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

Let $\tau$ consist only of $P$ with arity $1$ and ${\cal K}$ of two $1$-element structures, one in which $P$ is true and one in which it is false. Then ${\cal K}$ satisfies HP and AP (in my opinion) but not JEP.

Where is the mistake?

fweth
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    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 '16 at 03:07
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    It's worth noting that if we also allow $0$-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 '16 at 03:19
  • 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 Oct 21 '16 at 09:31
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    @fweth you might be interested in the discussion here: http://math.stackexchange.com/q/45198/7062 – Alex Kruckman Oct 21 '16 at 11:42

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