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:



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> 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.
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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.



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


