Functorial Representation Theorems for MVΔ Algebras with Additional Operators

We prove functorial representation theorems for MVΔ algebras, and for varieties obtained from MVΔ algebras by the adding of additional operators corresponding to natural operations in the real interval [0, 1], namely PMVΔ algebras, obtained by the adding of product, and Π algebras, obtained by the adding of product and of its residuum. Our first result is that the category of MVΔ algebras is equivalent to that of lattice ordered abelian groups with strong unit and with some kind of characteristic function. Our second result is that the category of PMVΔ algebras is equivalent to a category of commutative f-rings with strong unit, again with a suitable characteristic function, whose members (modulo a forgetful functor) are isomorphic to subdirect products of linearly ordered domains of integrity. Our third result (in our opinion, the most interesting) is that the category of Π algebras is equivalent to a category whose members are regular commutative f-rings with unit, equipped with an ideal with suitable properties.