joint embedding property
Let $K$ be a class of models (structures^{}) of a given signature^{}. We say that $K$ has the joint embedding property (abbreviated JEP) iff for any models $A$ and $B$ in $K$ there exists a model $C$ in $K$ such that both $A$ and $B$ are embeddable in $C$. [1, 2]
0.0.1 Examples
Examples include [2]:

•
The class of all groups.

•
The class of all monoids.

•
The class of all nontrivial Boolean algebras^{}.
As is the case with the above examples, classes having the joint embedding property often satisfy an even stronger condition  for every indexed family of models in the class there is a model in the class into which each member of the family can be embedded. This is known as the strong joint embedding property (abbreviated SJEP). [3]
In general any factor embeddable class closed under^{} products^{} will have the strong joint embedding property. [2]
0.0.2 Characterizations
Elementary classes with the joint embedding property may be characterized syntactically and semantically:
Let $T$ be a first order theory in a language^{} $L$ and let $K$ be the class of models of $T$ then:

1.
$K$ has the joint embedding property iff for all universal sentences $\varphi $ and $\psi $ in $L$, $T\u22a2\varphi \vee \psi $ implies either $T\u22a2\varphi $ or $T\u22a2\psi $. [1]

2.
If $T$ is consistent, then $K$ has the joint embedding property iff $T$ has an ultrauniversal model. [2]
References
 1 Abraham Robinson: Forcing^{} in model theory^{}, Actes du Congrès International des Mathématiciens (Nice, 1970) GauthierVillars, Paris, 1971, pp. 245250
 2 Colin Naturman, Henry Rose: Ultrauniversal models, Quaestiones Mathematicae, 15(2), 1992, 189195
 3 Colin Naturman: Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Mathematics, 1991
Title  joint embedding property 
Canonical name  JointEmbeddingProperty 
Date of creation  20130322 19:36:14 
Last modified on  20130322 19:36:14 
Owner  Naturman (26369) 
Last modified by  Naturman (26369) 
Numerical id  25 
Author  Naturman (26369) 
Entry type  Definition 
Classification  msc 03C52 
Synonym  JEP 
Synonym  SJEP 
Related topic  ultrauniversal 
Related topic  amalgamationproperty 
Related topic  UltraUniversal 
Related topic  FactorEmbeddable 
Defines  joint embedding property 
Defines  strong joint embedding property 