On the last question of Stefan Banach

TL;DR

This paper explores Banach's final question regarding the extension of ternary maps, contextualizes it historically within Polish mathematics, and connects it to subsequent developments in logic, algebra, and Hilbert's 13th problem, ultimately linking it to Jacobson's 1949 ideas.

Contribution

It provides a comprehensive analysis of Banach's last question and demonstrates how Jacobson's 1949 concept can be used to address variants of this problem.

Findings

Banach's question relates to extending ternary maps to superpositions of binary maps.

Jacobson's 1949 idea of envelopes of Lie triple systems offers a solution framework.

Historical context links Banach's question to modern algebra and logic developments.

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.