The variety (in the sense of universal algebra) of Boolean algebras, for example, has the property that finitely generated free algebras have finite cardinality; in that case specifically $|F_n|=2^{2^n}$, in the obvious notation.
Can one usefully characterize varieties whose finitely generated free algebras have finite cardinality?
Can one characterize natural number sequences arising as $|F_n|$ in association with such varieties?
The Burnside problem for groups asks whether the variety $x^n=1$ is locally finite. By work of Adian and Novikov they are not locally finite for $n$ odd and large enough (I think at least 667) and in the even case results are by Ivanov and Lysenok. For n=2,3,4,6 local finiteness is known. For n=5 it is unknown. Mark Sapir classified locally finite semigroup varieties modulo the group case.
Varieties generated by a finite algebra are locally finite by a result of Birkhoff.
Added. By Zelmanov's solution to the restricted Burnside problem a variety of groups is locally finite iff it is generated by a set of finite groups with uniformly bounded exponent. The analogue is false for semigroups.
I asked George McNulty, and here is his answer. It partially coincides with my answer here, but is much more complete.
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I think the first result of this kind is an immediate if unstated consequence of a result in Peter Perkins dissertation
P. Perkins, ``Decision Problems of Equational Theories of Semigroups and General Algebras'', University of California, Berkeley 1966.
In a signature with two binary operation symbols and two constant symbols, Perkins proves (his Theorem 36) that the collection of finite sets of equations that are bases of finite algebras is undecidable. He does this by reducing the word problem on a particular finitely presented semigroup to this question. Loosely, if the word w is a consequence of the semigroup presentation, then the associated finite set of equations will be a base of a finite algebra, whereas if w is not a consequence then the free algebra on one generator in the variety based on the set of equations will be infinite. Of course, this also shows that the locally finiteness problem is also undecidable. This part of Perkins dissertation was published as
P. Perkins, ``Unsolvable problems for equational theories'', Notre Dame Journal of Formal Logic, vol. 8 (1967) 175--185.
One of the things I did in my 1972 dissertation was to establish various extensions of Perkins work on this topic. In particular, I showed that the above result holds for any finite signature that has an operation symbol of rank at least two. I had a rather long list of properties of finite sets of equations or of the varieties based on finite sets of equations that I could prove to be undecidable, but I didn't put all the proofs even in my dissertation. By the time I came to write it up for publication, I had figured out a handful of results from which most of the undecidability results I knew would follow. I published these in
G. McNulty, ``Undecidable properties of finite sets of equations''. Journal of Symbolic Logic, vol. 41 (1976) 589-604.
You can find in that paper a long list of such properties, but local finiteness is not on the list, while being the base of a finite algebra is. The local finiteness business follows in the same way as it did from Perkins result.
There are only a handful of other papers that address undecidable properties of finite sets of equations (mostly, I think, because undecidability seems to prevail---although some result like the Adjan-Rabin Theorem is unknown). Here they are:
V.L. Murskii, ``Nondiscernible properties of finite system of identity relations'', Doklady Akademii Nauk SSSR vol. 196 (1971) 520--522.
This paper is independent of my work or Perkins work. There is a large overlap between Murskii's findings and what is in my dissertation, although this 3 page account of Murskii's work is, of course, very terse. I don't think Muskii's work covers either being the base of a finite algebra or being the base of a locally finite variety, but it is very interesting. Murskii was the first to frame a general condition on collections of finite sets of equations that would ensure undecidability. It was Murskii's paper that spurred me to frame other general conditions that you can find in the paper of mine above. (I also include there a second proof of Murskii's general condition.
Douglas Smith in his 1972 Penn State dissertation found another undecidability result. It is in
D. Smith, ``the non-recursiveness of the set of finite sets of equations whose theories are one-based.'' Notre Dame Journal of Formal Logic, vol. 13 (1972) 135--138.
Ralph McKenzie wrote
R. McKenzie, ``On spectra and the negative solution of the decision problem for identities having a nontrivial finite model.'' Journal of Symbolic Logic, vol. 40 (1975) 186--196.
Don Pigozzi wrote
D. Pigozzi, ``Base-undecidable properties of universal varieties.'' Algebra Universalis, vol. 6 (1976), no. 2, 193–223.
Among other things, Pigozzi shows that it is undecidable whether the variety based on a finite set of equations has the amalgamation property or the Schreier property (subalgebras of free algebras are free).
C. Kalfa, ``Decision problems concerning properties of finite sets of equations. Journal of Symbolic Logic, vol. 51 (1986) 79--87.
Here Cornelia Kalfa shows that the joint embedding property is undecidable, as is whether the elementary theory of the infinite models is model complete.
The latest paper I know about is
C. O'Dunlaing, ``Undecidable questions related to Church-Rosser Thue systems.'' Theoretical Computer Science, vol. 23 (1983) 339--345.
Here it is shown that it is undecidable whether of finite set of equations is logically equivalent of a finite confluent set of equations.
Recently Ralph Freese and some collaborators have found fairly quick algorithms for a lot of the kind of properties above when the equations examined have certain restricted forms.
That's about what I know about this line of research. Hope some of it is useful.
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George also wrote:
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Also I noticed the interest expressed about free spectra in the original posting. There are a lot of papers on this (and related topics). Perhaps Joel Berman is the person who knows all about it.
In general a variety can be given in two different ways. First - by a finite (or recursive) set of identities and second - by a generating algebra. In the first case, the local finiteness of a variety is undecidable in general. But in some cases (for example, for semigroups with "nice" subgroups) the algorithm exists. In the second case, the generating algebra should be "uniformly locally finite" (say, finite, as in the case of Boolean algebras). See the survey "Algorithmic problems in varieties" here http://www.math.vanderbilt.edu/~msapir/ftp/pub/survey/survey.pdf .
Edit. The undecidability result not exactly in my survey but can be deduced from it. Here is a correct reference: Perkins, Peter Unsolvable problems for equational theories. Notre Dame J. Formal Logic 8 1967 175–185. Perkins proves that there is no algorithm that, given a finite system of identities, says whether it is a basis of identities of a finite algebra (theorem 13). In fact he proves more. He constructs an algebra $E$ with undecidable word problem, and for every two terms $u,v$ of $E$ he constructs a finite set of identities $I(u,v)$ such that if $u=v$ in $E$ then the set $I(u,v)$ is the set of identities of a finite algebra, and if $u\ne v$, $I(u,v)$ holds on an infinite 1-generated algebra. Since we cannot decide whether $u=v$, we cannot decide, given a finite set of identities, it is a basis of a locally finite variety.
If you are interested in just one free object, the situation is even easier. It is known (Markov) that the finiteness of a 2-generated semigroup is undecidable. Now consider the signature consisting of the semigroup operation plus two 0-ary operations giving the generators. Then any finitely presented semigroup becomes a relatively free object in a variety given by a finite number of identities (involving the 0-ary operations). Thus it is undecidable, given a finite number of identities in that signature whether the 2-generated free algebra in the variety given by these identities is finite (that is almost exact quote from the survey).