Inverse ambiguous functions on fields

Abstract

If \(f\,{:}\,G \rightarrow G\) is a bijection from a group G to itself, then the superscript \(-1\) written in proximity of f could, depending on its placement, indicate the inverse function of f or the function whose outputs are the inverses (in G) of the corresponding outputs of f. In this paper we investigate which groups G admit functions f where these two interpretations lead to the same outcome, that is, when \(f^{-1}(x) = (f(x))^{-1}\) for every \(x \in G\). We call such functions inverse ambiguous. After deriving some preliminary results, we turn our attention to the existence of inverse ambiguous functions defined on various fields F with respect to the underlying additive structure and multiplicative structure. Our study of this notational curiosity will lead to a surprising finding: namely, in many of the standard cases, there exist either continuous inverse ambiguous functions on F with respect to addition or on \(F^\times \) with respect to multiplication, but not both.