Strongly minimal theory
Encyclopedia
In model theory
—a branch of mathematical logic
—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
. A strongly minimal theory is a complete theory
all models of that 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 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.
A strongly minimal set, equipped with the closure operator
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 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
, who developed a method known as the "Hrushovski construction" to build new strongly minimal structures from finite structures.
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
—a branch of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
—a minimal structure is an infinite one-sorted structure
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
such that every subset of its domain that is definable with parameters is either finite or cofinite
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...
. A strongly minimal theory is a complete theory
Complete theory
In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent...
all models of that 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
Morley's categoricity theorem
In model theory, a branch of mathematical logic, a theory is κ-categorical if it has exactly one model of cardinality κ up to isomorphism....
on totally categorical structures.
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.
A strongly minimal set, equipped with the closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
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 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
Ehud Hrushovski is a mathematical logician. He is a Professor of Mathematics at the Hebrew University of Jerusalem.His father, Benjamin Harshav, is Emeritus Professor in Yale University and Tel Aviv University to Comparative Literature and a poet....
, who developed a method known as the "Hrushovski construction" to build new strongly minimal structures from finite structures.