Valuation rings: A better algebraic treatment of Boolean algebras

This is Chapter 11 in my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

Let B be a Boolean algebra such as the set P(u) of all subsets of a set u.  The usual method of interpreting B as a ring is to define:

addition                 = exclusive or (= non-equivalence)

multiplication        = intersection (= meet)

unity (1)                = u

zero (0)                  = z (null set).

The resulting ring was a Boolean ring.  But there was always the asymmetrical oddity of an alternative way to render a Boolean algebra as a Boolean ring, namely with the definitions:

multiplication        = join

unity (1)                = z

zero (0)                  = u.

The usual Boolean ring-theoretic treatment of Boolean algebras made no particular sense out of this alternative definition; it was just an odd footnote.

In the approach using Gian-Carlo Rota’s valuation rings, a new ring V(B,2) is defined from the Boolean algebra B and the two-element ring 2 (= Z2).  On this valuation ring V(B,2) with multiplication defined using the meet, the “universe” u of B is the unity (1) of the ring but the “null set” z of B is distinct from the zero 0 of the ring.  The alternative join-multiplication can be defined using the same addition so V(B,2) is a module with two ring multiplications defined on it.  With the join-multiplication, z serves as the unity for that ring structure.  The usual Boolean ring is obtained from V(B,2) cum meet-multiplication by taking the quotient setting z equal to 0, and the other Boolean ring is obtained by taking the quotient of V(B,2) cum join-multiplication by setting u equal to 0.  Thus V(B,2) contains the information and structure of the two Boolean rings that can be obtained as quotients.

A Boolean algebra satisfies the Boolean duality principle that any theorem remains valid under the interchange of the meet and join, and the interchange of the null element z and the unit u.  Boolean duality generalizes to valuation rings as the “complementation” anti-isomorphism between the two ring structures that interchanges the two multiplications and the two elements u and z.  This formulation of Boolean duality is much more general since it applies to all valuation rings, and a valuation ring V(L,A) can be constructed starting with any distributive lattice L and any commutative ring A with unity.  Thus “Boolean” duality generalizes far beyond the two-valued case where the ring of coefficients is 2 = Z2.  For instance, the valuation ring V(B,Zn) might prove useful for an algebraic treatment of multi-valued logic.

The usual Boolean algebraic treatment of propositional logic uses the free Boolean algebra B on a set P of propositional variables.  We characterize the valuation rings for the free Boolean algebras as a certain class of special polynomial rings.  The polynomials are augmented with an extra variable z, which has the special property that it “acts like zero” with respect to the other variables in the sense that xz = z.  The unit 1 is the “u” of the valuation ring.  Thus each polynomial can have a “co-constant” term azz in addition to the usual constant term auu = au1 = au where the scalars az and au are from the ring of coefficients A.  In addition, all the variables are idempotent in the sense the x2 = x and z2 = z.  We show that the ring of augmented polynomials with idempotent variables and coefficients from a commutative ring A is isomorphic to the valuation ring of the free Boolean algebra on the set of variables (without z) with coefficients from A.

That characterization means that all the duality machinery of valuations rings can be defined on these augmented polynomial rings.  In particular, there is the second join-multiplication (with z as the ring unity) and the complementation anti-isomorphism between the two ring structures.  Thus we have a duality theory for these augmented polynomials.  Furthermore, propositional logic (i.e., free Boolean algebras) can be recast and generalized using these augmented polynomial rings.  We prove a generalization of the completeness theorem using elementary polynomial reasoning.