Strongly minimal theory
In model theory, a branch of mathematical logic, a minimal structure is an infinite one-sorted structure such that every subset of its domain which is definable with parameters is either finite or cofinite. A strongly minimal theory is a complete theory all models of which are minimal. A strongly minimal structure is a structure whose theory is strongly minimal.
Thus a structure is minimal if and only if the parametrically definable subsets of its domain are those which cannot be avoided, because they are already parametrically definable in the pure language of equality. Strong minimality was one of the early notions in the new field of classification theory and stability theory that was opened up by Morley's theorem.
The nontrivial standard examples for strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACFp of algebraically closed fields. As the example ACFp shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("curves").
More generally, a subset of a structure that is defined as the set of realizations of a formula φ('x') is called a minimal set if every parametrically definable subset of it is either finite or cofinite. It is called a strongly minimal set if this is true even in all elementary extensions.
See also
References
Baldwin, John T.; Lachlan, Alistair H. (1971), "On Strongly Minimal Sets", The Journal of Symbolic Logic, 36 (1): 79–96
