From: Ken.Pledger@vuw.ac.nz (Ken Pledger)
Subject: Re: Algebra of sets.
Date: Thu, 23 Sep 1999 15:03:21 +1200
Newsgroups: sci.math
Keywords: connections between rings and Boolean algebras
In article <7s0ii8$srq$1@uni00nw.unity.ncsu.edu>, "Oleg V. Poliannikov"
wrote:
> There is a definition of an algebra of sets that basically says
> that an algebra is a system of sets closed with respect to,
> say, intersection and complementation.
These satisfy the axioms for a Boolean algebra.
> There's also a definition of an abstract algebra as an abstract
> set with two binary operations satisfying some conditions.
This could mean any of several things, but I suspect your "abstract
algebra" means a ring. Rings are things like the integers, whose
addition, subtraction and multiplication behave in pretty much the usual
way. (The ring axioms make this more precise.)
> The Q: What are the operations on sets that make an algebra
> of sets be an abstract algebra?
> I remember it's symmetric difference (addition) and intersection
> (multiplication). Is it right?
Yes. As I understand it, you're observing that Boolean algebras are
defined differently from rings, and you're asking about a connection
between them. This is in various text-books, but seems to be surprisingly
little known among mathematicians in general.
Start with a Boolean algebra, say the subsets of a set. For any two
subsets A, B, define:
A + B = (A - B) U (B - A)
= (A U B) - (A intersection B) if you prefer.
This is the symmetric difference, as you mentioned. Also define:
AB = A intersection B.
It turns out that this addition and multiplication satisfy all the ring
axioms: associative, distributive, etc. Also, for every A, you have:
A^2 = A intersection A = A.
This makes these rings rather special.
On the other hand, you can start with any ring satisfying this
special extra axiom:
for every x, x^2 = x.
(This is called a Boolean ring.) Now define meet (intersection) and join
(union) operations by:
x meet y = xy,
x join y = xy + x + y.
It turns out that this meet and join satisfy all the Boolean algebra axioms.
Using these definitions you can re-interpret every Boolean _algebra_
as a Boolean _ring_, and vice versa. I've left out most of the details,
but you should find them in various algebra (or ring theory) books by
looking up "Boolean" in the index.
HTH. Ken Pledger.