A-type over a structure ($p(\bar{x})\text{ over }A\subseteq M$)

For $A\subseteq M$, an A-type over first-order structure $M$ is a set of formulas in a fixed finite tuple of variables, with parameters in $A$, consistent with the complete theory of $M$ expanded by constants for $A$. Every finite subset is realized in $M$. A complete type decides all formulas in its variables over those parameters.