about the solutions of the generalized associativity equation which are continuous and strictly monotonic in each variable.