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.