On the last question of Stefan Banach

Abstract

We discuss the last question of Banach, posed by him in 1944, shortly before his death, about extension of a ternary map to superposition of a binary map. We try to put things into the context of Polish mathematics of that time, and touch upon subsequent developments in such diverse areas as multivalued logics, binary and ternary semigroups, theory of clones, and Hilbert's 13th problem. Making almost a full circle in time, we show how variants of Banach's question may be settled using a 1949 idea of Jacobson about envelopes of Lie triple systems.