User:Hans Adler/Model theory and universal algebra

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[edit] DRAFT ARTICLE ON MODEL THEORY

Note: The greater part of the draft is now part of the article model theory.

[edit] The role of model theory

In a similar way as proof theory, model theory is situated in an area of interdisciplinarity between mathematics, philosophy, and computer science. [Insert picture here: Venn diagram depicting maths, logic and computer science, with model theory in the intersection.] The most important professional organization in the field of model theory is the Association for Symbolic Logic.

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[edit] Finite model theory

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[Picture showing how a relational database is a relational structure]


[Example of a non-trivial Datalog query and its translation into a normal first-order formula?]

[edit] ω-categoricity

  • reducts
  • back-and-forth [picture?]
  • Fraïssé's theorem
  • ω-categoricity

[edit] Classification theory

  • elementary embeddings
  • complete theories
  • Morley's theorem
  • indiscernibles
  • strong minimality
  • prime models
  • Baldwin-Lachlan theorem
  • stability and the spectrum of a theory
  • simplicity
  • independence property

[picture of stability, simplicity, NIP, superstability, strongly dependent, strong minimality, o-minimality]

[edit] Geometric stability theory

  • forking
  • heirs and coheirs
  • imaginaries
  • regular types
  • Zilber's conjecture
  • Hrushovski's amalgamation construction
  • group configuration [picture!]
  • modularity and triviality
  • Robinson theories / compact abstract theories
  • thorn-forking

[edit] Topological model theory

[Lots of pictures possible]

  • o-minimality
  • weak o-minimality
  • C-minimality and similar generalizations
  • classical results
  • topological structures
  • cell decomposition
  • o-minimal homology

[edit] Other areas of model theory

[edit] FRAGMENTS FOR LATER USE ELSEWHERE

[edit] Morphisms between structures

For every signature σ there is a category which has the σ-structures as objects. In fact, three such categories are in wide use and determine the flavours of different areas of model theory. Homomorphisms are the most general notion. They are the standard notion in finite model theory and universal algebra, and they specialize correctly in the case of many algebraic and other structures. Embeddings are used in algebraic model theory, and elementary embeddings are the most natural notion for abstract model theory.

[edit] Homomorphisms

Given two structures \mathcal A,\mathcal B of the same signature σ, a σ-homomorphism h: \mathcal A\rightarrow\mathcal B is

  • a map h:|\mathcal A|\rightarrow|\mathcal B| which
  • commutes with all (n-ary) functions f of σ:
h(f^{\mathcal A}(a_1,a_2,\dots,a_n))=f^{\mathcal B}(h(a_1),h(a_2),\dots,h(a_n))
for all a_1,a_2,\dots,a_n\in |\mathcal A|, and
  • preserves all (n-ary) relations R of σ:
\mathcal A\models R(a_1,a_2,\dots,a_n)\;\implies\;\mathcal B\models R(h(a_1),h(a_2),\dots,h(a_n))
for all a_1,a_2,\dots,a_n\in |\mathcal A|.

The notion of σ-homomorphism directly generalizes the usual notions of homomorphism for algebraic structures such as groups, rings, modules and lattice (mathematics)s, but also for relational structures such as graphs and partial orders, and for mixed structures such as linearly ordered groups. It is also the standard notion of morphism in universal algebra and in finite model theory.

The σ-structures and σ-homomorphisms form a category σ-Hom. The epimorphisms in this category are the surjective σ-homomorphisms, and the monomorphisms are the injective σ-homomorphisms. The isomorphisms in this category are bijective σ-homomorphisms, but the converse is true iff there are no function symbols in σ.

[edit] Quantifier-free embeddings

Given two structures \mathcal A,\mathcal B of the same signature σ, a σ-embedding h: \mathcal A\rightarrow\mathcal B is

  • an injective map h:|\mathcal A|\rightarrow|\mathcal B| which
  • commutes with all (n-ary) functions f of σ:
h(f^{\mathcal A}(a_1,a_2,\dots,a_n))=f^{\mathcal B}(h(a_1),h(a_2),\dots,h(a_n))
for all a_1,a_2,\dots,a_n\in |\mathcal A|, and
  • does not affect any (n-ary) relations R of σ:
\mathcal A\models R(a_1,a_2,\dots,a_n)\;\implies\;\mathcal B\models R(h(a_1),h(a_2),\dots,h(a_n))
for all a_1,a_2,\dots,a_n\in |\mathcal A|.

In other words, a σ-embedding is a σ-homomorphism is a σ-isomorphism with an induced substructure. The σ-structures and σ-embeddings form a subcategory σ-Emb of σ-Hom. These are the standard notions for a style of model theory that is often connected with Abraham Robinson and his school on the east coast of the United States. Hodges disputes this.

[edit] Elementary embeddings

Given two structures \mathcal A,\mathcal B of the same signature σ, an elementary σ-embedding h: \mathcal A\rightarrow\mathcal B is

  • an map h:|\mathcal A|\rightarrow|\mathcal B| which
  • commutes with all first-order σ-formulas φ(x_1,x_2, ... ,x_n):
\mathcal A\models\varphi(a_1,a_2,\dots,a_n)\iff\mathcal B\models\varphi(h(a_1),h(a_2),\dots ,h(a_n))
for all a_1,a_2,\dots,a_n\in |\mathcal A|.

The σ-structures and elementary σ-embeddings form a subcategory σ-Elem of σ-Hom. There is a tradition, questioned by Hodges, that connects the style of model theory which works with elementary embeddings to Alfred Tarski and his school on the west coast of the United States.

[edit] Biographies related to model theory

[edit] Resources

  • Stanford Encyclopedia of Philosophy: Model Theory [1]
  • Stanford Encyclopedia of Philosophy: First-order Model Theory [2]
  • MathWorld: Model Theory [3]
  • MathWorld: Universal Algebra [4]
  • Springer Encyclopaedia of Mathematics: Universal Algebra [5]
  • Springer Encyclopaedia of Mathematics: Algebraic System [6]
  • Anand Pillay: Lecture Notes - Model Theory [7]
  • Anand Pillay: Model Theory [8]
  • Jouko Väänänen: A Short Course in Finite Model Theory [9]
  • Burris & Sankappanavar: A Course in Universal Algebra [10]
  • Basic Notation of Universal Algebra [11]
  • Terms over Many Sorted Universal Algebra [12]
  • Many Sorted Algebras [13]
  • Minimal Signature for Partial Algebra [14]
  • George M. Bergman: An Invitation to General Algebra [15]
  • Peter Burmeister: A Model Theoretic Oriented Approach to Partial Algebras [16]
  • Peter Burmeister: Lecture Notes on Universal Algebra – Many Sorted Partial Algebras [17]
  • [18]

[edit] LIST OF THINGS TO DO

[edit] Organisational things

[edit] Existing articles

[edit] Wishlist for new articles

[edit] Strange things

[edit] Non-first-order model theory

[edit] DRAFT ARTICLE ON TOPOLOGICAL MODEL THEORY

[edit] First-order topological structures and theories

Given a signature σ, a first-order topological σ-structure is a σ-structure M, equipped with a first-order σ-formula (without parameters) Ω(x;y1…yn) such that the realisation sets of the instances Ω(x;b1…bn) form the basis of a topology. In other words, Ω is required to satisfy the axiom \forall x\forall\bar y\forall\bar z\exists\bar w [\Omega(x;\bar w) \wedge \forall x'(\Omega(x;\bar w)\to\Omega(x;\bar y)\Omega(x;\bar z))]. A first-order topological σ-theory is a first-order σ-theory T equipped with a first-order σ-formula Ω(x;y1…yn) such that all models of T are first-order topological σ-structures in the obvious way.

t-minimal structures and theories. (Follow Pillay or Schoutens?)