Let $\mathbb{V}$ be a non-trivial variety of algebras, and let $F_S\in\mathbb{V}$ be a free algebra on a set $S$. I want to know what is known about the model theory of $F_S$; I know these objects are very well-studied in universal algebra, but I want to know about the model theory of these structures as it fits into the broader picture of Saharon Shelah's **classification theory**.

Some examples: I know that, when $\mathbb{V}$ is the variety of groups, then all the $F_S$ have the same complete theory and it is stable, which is a very hard theorem. Also if $\mathbb{V}$ is the variety of abelian groups then they are again all stable.

It also seems it should be easy to find unstable examples. If $F_S$ has a binary operation $\odot$ and a non-$\odot$-unit $a$ so that $a\odot a\neq a$ then I think $F_S$ is always unstable, because $\odot$-divisibility has the order property for powers of $a$.

I am looking for any of the following:

- General results about the model theory of free algebras. For example, when might they be stable? (Or $\omega$-stable or dependent or simple etc etc.) Can you classify complete types with no parameters?
- Results about $\mathrm{Th}(F_S)$ for a fixed choice of $\mathbb{V}$ and $S$. I especially care about the case where $S$ is finite or countable but bigger examples will be fine. I guess some examples like rings will be too complicated, since you get arithmetic. But how about if $\mathbb{V}$ is the variety of semigroups or monoids or meadows?

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