We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion $(M,A)$ by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct, and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.
"Mutually algebraic structures and expansions by predicates." J. Symbolic Logic 78 (1) 185 - 194, March 2013. https://doi.org/10.2178/jsl.7801120