From rectangular bands to k -primal algebras

We begin by giving a new proof that every finite rectangular band is naturally dualisable. Motivated by the dualising structure arising from this proof, we call an algebra k-primal if it is (isomorphic to) a product of k independent primal algebras. For each k \geq 2 we exhibit a strong duality between the quasi-variety generated by a k -primal algebra and the topological quasi-variety \lilcat D _k of Boolean topological k-dimensional diagonal algebras. The category \lilcat D ₂ is the category of compact, totally disconnected rectangular bands. This duality extends Hu’s duality for varieties generated by a primal algebra to the k -dimensional case. We find that Hu’s ``uniqueness principle’’ for such varieties also extends to the k -dimensional case, namely, we show that a quasi-variety is equivalent as a category to the quasi-variety generated by a k -primal algebra if and only if it is itself generated by a k -primal algebra.