A Categorical Equivalence for Product Algebras

Abstract

In this paper we provide a categorical equivalence for the category ${\mathcal{P}}$ of product algebras, with morphisms the homomorphisms. The equivalence is shown with respect to a category whose objects are triplets consisting of a Boolean algebra B, a cancellative hoop C and a map ${\vee_e}$ from B × C into C satisfying suitable properties. To every product algebra P, the equivalence associates the triplet consisting of the maximum boolean subalgebra B(P), the maximum cancellative subhoop C(P), of P, and the restriction of the join operation to B × C. Although several equivalences are known for special subcategories of ${\mathcal{P}}$, up to our knowledge, this is the first equivalence theorem which involves the whole category of product algebras. The syntactic counterpart of this equivalence is a syntactic reduction of classical logic CL and of cancellative hoop logic CHL to product logic, and viceversa.