mathematical logic Mathematical logic is the study of logic, 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 for ...

, Morley rank, introduced by , is a means of measuring the size of a subset 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 ''modulus'', a measure. Models c ...

of a

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...

, generalizing the notion of dimension in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

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. Properties Every ordinal other than 0 is either a successor ordin ...

, 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 an ...

, 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 Saturation, saturated, unsaturation or unsaturated may refer to: Chemistry * Saturation, a property of organic compounds referring to carbon-carbon bonds ** Saturated and unsaturated compounds **Degree of unsaturation ** Saturated fat or fatty ac ...

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 fini ...

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 diffe ...

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 may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a dat ...

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 . Examples As an example, the field of real numbers is not algebraically closed, because ...

''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 t ...

. 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 is an algebraic subset that is irreducible and maximal (for ...

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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...

, considered as an

ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...

, has Morley rank ∞, as it contains a countable

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...

of definable subsets isomorphic to itself.

References

Alexandre Borovik Alexandre V. Borovik (born 1956) is a Professor of Pure Mathematics at the University of Manchester, United Kingdom. He was born in Russia and graduated from Novosibirsk State University in 1978. His principal research lies in algebra Alge ...

Ali Nesin ʿAlī ibn Abī Ṭālib ( ar, عَلِيّ بْن أَبِي طَالِب; 600 – 661 CE) was the last of four Rightly Guided Caliphs to rule Islam (r. 656 – 661) immediately after the death of Muhammad, and he was the first Shia Imam. ...

, "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