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Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every finitely generated algebra in $\mathcal{V}$ is finite. Equivalently, the finitely generated free algebras $F_{\mathcal{V}} \, n$ are finite.

Given any finite $\Sigma$-algebra $A$ then the variety $V(A)$ generated by $A$ is a locally finite variety. Let us call such varieties finitely generated. Then my question is as follows:

What are natural examples of locally finite varieties $\mathcal{V}$ which are not finitely generated? I am particularly interested in varieties of semigroups or monoids, although other examples are welcome.

Let me mention that one can characterise the finitely generated varieties amongst the locally finite varieties as follows. There is some fixed $n \in \omega$, such that for every $m \geq n$ and $x \neq y \in F_\mathcal{V} m$, there exists a function $f : m \to n$ such that $F_\mathcal{V} f(x) \neq F_\mathcal{V} f(y)$. In other words, if an equation fails to hold then we can already deduce this using at most $n$ variables.

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The variety of semigroups all of whose elements are idempotents (also called bands) is locally finite but not finitely generated.

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  • $\begingroup$ Of course the corresponding monoid variety is also locally finite but not finitely generated. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 19, 2014 at 20:58
  • $\begingroup$ That's a great example, but do you know of any others? I believe that the finitely generated varieties of monoids amount to the compact elements in the lattice of pseudovarieties of monoids, whereas the locally finite varieties of monoids correspond to pointwise finite varieties of languages in the sense of Eilenberg. I'm trying to understand their relationship, so any such examples would be greatly appreciated. $\endgroup$
    – Rob Myers
    Commented Mar 19, 2014 at 21:32
  • $\begingroup$ There are many others but they are more complicated to describe. Locally finite varieties are closed under semidirect product and Malcev product but finitely generated varieties rarely are. Compact elements of the lattice of pseudovarieties are finitely generated. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 19, 2014 at 21:47
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The variety of Gödel–Dummett algebras (Heyting algebras satisfying $(x\to y)\lor(y\to x)=1$). The varieties of $\mathrm{BD}_n$ Heyting algebras or $\mathrm{K4BD}_n$ modal algebras for any constant $n>1$.

(There are more examples like this. In case it helps with searching, algebraizable logics whose equivalent variety is locally finite go by the name “locally tabular” in logic literature, and logics whose variety is finitely generated are “tabular”.)

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Many small semigroups generate varieties that contain non-finitely generated subvarieties. For instance, the 3-element semigroup $\langle a,1\,|\,a^2=0\rangle$ and the 4-element semigroup $\langle a,b\,|\,a^2=a,b^2=b,ab=0\rangle$.

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