Strongly minimal theory

Remarks

Example: the theory of infinite sets in the empty language is strongly minimal. By quantifier elimination, a formula $\varphi(x, \bar b)$ is equivalent to a Boolean combination of equations $x = b_i$, so it defines a finite or a cofinite set. The theory of algebraically closed fields of a fixed characteristic is strongly minimal for a similar reason: a nonzero polynomial in one variable has finitely many roots.

Non-example: the theory of dense linear orders without endpoints is not strongly minimal. In $(\Q, <)$ the formula $x > 0$ defines a set that is infinite and has an infinite complement.

A structure can be minimal while its theory is not strongly minimal, since the condition must hold in every model of the theory. In a strongly minimal theory, algebraic closure satisfies the exchange property, so every model has a dimension. If $L$ is countable, a strongly minimal theory is categorical in every uncountable cardinality.