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Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a canonical presentation via the counit i.e. take the carrier of $A$ as generators and take all those relations that hold in $A$.

However I know of at least five varieties where each finite algebra $A \in \mathcal{V}_f$ has another `canonical' set of generators, in the sense that they induce an equivalence functor $G : \mathcal{V}_f \to \mathcal{W}$. Specifically:

  1. Let $\mathcal{V}$ be the variety of pointed sets i.e. one has a single constant and no equations. Then $\mathcal{V}_f$ is equivalent to the category of finite sets and partial functions, where $GA = A \setminus \{ 0 \}$ removes the point.

  2. Let $\mathcal{V}$ be the variety of boolean algebras. Then $\mathcal{V}_f$ is equivalent to the category whose objects are the finite sets and whose morphisms are the converse-functional relations, composition being relational composition. $GA$ takes the set of atoms of $A$ and $Gf = \{ (z,z') : z' \leq_A f(z) \}$.

  3. Let $\mathcal{V}$ be the variety of vector spaces over $\mathbb{Z}_2$. Then $\mathcal{V}_f$ is equivalent to the category of finite sets and relations, where now relational composition corresponds to multiplication of matrices over $\mathbb{Z}_2$. $GA$ chooses some basis of $A$, whereas $Gf$ is the relation corresponding to the matrix representation of $f$.

  4. Let $\mathcal{V}$ be the variety of join-semilattices with bottom. Then $\mathcal{V}_f$ is equivalent to a category whose objects are the finite strict closure spaces (finite sets with a closure operator that preserves emptyset) and whose morphisms are relations that are somehow `continuous' relative to the closures. Composition is defined by first taking that relational composition and then forming the closure. $GA$ is the set of join-irreducibles of $A$ endowed with a suitable closure. $Gf$ is defined as in it was for boolean algebras.

  5. Let $\mathcal{V}$ be the variety of bounded distributive lattices. Then $\mathcal{V}_f$ is equivalent to a category whose objects are the finite posets and whose morphisms are relations satisfying a few conditions relative to the ordering. Composition is relational composition. $GA$ is again the set of join-irreducibles but now viewed as a subposet. $Gf$ is defined as before.

My question is:

Is there a uniform explanation of these examples, to the extent that one can derive a canonical choice of generators given some variety satisfying suitable properties?

It seems that this is strongly related to Stone duality. For each of the varieties above we have $\mathcal{V}^{op} \cong \mathcal{U}(\mathsf{Stone})$ e.g. for boolean algebras $\mathcal{U} = \mathsf{Set}$ and for distributive lattices $\mathcal{U} = \mathsf{Poset}$. Restricting to $\mathcal{V}_f$ we obtain a concrete representation of its dual, and it seems that we are using something like the self-duality of the category of finite sets and relations to turn this dual-equivalence into a plain equivalence.

However I don't quite understand exactly why these particular examples work and the exact relationship to the canonical set of generators.

Any help much appreciated.

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