On quotients of subset rings over infinite sets

Question

$\newcommand{\S}{\mathcal{S}}$ For a set $X$ we definte the subset ring $\S_X$ as the set of subsets of $X$ equipped witht the two operations $$\begin{align} A + B &:= (A\setminus B) \cup (B \setminus A),\\ A \cdot B &:= A\cap B. \end{align} $$ It's easy enough to show that $(\S_X, +, \cdot)$ defines a ring, and that for finite $X$ with $|X|=n$, we have the isomorphism $\S_X\cong \mathbb{F}_2^n$ where addition and multiplication in $\mathbb{F}_2^n$ are performed componentwise.

One can also show that given a (not necessarily finite) set $X$ with $Y\subseteq X$, that $\S_Y$ is an ideal of $\S_X$, and the quotient ring $\S_X/\S_Y \cong \S_{X\setminus Y}$ by considering the homomorphism $f:\S_X\to\S_X$ given by $f(A) = A\setminus Y$ and apply the FIT.

But there are other possibly more interesting ideals of $\S_X$, particularly in the case that $X$ is infinte:

Let $\mathcal{F}_X = \{A\subset X \mid |A|<\infty\}$ be the set of finite subsets of $X$. It's trivial to show that $\mathcal{F}_X$ is an ideal of $\S_X$. Therefore, by the FIT, it must arise as the kernel of some homomorphism $\phi:\S_X\to R$, which we can hopefully use to understand what the quotient ring $\S_X/\mathcal{F}_X$ "looks like".

The elements $A+\mathcal{F} \in \S_X/\mathcal{F}$ are equivalence classes of subsets of $X$ up to the addition or removal of finitely many elements. But it's not clear to me what exactly this means - can we view the elements as "single points" of some space? In the case that $X$ is countable (e.g. $X=\mathbb{N}$) then it's a well known result that $|\S_{\mathbb{N}}|=|\mathbb{R}|$, but what about $|\S_\mathbb{N}/\mathcal{F}_\mathbb{N}|$? I feel like we want to think of a space of objects that are invariant under a finite number of small changes (corresponding to adding/removing a single element), but suddently distinct when the number of changes is infinite... though I haven't been able to come up with such a space!

I've had many questions arise in this discussion - and there are likely many other things that I haven't even thought about, but to highlight the main ones:

• Is there a more direct way to think about the elements of $\S_X/\mathcal{F}_X$ for infinite sets $X$, in general or in the specific case when $X$ is countable?

• What are some appropriate rings $R$ and homomophisms $\phi$ that might give rise to more "understandable" representations?

• Can we hence immediately determine the cardinality of $\S_\mathbb{N}/\mathcal{F}_\mathbb{N}$?

• Are ther other "interesting" ideals of $\S_X$ that do not arise as $\S_Y$ for some subset $Y\subseteq X$?

Answer 1

Here are some relevant facts.

• For arbitrary sets $X$, this ring is isomorphic to the infinite direct product $\mathbb{F}_2^X$ of $|X|$ copies of $\mathbb{F}_2$. It is a Boolean ring. There is an equivalence of categories between Boolean rings and Boolean algebras.
• Ideals of $\mathbb{F}_2^X$ are in one-to-one correspondence with objects called filters (given by taking their complements). Maximal (equivalently, prime) ideals correspond to objects called ultrafilters. If $X$ is finite then filters are in one-to-one correspondence with subsets of $X$ and ultrafilters correspond to points, where the corresponding quotients are the evaluation homomorphisms at each $x \in X$.
• However, if $X$ is infinite the situation is much more interesting, and (in the presence of the axiom of choice) there are non-principal ultrafilters which give "exotic" maximal ideals $m$, namely those containing the ideal of all finite subsets. The collection of all ultrafilters on an infinite set $X$ has a topology (essentially the Zariski topology) making it compact, Hausdorff, and totally disconnected, and in fact it is exactly the Stone-Cech compactification $\beta X$ of $X$.
• Assigning to a Boolean ring its collection of maximal ideals, together with the Zariski topology, produces a contravariant equivalence of categories between Boolean rings and what are called Stone spaces, which can be defined either as profinite sets or as compact Hausdorff totally disconnected spaces; this is Stone's representation theorem. The functor going the other direction is given by taking equivalently either the ring of continuous $\mathbb{F}_2$-valued functions, or the collection of clopen subsets.
• From this point of view, $\mathbb{F}_2^X$ itself is the ring of continuous $\mathbb{F}_2$-valued functions on the Stone-Cech compactification $\beta X$, and $\mathbb{F}_2^X/I$ is the ring of continuous $\mathbb{F}_2$-valued functions on the complement $\beta X \setminus X$. Generally, Stone's representation theorem implies that ideals of $\mathbb{F}_2^X$ are in one-to-one correspondence with closed subspaces of $\beta X$. The "obvious" ideals $\mathbb{F}_2^Y$, where $Y \subseteq X$ is a subset, correspond to the closed subspaces $\beta (X \setminus Y) \subseteq \beta X$, and there are many others as we've seen.

I am not really aware of a more concrete description of subsets-modulo-finite-subsets; the above description is really not any more concrete. It's at least straightforward to see that its cardinality is still the continuum, because the ideal of finite subsets is countable. The situation is somewhat analogous to the construction of $L^p$ spaces, where we quotient by measurable functions of $L^p$ norm zero and hence have the freedom to freely modify functions on any set of measure zero. Generally speaking elements of an $L^p$ space simply don't have any canonical representatives as actual functions and we just get used to that.

For $X = \mathbb{N}$ here is a description of $\mathbb{F}_2^{\mathbb{N}}/I$ that you might find easier to think about. You can think of its elements as equivalence classes of sequences of $0$s and $1$s, where we identify two sequences if they agree after their first $n$ terms, for some $n$. Loosely speaking this means we only care about the behavior of such a sequence "at infinity" (and we can give "infinity" a precise meaning here, it means the complement $\beta \mathbb{N} \setminus \mathbb{N}$). This may not seem any more concrete but it's at least somewhat analogous to the description of, say, $\mathbb{R}$ as equivalence classes of Cauchy sequences of rational numbers; in fact this description is equivalent to requiring that the difference between two sequences converges to zero in the discrete topology.