I am taking an introductory course on model theory and am looking at Fraïssé Limit, as this is defined by Tent and Ziegler in A Course in Model Theory.
They prove that for a countable language $L$, a class of $L$-Structures has a Fraïssé limit if and only if the class has hereditary, joint embedding and amalgamation properties. I do not see why joint embedding does not follow from amalgamation.
Does anyone know of a good example which shows this?
Thank you very much!
The classic example for this is the class $K$ of finite fields. $K$ fails to have JEP, because field embeddings preserve the characteristic. But $K$ does have AP: If $A$ embeds in $B$ and $C$, then all three fields have the same characteristic $p$, and they amalgamate as $\mathbb{F}_{p^n}$ for some $n$.
What's going on is that AP only implies JEP if for any two structures in the class, there is a third structure that embeds in both of them.
This condition is satisfied, and AP implies JEP, when you're in a relational language, you allow empty structures, and you don't have any $0$-ary relation symbols ("proposition symbols", which are either true or false in a model). E.g. classes of graphs, partial orders, etc. Indeed, under these conditions there is a unique empty structure, which embeds in every structure in the class.
But as soon as you have function or constant symbols or $0$-ary relations, there's likely to be multiple "prime structures" in the class up to isomorphism. Assuming HP, every structure in the class will uniquely embed one of these structures (as the substructure generated by the empty set), but two structures with different prime substructures won't ever appear together in an AP diagram.