Associativity and Distributivity of Analytic Functions

It is proved that, up to natural equivalence, there exist exactly five locally associative analytic functions of two variables: $$0$$ , $$x$$ , $$y$$ , $$(x+y)$$ , and $$xy$$ . A description of all associative (nonlocally) polynomials is given. All distributive pairs where at least one of the functions is locally associative are also described.