Rings and Boolean Algebras as Algebraic Theories

Abstract

We present a unified framework for representing commutative rings through affine algebraic theories and Boolean rings through hyperaffine algebraic theories. This yields categorical equivalences between these theories and, respectively, commutative rings and Boolean rings. We then analyse models of affine theories over a Boolean ring $B$, comparing them with the models of hyperaffine theories, the well-known $B$-sets. Two novel characterisations are presented: the first defines these models as Boolean vector spaces equipped with an action of the Boolean ring; the second provides a representation in terms of sheaves, in analogy with $B$-sets. Finally, we establish a connection between hyperaffine theories and multidimensional Boolean algebras, a recently introduced generalisation of Boolean algebras.