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

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