On a method of Pi-Calleja for describing additive generators of associative functions (Q1188293)

The author reviews a method introduced by \textit{P. Pi-Calleja} [\(2^\circ\) Sympos. Probl. Mat. Latino Améria, Villavicencio-Mendoza 21--25 Julio 1954, 199--280 (1954; Zbl 0058.33602)] for describing additive generators of some associative functions on closed intervals, i.e., given \(T\) such that \(T(x,T(y,z))=T(T(x,y),z)\) one looks for functions \(\varphi\) such that \(T(x,y)=\varphi(\varphi^{-1}(x)+\varphi^{-1}(y))\). Then the author proves a theorem extending the results of Pi-Calleja. The essence of this theorem is that assuming some rather weak conditions on \(T\) the existence of some additive generator \(\varphi\) is proved.