From Rectangular Bands to k -Primal Algebras

noindent 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. June 18, 1999