# Free algebras from model theory perspective

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?
• The variety of Jonsson-Tarski algebras (set $X$ endowed with bijective map $X^2\to X$) has been studied from the model-theoretic point of view too (model theorists call them "set with pairing function, if I remember correctly).
– YCor
May 15, 2022 at 13:39
• That sounds like exactly what I'm looking forward, thank you. I haven't heard of those algebras, do you have any papers in mind? May 15, 2022 at 13:52
• When $\mathbb V$ is the variety of all algebras of a given signature, or more generally, a variety axiomatized by “commutativity” equations postulating that some of the functions are symmetric wrt specific permutation groups, the first-order theory of $F_S$ has been described and studied by Malcev, Axiomatizable classes of locally free algebras of several types (mi.mathnet.ru/eng/smj/v3/i5/p729, doi.org/10.1016/S0049-237X(08)70560-3). These theories are all stable (though this is not in Malcev’s paper, which predates the definition of stability). May 15, 2022 at 16:21
• @arunpatel: regarding your request for a paper about free algebras in the variety of Jonsson-Tarski algebras, a good one is "Bouscaren, Elisabeth; Poizat, Bruno, Des belles paires aux beaux uples. J. Symbolic Logic 53 (1988), no. 2, 434-442". They show that the theory of the free algebras in this variety has QE, is stable, 1-based, and has the DOP. May 15, 2022 at 16:43
• The number of generators of a free semigroup or monoid is determined by the first order theory. The generators are the irreducible elements. The first order theory encodes arithmetic. May 15, 2022 at 18:09

Here are some papers.

(1)
Baldwin, J. T.; Shelah, S.
The structure of saturated free algebras.
Algebra Universalis 17 (1983), no. 2, 191-199.

From the Math Review (written by Steve Comer):

The authors investigate the structure of an algebra $$M$$ in a variety $$V$$ (with a countable similarity type) such that $$M$$ is free and ℵ1-saturated. The main result says that "every model of Th($$M$$) is `generated' by the union of a finite set of indiscernible sequences.

(2)
Model theory for locally free algebras.
Trudy Inst. Mat. (Novosibirsk) 8 (1988), Teor. Model. i ee Primenen., 3-25, 184.

From the abstract:

In this paper we study model-theoretic properties of locally free algebras. It is proved that every complete theory of locally free algebras is stable and normal.

(3)
Mekler, Alan H.; Shelah, Saharon
Almost free algebras.
Israel J. Math. 89 (1995), no. 1-3, 237-259.

From the Math Review (written by John Baldwin):

An algebra $$A$$ is "almost free'' if most of its subalgebras of smaller cardinality are free. A is essentially free if the free product of $$A$$ and $$F$$ is free for some free algebra $$F$$; otherwise, $$A$$ is essentially non-free. The authors characterize the class of cardinals in which a variety $$V$$ has an essentially non-free algebra which is almost free.

More recently,

(4)
Kucera, Thomas G.; Pillay, Anand
Saturated free algebras and almost indiscernible theories.
Algebra Universalis 83 (2022), no. 1, Paper No. 11, 25 pp.

This is a very recent paper and the Math Review is not yet posted. But Kucera spoke about this in our seminar last month and you will find links to slides and video for his talk at the foot of the page for the abstract of his talk.