I apologize if this question is meaningless or trivial:

What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?

I need to say some words about my motivation: last week I was proved some results concerning relatively free algebras in varieties, as an example the following result was proved:

Theorem (false) Let $\mathbf{V}$ be a variety of algebras and $X$ be a set. Then the relatively free algebra $F_{\mathbf{V}}(X)$ is $u_{\omega}$-compact.

A communication with Anton Klaychko revealed to me that my argument had mistake. The mistake was funny: I was replaced $\forall x \varphi(x) \Leftrightarrow \forall x \psi(x)$ by $\forall x( \varphi(x)\Leftrightarrow \psi(x))$ implicitly in my proof. Today, I learned some elementary facts about quantifier elimination and assuming that $\mathbf{V}=Var(A)$ with $A$ an algebra admitting quantifier elimination, I modified my proof. Now I have a correct proof of the above theorem with the extra assumption:

1- $\mathbf{V}=Var(A)$,

2- $A$ admits quantifier elimination.

I need some examples of such algebras to complete my work. Before I learned about some well-known examples such as algebraically closed fields and some Boolean algebras (and some other examples which are not algebras and so are not relevant).

• Any structure can be encoded as an algebra while preserving quantifier elimination. Just fix two constants and replace all predicates with their characteristic functions. – Emil Jeřábek Jan 25 '14 at 22:45
• It's worth noting that in general, quantifier elimination and completeness go hand in hand... – cody Jul 7 '15 at 1:02
• And decidability of course. – cody Jul 7 '15 at 1:07