# Hereditary Property and Almagamation Property implying Joint Embedding Property?

I’m reading the lecture on Fraأ¯ssأ©’s theorem in Thomas Scanlon’s script (p. 115). He mentions that ‘if τ\tau is a relational signature, then HP and AP does imply JEP’ but I think I don’t get it.

Let τ\tau consist only of PP with arity 11 and K{\cal K} of two 11-element structures, one in which PP is true and one in which it is false. Then K{\cal K} satisfies HP and AP (in my opinion) but not JEP.

Where is the mistake?

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5

On page 6 he states that he allows structures to be empty. In a relational signature, the empty structure is a substructure of every structure. JEP is just amalgamation over the empty structure.
– Keith Kearnes
Oct 21 at 3:07

1

It’s worth noting that if we also allow 00-ary relations (i.e. propositional symbols), then once again HP+AP fails to imply JEP, since there is not a unique empty structure.
– Alex Kruckman
Oct 21 at 3:19

Thanks a lot for pointing that out (both of you). I was completely fixated on non-empty structures, which is what you assume when you are dealing with first order logic, but apparently model theory has a much wider range of applications than modeling logical theories!
– fweth
2 days ago

1

@fweth you might be interested in the discussion here: math.stackexchange.com/q/45198/7062
– Alex Kruckman
2 days ago

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