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