mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, Morley rank, introduced by , is a means of measuring the size of a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of a

model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

of a

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

, generalizing the notion of dimension in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

Definition

Fix a theory ''T'' with a model ''M''. The Morley rank of a formula ''φ'' defining a definable (with parameters) subset ''S'' of ''M'' 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 at least 0 if ''S'' is non-empty. *For ''α'' a

successor ordinal In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. The ordinals 1, 2, and 3 are the first three successor ordinals ...

, the Morley rank is at least ''α'' if in some

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 ...

''N'' of ''M'', the set ''S'' has countably infinitely many disjoint definable subsets ''S_i'', each of rank at least ''α'' − 1. *For ''α'' a non-zero

limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...

, 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 definable 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 In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a fi ...

and in the larger area of model theoretic

stability theory In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differ ...

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 set Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

in ''K''^''n'', for an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

''K'', then the Morley rank of ''V'' is the same as its usual

Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...

. The Morley degree of ''V'' is the number of

irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irred ...

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

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...

, considered as an

ordered set In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; ...

, has Morley rank ∞, as it contains a countable

disjoint union In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...

of definable subsets isomorphic to itself.

References

* Alexandre Borovik,

Ali Nesin Hüseyin Ali Nesin (born 18 November 1956, Istanbul) is a Turkish people, Turkish Prof. Mathematician and scientist He was born in November 18 1956 in Istanbul, İstanbul. His father is the well known writer Aziz Nesin, and his mother is Meral Ç ...

, "Groups of finite Morley rank", Oxford Univ. Press (1994) *B. Har
Stability theory and its variants
(2000) pp. 131–148 in ''Model theory, algebra and geometry'', edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000. Contains a formal definition of Morley rank. *David Marke
Model Theory of Differential Fields
(2000) pp. 53–63 in ''Model theory, algebra and geometry'', edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000. * * *{{springer, id=M/m110200, first=Anand , last=Pillay Model theory

Definition

Examples

See also

References