# 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 non-trivial 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. 1.

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

2. 2.

If $T$ is consistent, then $K$ has the joint embedding property iff $T$ has an ultra-universal model. [2]

## References

• 1 Abraham Robinson: Forcing in model theory, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 245-250
• 2 Colin Naturman, Henry Rose: Ultra-universal models, Quaestiones Mathematicae, 15(2), 1992, 189-195
• 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 2013-03-22 19:36:14 Last modified on 2013-03-22 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