Morley rank
Encyclopedia
In mathematical logic
, Morley rank, introduced by , is a means of measuring the size of a subset of a model
of a theory, generalizing the notion of dimension in algebraic geometry
.
is an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least α for some ordinal α.
The Morley rank is then defined to be α if it is at least α but not at least α + 1, and is defined to be ∞ if it is at least α for all ordinals α, and is defined to be −1 if S is empty.
For a subset of a model M defined by a formula φ the Morley rank is defined to be the Morley rank of φ in any ℵ0-saturated elementary extension of M. In particular for ℵ0-saturated models the Morley rank of a subset is the Morley rank of any formula defining the subset.
If φ defining S has rank α, and S breaks up into no more than n < ω subsets of rank α, then φ is said to have Morley degree n. A formula defining a finite set has Morley rank 0. A formula with Morley rank 1 and Morley degree 1 is called strongly minimal
. A strongly minimal structure is one where the trivial formula x = x is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity theorem
and in the larger area of stability theory (model theory).
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...
, Morley rank, introduced by , is a means of measuring the size of a subset of a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of a theory, generalizing the notion of dimension in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
Definition
Fix a theory T with a model M. The Morley rank of a formula φ defining a definable subset S of Mis an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least α for some ordinal α.
- The Morley rank is at least 0 if S is non-empty.
- For α a successor ordinal, the Morley rank is at least α if in some elementary extension N of M, S has countably many disjoint definable subsets Si, each of rank at least α − 1.
- For α a non-zero limit ordinal, the Morley rank is at least α if it is at least β for all β less than α.
The Morley rank is then defined to be α if it is at least α but not at least α + 1, and is defined to be ∞ if it is at least α for all ordinals α, and is defined to be −1 if S is empty.
For a subset of a model M defined by a formula φ the Morley rank is defined to be the Morley rank of φ in any ℵ0-saturated elementary extension of M. In particular for ℵ0-saturated models the Morley rank of a subset is the Morley rank of any formula defining the subset.
If φ defining S has rank α, and S breaks up into no more than n < ω subsets of rank α, then φ is said to have Morley degree n. A formula defining a finite set has Morley rank 0. A formula with Morley rank 1 and Morley degree 1 is called strongly minimal
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 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 one where the trivial formula x = x is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity 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....
and in the larger area of stability theory (model theory).
Examples
- The empty set has Morley rank −1, and conversely anything of Morley rank −1 is empty.
- A subset has Morley rank 0 if and only if it is finite and non-empty.
- If V is an algebraic setAlgebraic setIn mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
in Kn, for an algebraically closed fieldAlgebraically closed fieldIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
K, then the Morley rank of V is the same as its usual Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
. The Morley degree of V is the number of irreducible componentIrreducible componentIn mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equationis the union of the two linesandThe notion of irreducibility is stronger than connectedness.- Definition :...
s of maximal dimension; this is not the same as its degree in algebraic geometry, except when its components of maximal dimension are linear spaces. - The ordinal ωα has Morley rank α when considered as an ordered set.
- The rational numbers, considered as an ordered set, has Morley rank ∞, as it contains a countable disjoint union of definable subsets isomorphic to itself.