We introduce the notion of a coordinate k-algebra scheme and the corresponding notion of a B-operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it also generalizes the operators related to D-rings from [15]. We classify the (coordinate) kalgebra schemes for a perfect field k and we also discuss the model-theoretic properties of fields with B-operators.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

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.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory DCF p and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving "one-dimensional" axiomatizations for model companions of some theories of fields with operators.