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Suppose one has two locally finite quasivarieties $\mathcal{V}$ and $\mathcal{W}$.

Further suppose that:

  1. $\mathcal{V}$ is a variety.

  2. The finite algebras $\mathcal{V}_f$ are dually equivalent to $\mathcal{W}_f$.

Does it follow that $\mathcal{W}$ is also a variety?


Background: This question fits into the area of `Natural duality'. I am hoping someone can provide a counterexample, or perhaps provide additional conditions.

Here is a way to manufacture such examples. Let $\mathcal{V}$ be a locally finite variety such that (i) $\mathcal{V}$-epis are surjective, (ii) $\mathcal{V}_f$ has an injective cogenerator $A$. Define the algebra $B$ to have the same carrier as $A$, its finitary operations between all $\mathcal{V}$-morphisms $A^n \to A$. Further define $\mathcal{W} := \mathcal{Q}(B) = \mathbb{S}\mathbb{P}(B)$ to be the quasivariety generated by $B$. Then $\mathcal{V}_f$ is dually equivalent to $\mathcal{W}_f$ via the contravariant hom-functors for $A$ and $B$, endowed with additional algebraic structure.

e.g. Let $\mathcal{V} = \mathsf{Vect}(\mathbb{F})$ for a finite field $\mathbb{F}$ and take $A = \mathbb{F}$. Then $\mathcal{W}$ is essentially $\mathcal{V}$ again, where every linear combination of variables is viewed as an operation.

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1 Answer 1

up vote 3 down vote accepted

Here is a counterexample.

Let $A = \langle \{0,1\}; d, s\rangle$ be the algebra on $\{0,1\}$ whose defining operations are the discriminator operation $d$ on $\{0,1\}$ and the "switch" ($s(0)=1$, $s(1)=0$). According to the NU Duality Theorem of natural duality theory, $A$ is dualizable, and the topological quasivariety dual to $\mathcal V:=SP(A)$ is $\mathcal W:=S_cP(\langle \{0,1\}; s, {\mathcal T}\rangle)$, where $\mathcal T$ is the discrete topology.

I am allowing the empty algebra, else the duality doesn't work out perfectly (empty is dual to singleton in both directions).

It follows from Jonsson's Lemma that $\mathcal V:=SP(A)$ is a variety, but it is easy to see that $\mathcal W_f$ is not the finite part of a variety. $\mathcal W_f$ is the finite part of the quasivariety of $\mathbb Z_2$-sets having no fixed points, while the variety generated by $\mathcal W_f$ contains all $\mathbb Z_2$-sets.

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