Existentially closed fields with operators in various categories

Abstract

We deal with fields with certain operators - introduced by the author and Kowalski - which we call $\mathcal{B}$-fields. We are mostly interested in $\mathcal{B}$-fields which are existentially closed, possibly in some restricted category of $\mathcal{B}$-fields. This in particular includes seeking for a model companion, but also is related to so-called pseudo algebraically closed structures. We prove a very general result saying that in many cases being existentially closed (in a generalized sense) is an elementary property. This encompasses, generalizes and simplifies many results from the literature. We study the resulting first-order theories, most importantly we study dividing lines and quantifier elimination.