  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.
