Stone’s Representation Theorem I
Today we start in on the representation theorem proved by Marshall Stone: every boolean ring is isomorphic (as a ring) to a ring of subsets of some set
. That is, no matter what
looks like, we can find some space
and a ring
of subsets of
so that
as rings.
We start by defining the “Stone space” of a Boolean ring
. This is a representable functor, and the representing object is the two-point Boolean ring
. That is,
, the set of (boolean) ring homomorphisms from
to
. To be clear,
consists of the two points
, with the operations
for addition and
for multiplication, and the obvious definitions of these operations. This is a contravariant functor — if we have a homomorphism of Boolean rings
we get a function between the Stone spaces
, which takes a function
to the function
.
The Stone space isn’t just a set, though; it’s a topological space. We define the topology by giving a base of open sets. That is, we’ll give a collection of sets — closed under intersections — which we declare to be open, and we define the collection of all open sets to be given by unions of these sets. For each element , we define a basic set like so:
To see that this collection of sets is closed under intersection, consider two such sets and
. I say that the intersection of these sets is the set
. Indeed, if
and
, then
Conversely, if , then
. Thus
and so , and
. Similarly,
. Thus we see that
.
In fact, this map from to the basic sets is exactly the mapping we’re looking for! We’ve already seen that our base is closed under intersection, and that the map
preserves intersections. I say that we also have
. If
but
, then
and
. Then
and similarly if but
. Thus
. Conversely, if
, then
and so either and
or vice versa. Thus
.
So we know that is a homomorphism of (boolean) rings. However, we don’t know yet that it’s an isomorphism. Indeed, it’s possible that
has a nontrivial kernel —
could be
for some
. We must show that given any
there is some
so that
.
For a finite boolean ring this is easy: we pick some minimal element
and define
if and only if
. Such a
exists because there’s at least one element below
—
itself is one — and there can only be finitely many so we can just take their intersection. Clearly
by definition, and it’s straightforward to verify that
is a homomorphism of boolean rings using the fact that
is an atom of
.
For an infinite boolean ring, things are trickier. We define the set of all functions
, not just the ring homomorphisms. This is the product of one copy of
for every element of
. Since each copy of
is a compact Hausdorff space, Tychonoff’s theorem tells us that
is a compact Hausdorff space. If
is any finite subring of
containing
, let
be the collection of those functions
which are homomorphisms when restricted to
and for which
.
I say that the class of sets of the form has the finite intersection property. That is, if we have some finite collection of finite subrings
and the finite subring
they generate, then we have the relation
Indeed, is clearly contained in the generated ring
. Further, if
is a homomorphism on
then it’s a homomorphism on each subring
.
Okay, so since is a finite boolean ring, the proof given above for the finite case shows that
is nonempty. Thus the intersection of any finite collection of sets
is nonempty. And thus, since
is compact, the intersection of all of the
is nonempty.
That is, there is some function which is a homomorphism of boolean rings on any finite boolean subring containing
, and with
. Given any other two points
and
there is some finite boolean subring containing
,
, and
, and so we must have
and
within this subring, and thus within the whole ring. Thus
is a homomorphism of boolean rings sending
to
, which shows that
.
Therefore, the map is a homomorphism sending the boolean ring
isomorphically onto the identified base of the Stone space
.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Stone.html
Marshall Stone’s mother was Agnes Harvey. His father, Harlan Fiske Stone, was a distinguished lawyer who served as dean of Columbia Law school from 1910 to 1923 and was on the supreme court for 21 years, serving as chief justice for the last five of these from 1941 to 1946. The family tradition would have had Marshall follow his father’s into a law career and while he attended public schools in Englewood, New Jersey, it was assumed that he would become a lawyer. He entered Harvard in 1919 still intending to continue his studies at Harvard law school after first taking his bachelor’s degree, but he was so enthused by the mathematics courses that he took that, by the time he graduated in 1922, he began to think that it was mathematics, not law, that would be his life. An inspired, but extraordinary, arrangement by the Harvard Mathematics Department saw him appointed as an instructor for session 1922-23 to see whether he would enjoy teaching mathematics and whether he would decide to take his mathematical studies further….
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kindly update stone representation theorem for boolean algebra
kindly update proof of stone representation theorem for boolean algebra