# Locally finite varieties which are not finitely generated

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.

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 $n>1$.
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 $\langle a,b\,|\,a^2=a,b^2=b,ab=0\rangle$.