Examples of Infinite Boolean Rings

Question

I'm attempting to list examples of infinite boolean rings and I need some clarification.

Firstly, is it possible to take an infinite direct product of the integers mod $2$ to get a boolean ring? (i.e. is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\times\cdots$ an infinite boolean ring?)

The only other example of an infinite boolean ring that I can think of is the ring $\mathcal{P}(X)$, the set of all subsets of some set X, with addition defined to be symmetric difference and multiplication defined to be intersection.

What are some other examples of infinite boolean rings?

Answer 1

The answer to the first question is yes, any (finite or infinite) direct product of Boolean rings is a Boolean ring. This is because the class of all Boolean rings is an instance of what's called an equational class or a variety, i.e., the class of algebraic structures satisfying a given set of identities. The example you mentioned is isomorphic to the power set of $\mathbb Z$.

A Boolean ring does not have to be a power set, but every Boolean ring is isomorphic to a subring of a power set (Stone's representation theorem). To see more examples, start with the ring of all subsets of the real line, and consider the subring consisting of the (a) finite sets, (b) sets which are finite or cofinite, (c) countable sets, (d) Borel sets, (e) Lebesgue measurable sets. Or consider the ring of all clopen (closed and open) sets in any topological space, say the Cantor set.

P.S. Examples (a) and (c) are Boolean rings but not Boolean algebras. Lacking an identity element, they allow relative but not absolute complementation. Some people would not count them as Boolean rings.

Answer 2

Every axiomatizing equation is preserved by the direct product, subalgebra and quotient algebra operations. It can be also seen directly, that each element of ${\Bbb Z_2}^{\Bbb N}$ is idempotent.

All Boolean rings define a Boolean algebra, hence arise as a subring of $P(X)$ (with symmetric difference and intersection).

Answer 3

Let $X$ be any set and $P(X)$ is its power set. On $P(X)$ define $\triangle$ by $$A\triangle B=(A\setminus B)\cup (B\setminus A) $$

for every $A,B \subset X$. Then $(P(X),\triangle ,\cap)$ is a Boolean ring.