Example of a function of two variables that cannot be an $R$-function

Abstract

We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an $R$-mapping for a proper choice of a surjection.