# Is there an algebraic ω-categorical structure with quantifier eli., that is not ultrahomogenous?

The following result holds for relational structures:

If $A$ is a countable structure, with an $\omega$−categorical theory $Th(A)$, that admits quantifier elimination, then $A$ is ultrahomogeneous.

I am looking for an algebraic counterexample. So: an algebraic structure with an $\omega-$categorical theory, that admits QE , but is not ultrahomogenous.

(Maybe some abelian group??)

Thank you for any help!

When thinking about Fraïssé limits and ultrahomogeneous structures, the benefit to working in a finite relational language is that there are only finitely many possible quantifier-free types in $n$ variables for each $n$. When you move to a language with function symbols, this is no longer true, since a structure generated by $n$ elements can be arbitrarily large or even infinite. This is the main difference: there can be infinitely many quantifier-free types in $n$ variables for a fixed $n$. For this reason, working with a language with function symbols is very much like working with an infinite relational language.
To regain the benefits of a finite relational language, one often assumes uniform local finiteness: a finite bound on the size of substructures generated by $n$ elements will ensure only finitely many quantifier-free types in $n$ variables, as long as your language is finite.
However, if you assume $\aleph_0$-categoricity of the countable structure at the outset, then there are only finitely many complete types, hence only finitely many quantifier-free types! In this case, whether the language is finite or infinite or has function symbols or not is fairly irrelevant.
• One more question, that is close related to that one: Is there an ultrahomogenous algebraic structure in a finite language, that is not $\omega$-categorical, hence doesn't admit QE? Because I also read that the result ultrahomogeneous + finite language $\Rightarrow$ $\omega-$categorical only holds in relational structures. But maybe in that case the function symbols are irrelevant aswell. Jul 18, 2014 at 18:20
• Consider the Fraïssé class of finitely generated (not finite) fields of characteristic $0$. The Fraïssé limit is the algebraic closure of $\mathbb{Q}$. This theory is not $\aleph_0$-categorical. However, it does have QE (you have your implication backwards: ultrahomogeneous and $\aleph_0$-categorical implies QE, but there's no reason why an ultrahomogeneous but not $\aleph_0$-categorical structure can't have QE! Jul 19, 2014 at 0:26