Operators coming from ring schemes

Abstract

We introduce the notion of a coordinate 𝐤-algebra scheme and the corresponding notion of a -operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it generalizes the operators related to -rings from Moosa and Scanlon (J.Math. Log. 14 (2014), no. 02, 1450009) as well. We classify the (coordinate) 𝐤-algebra schemes for a perfect field 𝐤 and we also discuss the model-theoretic properties of fields with -operators.