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(x1,x2),x3),…,xn)=f(-1)(f(x1)+⋯+f(xn)) for all n∈N,n⩾2 and for all x1,…,xn∈[0,1]. Some constructions of non-continuous additive generators of associative functions are presented.