In model theory—a branch of

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

—a minimal structure is an infinite one-sorted structure such that every subset of its domain that is definable with parameters is either finite or

cofinite In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is coco ...

. A strongly minimal theory is a

complete theory In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or it ...

all models of which are minimal. A strongly minimal structure is a structure whose theory is strongly minimal. Thus a structure is minimal only if the parametrically definable subsets of its domain 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 on totally categorical structures. The nontrivial standard examples of strongly minimal theories are the one-sorted theories of infinite-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

s, and the theories ACF_''p'' of algebraically closed fields of characteristic ''p''. As the example ACF_''p'' 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 extension In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one oft ...

s. A strongly minimal set, equipped with the

closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...

given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem.

Boris Zilber Boris Zilber (russian: Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a professor of mathematical logic at the University of Oxford. H ...

conjectured that the only pregeometries that can arise from strongly minimal sets are those that arise in vector spaces, projective spaces, or algebraically closed fields. This conjecture was refuted by

Ehud Hrushovski Ehud Hrushovski ( he, אהוד הרושובסקי; born 30 September 1959) is a mathematical logician. He is a Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He was also Professor of M ...

, who developed a method known as "Hrushovski construction" to build new strongly minimal structures from finite structures.

References

* * {{Mathematical logic Model theory

See also

References