Additive generators of associative functions

Associativity of a two place function T(x,y)=f^(^-^1^)(f(x)+f(y)) where f:[0,1]->[0,~] is a strictly monotone function andf^(^-^1^):[0,~]->[0,1] is the pseudo-inverse of f depends only on properties of the range of f. The following question is answered: what property of the range of an additive generator f is necessary and sufficient for associativity of the corresponding generated function T? We also introduce the characterization of all additive generators f of T with property T(...T(T(x"1,x"2),x"3),...,x"n)=f^(^-^1^)(f(x"1)+...+f(x"n)) for all [email protected]?N,n>=2 and for all x"1,...,x"[email protected]?[0,1]. Some constructions of non-continuous additive generators of associative functions are presented.