Joint embedding property

Last updated January 16, 2022

In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for all structures A and B in K, there is a structure C in K such that both A and B have embeddings into C.

It is one of the three properties used to define the age of a structure.

A first-order theory has the joint embedding property if the class of its models of has the joint embedding property.^[1] A complete theory has the joint embedding property. Conversely a model-complete theory with the joint embedding property is complete.^[1]

A similar but different notion to the joint embedding property is the amalgamation property. To see the difference, first consider the class K (or simply the set) containing three models with linear orders, L₁ of size one, L₂ of size two, and L₃ of size three. This class K has the joint embedding property because all three models can be embedded into L₃. However, K does not have the amalgamation property. The counterexample for this starts with L₁ containing a single element e and extends in two different ways to L₃, one in which e is the smallest and the other in which e is the largest. Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of e.

Now consider the class of algebraically closed fields. This class has the amalgamation property since any two field extensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embedded into a common field when the characteristic of the fields differ.

Notes

1 2 Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover Publications. pp. 672 pages.

Related Research Articles

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, particularly in algebra, a field extension is a pair of fields $such that the operations of E are those of F restricted to E . In this case, F is an extension field of E and E is a subfield of F . For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.$

In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A of finite arity, and a finite set of identities, known as axioms, that these operations must satisfy.

In number theory, the ideal class group of an algebraic number field $K$ is the quotient group $J K / P K$ where $J K$ is the group of fractional ideals of the ring of integers of $K$ , and $P K$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of $K$ . The order of the group, which is finite, is called the class number of $K$ .

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

In mathematics, class field theory is the branch of algebraic number theory concerned with describing the Galois extensions of local and global fields. Hilbert is often credited for the notion of class field. But it was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. This theory has its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.

In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield.

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.

In the mathematical field of model theory, the amalgamation property is a property of collections of structures that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one.

In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.

In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah.

In mathematics, an algebraic number field is an extension field $of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .$

In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit is a method used to construct (infinite) mathematical structures from their (finite) substructures. It is a special example of the more general concept of a direct limit in a category. The technique was developed in the 1950s by its namesake, French logician Roland Fraïssé.

References

Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.

This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[ChangJerome-1] 1 2 Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover Publications. pp. 672 pages.

[1]