Re: equivalent varieties

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Fri, 25 Apr 2003, Peter Freyd wrote:

> Varieties of algebras when viewed as categories can be unexpectedly
> equivalent. For a reason explained at the end, I was looking at
> varieties of unital rings satisfying the equations  p = 0  and
> x^p = x, one such variety for each prime integer  p.
>
> The equivalence type of these categories is independent of  p. The
> easiest way of establishing that is to show that each is equivalent to
> the category of Boolean algebras (a well-known fact when  p = 2) and
> all the equivalences can by established by just one functor. Given a
> unital ring, R, define  B(R)  to be the boolean algebra of its central
> idempotents where the meet of  a  and  b  is  ab  and the join is
> a + b - ab. Then the restriction of  B  to the  p'th variety described
> above is always an equivalence of categories.
>
> The fastidious will note (one would certainly hope) that  B  is not a
> functor in general (homomorphisms don't preserve centrality). But in a
> ring "without nilpotents" (that is, in which  x^2 = 0  implies  x = 0)
> all idempotents are central. The equations  x^p = x, of course, imply
> the absence of nilpotents.
>
> (Given  p  the inverse functor to  B  can be described as follows: for
> a Boolean algebra  C  consider the set of  "p-labeled partitions of
> unity", that is, the set of functions  f:Z_p  ->  C  whose values are
> pairwise disjoint and have unity as their join. Given two such, f  and
> g, define their sum by setting  (f+g)i  to be the join of the set
> { fj ^ gk | j+k = i }  and their product by setting  (fg)i  to be the
> join of  { fj ^ gk | jk = i }.)
>
The equivalence of these varieties for all p is well known. It's best
understood by seeing that they are all dual to the category of Stone
spaces: given a Stone space, the ring of continuous Z_p-valued functions
on it (where Z_p is given the discrete topology) is a ring satisfying
p1=0 and x^p=x; conversely, given such a ring, its prime (=maximal)
ideal spectrum is a Stone space.

Not having my copy of "Stone Spaces" to hand as I write this, I can't
remember whether this fact was in the book. But it certainly should have
been.

Peter Johnstone