[FOM] Re: FOM Digest, Vol 4, Issue 33

[Vaughan Pratt]

Alasdair is quite right about my getting the wrong pointer, I was too hasty
in digging it up.

However the full picture is actually quite complicated, since there are
four clearly distinguishable theorems here (along with several closely
related side theorems that needn't distract us).  In chronological order
of publication date they are

1.  Representation of distributive lattices, Birkhoff, 1933 (Thm. 25.2,
p461, as acknowledged by Stone 1937).

2.  Representation of Boolean algebras, Stone, 1934 (as acknowledged by
Birkhoff 1935).

3.  Duality of Boolean algebras and Stone spaces, Stone, principally 1936
with earlier results in 1934.

4.  Duality of distributive lattices and ordered Stone spaces, Stone, 1937.
  (The name and insight "ordered Stone space" is from Priestley 1970.)

For each of Boolean algebras and distributive lattices (so named by Mac Neille
in his 1935 Harvard dissertation; Birhoff had been calling them C-lattices and
Ore arithmetic structures), the duality result is the tight version of the
"sloppy" representation result.  Representation (as I'm using it here) only
worries about capturing the lattice structure in terms of inclusions between
subsets of a given set X.  Duality refines this by fine tuning the structure
on the representing set of sets to make it a suitable object from which the
represented object can be faithfully reconstructed; there is in addition the
notion, overlooked in those days, that the homomorphisms f:A->B of Boolean
algebras are in bijection with the continuous functions f':B'->A' between
the dual Stone spaces (and mutatis mutandis for distributive lattices).

There is no dispute that Birkhoff's representation of distributive lattices
as a ring of sets (Hausdorff's term for a set of sets closed under binary
union and intersection) was first, indeed Stone acknowledges this in footnote
4 on page 2 of his 1937 paper.  Moreover this is a nontrivial result on which
an introductory course on algebraic logic might spend half a lecture or more.

As a stand-alone result studied independently of Birkhoff's theorem, Stone's
theorem is equally nontrivial.  But if you apply Birkhoff's theorem to a
given distributive lattice L that happens to be complemented, then you are
one trivial step away from Stone's theorem: simply remove from every subset
in Birkhoff's representation of L any elements that Birkhoff might for some
reason have put in the least subset.  This step preserves the property of
being a ring of sets, i.e. a Birkhoff representation of L, while ensuring
that algebraic complement is now represented as set complement (with
respect to the subset of X representing the top element of L).  That is,
it now qualifies as a field of sets (the name given to a ring of sets closed
under complement) representing L qua Boolean algebra.

This should make it clear that, as far as the representation of either Boolean
algebras or distributive lattices as sets of sets under the appropriate
Boolean set operations is concerned, Birkhoff has clear title to the
whole development save that one small step described above for the case of
Boolean algebras.  (That's in fact how I've been proving the theorem in my
algebraic logic course for the past two decades: do it for distributive
lattices and then point out at the end how to bring in complement when
specializing Birkhoff's theorem to Boolean algebras.)

If one is going to attach any serious importance to the Boolean representation
theorem at all, it does Birkoff a grave injustice to uniquely credit Stone
with a result for which essentially all the necessary detailed and quite
nontrivial work had already been accomplished by Birkhoff in a paper received
by the Cambridge Philosophical Society in May 1933.

There does remain the awkward question why Birkhoff did not take this last
little step himself.  In acknowledging Stone's representation theorem in
his 1935 paper Birkhoff makes clear he is not questioning Stone's priority.
But if the step was so easy how did Birkhoff come to overlook it?  It is
not as though he did not have Boolean algebras in mind at that moment;
the immediately preceding Theorem 25.1 in Birkhoff 1933 states the Boolean
representation theorem for the finite case!  Furthermore Stone had been
working on these representation issues simultaneously with Birkhoff during
1932-3 (Johnstone, "Stone Spaces", p.xv).  And lastly there is Stone's
deeper contribution, the duality theorems.

So in light of all this, how should the credit for these various
representation theorems be best divided?  While I won't attempt to answer
that here, I hope I've at least made clear that attributing it to Stone
does not do Birkhoff justice.

Vaughan Pratt

	@Article(
Bir33,	Author="Birkhoff, G.",
	Title="On the combination of subalgebras",
	Journal="Proc. Cambridge Phil. Soc",
	Volume=29, Pages="441-464", Year=1933)

	@Article(
Birk35,	Author="Birkhoff, G.",
	Title="On the structure of abstract algebras",
	Journal="Proc. Cambridge Phil. Soc",
	Volume=31, Pages="433-454", Year=1935)

	@Book(
Joh82,	Author="Johnstone, P.T.",
	Title="Stone Spaces",
	Publisher="Cambridge University Press", Year=1982)

	@Article(
Pri70,	Author="Priestley, H.A.",
	Title="Representation of distributive lattices",
	Journal="Bull. London Math. Soc.", Volume=2,
	Pages="186-190", Year=1970)

	@Article(
Sto34,	Author="Stone, M.",
	Title="{B}oolean algebras and their applications to topology",
	Journal="Proc. Nat. Acad. Sci.", Volume=20, Pages="197-202", Year=1934)

	@Article(
Sto36,	Author="Stone, M.",
	Title="The theory of representations for {B}oolean algebras",
	Journal="Trans. Amer. Math. Soc.", Volume=40, Pages="37-111", Year=1936)

	@Article(
Sto37,	Author="Stone, M.",
	Title="Topological representations of distributive lattices and
	Brouwerian logics",
	Journal="\v Casopis P\v est. Math.", Volume=67, Pages="1-25",
	Year=1937)


-----Original Message-----
From: Alasdair Urquhart <urquhart at cs.toronto.edu>
Birkhoff's paper of 1935 cited by Vaughan Pratt
does indeed contain a statement of the representation
theorem for Boolean algebras.  However, the footnote
to this result reads:

"Proved by M.H. Stone, "Boolean algebras and
their application to topology", Proc. U.S.A. Acad. 
20 (1934), 197-202."

Hence, the usual attribution of the theorem to 
Marshall Stone appears to be correct.