Functions on universal algebras (Q1076129)

This paper contains a short proof of the result that the generic model (in the topos-theoretic sense) U of an arbitrary algebraic theory is functionally complete, in the sense that every morphism \(U^ n\to U\) (in the topos in which U lives) is given by an n-variable polynomial in the theory (in fact by a unique such polynomial). This result has been known to topos-theorists for some time [for the particular theory of commutative rings, it underlies the earliest construction of a ''ring of line type'' - see \textit{A. Kock}, Math. Scand. 40, 183-193 (1977; Zbl 0375.12029)], but it has not been explicitly stated in print before. An application is given to the theory of combinatory algebras.