Finite free objects

Question

Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements.

Is there other examples of free objects that are finite and have at least ten elements?

Answer 1

An example: any finitely generated free idempotent semigroup is finite.

Answer 2

One notion you might be interested in is that of locally finite varieties; they satisfy an apparently stronger finiteness requirement, namely: Every finitely generated algebra in it is finite. Indeed, this is equivalent to having all finitely generated free algebra finite.

A prominent example is the class of Boolean algebras.

There is a great deal of study of this varieties, in particular with the question of which of them are decidable, meaning that there is an effective procedure to determine if a first-order formula is a consequence of the axioms. The book The structure of decidable locally finite varieties by McKenzie and Valeriote has deep results in this direction. I suggest that you take a look into its zbMATH review.

Answer 3

One classical example, still unsolved in generality, is Burnside groups.

The free Burnside group $B(m,n)$ is defined to be the free group on $m$ generators with the relations $g^n=e$ for all $g\in B(m,n)$.

In general, it is unknown whether or not $B(m,n)$ is a finite group (though the answer is known in many cases). In fact, we already do not know whether $B(2,5)$ is finite.