Rings and Boolean Algebras as Algebraic Theories
Arturo De Faveri
TL;DR
The paper develops a unified algebraic-theory framework that represents commutative and Boolean rings as affine and hyperaffine theories, respectively, establishing a precise correspondence between these theories and the underlying rings. It proves that the full subcategory of affine theories (excluding the degenerate $U'$) is equivalent to the category of commutative rings, with $A_R(2)\cong ext{R}$, and that hyperaffine theories are equivalent to Boolean rings, via their two-ary operations forming a Boolean algebra. The models of these theories over a Boolean ring $B$ correspond to Boolean-vector spaces with a compatible $B$-action, and admit a sheaf representation as $S: B^{ ext{op}} o ext{Vect}_{\(mathbb{F}_2 ext{)}}$, yielding a bridge to Bergman-style $B$-set semantics for the if-then-else construct. This framework clarifies how affine and hyperaffine algebraic theories capture the algebraic semantics of programming language constructs and points toward generalizations to broader algebraic settings and higher-dimensional affine structures.
Abstract
We present a unified framework for representing commutative and Boolean rings through affine and hyperaffine algebraic theories. This yields categorical equivalences between these theories and rings, and leads to a new analysis of certain classes of modules over Boolean rings. The resulting structures naturally capture the algebraic semantics of the if-then-else construct in programming languages.