# Looking for a uniform explanation of algebras with canonical generators.

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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