Functions commuting with ternary operations

Abstract

The problem of finding all functions which commute with a given ternary operation leads to a general functional equation. In this note we solve this problem in the case when the ternary operation is given by [x, y, z] = (x *y - z)[-1], x,y,z ϵ F, x * y ≠ z where F is a field having at least four elements and in which x ϵ F, 2x = 0 imply x = 0, and * is any binary operation on F with a right-sided unit element.