universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as ...

and

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

, a class of

structures A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...

''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

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...

s into ''C''. It is one of the three properties used to define the

age Age or AGE may refer to: Time and its effects * Age, the amount of time someone or something has been alive or has existed ** East Asian age reckoning, an Asian system of marking age starting at 1 * Ageing or aging, the process of becoming olde ...

of a structure. A first-order theory has the joint embedding property if the class of its models of has the joint embedding property. Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover Publications. pp. 672 pages. A

complete theory In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its ...

has the joint embedding property. Conversely a model-complete theory with the joint embedding property is complete. A similar but different notion to the joint embedding property is the

amalgamation property 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. Thi ...

. To see the difference, first consider the class ''K'' (or simply the set) containing three models with

linear order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...

s, ''L''₁ of size one, ''L''₂ of size two, and ''L''₃ of size three. This class ''K'' has the joint embedding property because all three models can be embedded into ''L''₃. However, ''K'' does not have the amalgamation property. The counterexample for this starts with ''L''₁ containing a single element ''e'' and extends in two different ways to ''L''₃, 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

References

* {{mathlogic-stub Model theory