Show that every variety of mono-unary algebras is defined by a single identity.
Intuitively this makes sense but I am having trouble showing it. An example of a mono-unary algebra is $\langle \mathbb{N}, f \rangle$ where $f(n) = n + 1, \forall n \in \mathbb{N}.$ I'm not sure what identity would be used here, maybe $f^{n}(x) = f^{m}(y), n \le m; m,n \in \mathbb{N}$. Seems like some sort of 'least' identity is the one we are after since from the above we can generate many many more identities of the same 'flavour'.
I'm not sure how to generalize this more; it makes sense when I consider this specific mono-unary algebra.
I think a good approach to this (might be a bit of an overkilling but I don't find it easy to give a different explicit proof of this) is by using subdirectly irreducible algebras.
You might know which are the subdirectly irreducible mono-unary algebras (otherwise that can be another interesting exercise).
These are $$\mathbf{C}_{n^p}, \quad \mathbf{C}_{n^p}\dot\cup\mathbf{C}_{1}, \quad \mathbf{L}_n, \quad \mathbf{L}_{\infty},$$ where $p$ is a prime number, $\mathbf{C}_n$ is a cycle with $n$ elements, $\mathbf{L}_n$ is a chain with $n$ elements and $\mathbf{L}_{\infty}$ is an infinite chain with a last element (with respect to the operation) that is the image of itsef.
Like this:
Now we have: $$\mathbf{C}_{p^n}, \mathbf{C}_{p^n} \dot\cup \mathbf{C}_{1} \vDash f^{p^n}(x) \approx x,\; \mathbf{L}_n \vDash f^n(x) \approx f^n(y),\; \mathbf{L}_{\infty} \vDash x \approx x,$$ and $\mathbf{L}_{\infty}$ doesn't satisfy any nontrivial identity.
Let, for $n \geq 0$ and $m > 0$, $$U_n = Mod(f^n(x) \approx f^n(y)),\; U_{m,n} = Mod(f^n(x) \approx f^{n+m}(x)), \; U = Mod(x \approx x).$$ Then $V(\mathbf{C}_{p^n}) = V(\mathbf{C}_{p^n} \dot\cup \mathbf{C}_{1}) = U_{p^n,0}$, $V(\mathbf{L}_n) = U_n$ and $V(\mathbf{L}_{\infty}) = U$.
Then we have $$U_i \vee U_j = U_{\max\{i,j\}}, \; U_i \wedge U_j = U_{\min\{i,j\}},$$ $$U_i \vee U_{j,k} = U_{\max\{i,k\},j}, \; U_i \wedge U_{j,k} = U_{\min\{i,k\}},$$ $$U_{ij} \vee U_{kl} = U_{\max\{j,l\},\mathrm{lcm}\{i,k\}}, \; U_{ij} \wedge U_{kl} = U_{\min\{j,l\},\gcd\{i,k\}},$$ and so all varieties of mono-unary algebras belong to this family.
I might later look for a reference from where I got a hint to this, but I can't find it now and don't have much time, sorry...
Edit. Now I know why I couldn't find that reference: I never had it. I only had a review of a paper to which I couldn't get access, but perhaps you can. Anyway, I suppose it's not strictly necessary. It's the review of the paper The lattice of equational classes of algebras with one unary operation of Eugene Jacobs and Robert Schwabauer, published in The American mathematical monthly, vol. 71 (1964), pp. 151—155.