Does this topology have a name?

Does this topology have a name?

[Michael Barr]

Let A be a model of a finitary equational theory and let X be the set of
congruences on A.  For a,b in A, let M(a,b) = {E} such that E is a
congruence on A and aEb.  Does this topology have a name?  It turns out
that this topology is coherent which means, among other things, that if we
make the M(a,b) clopen, the result is a Stone space.

Obviously in a ring, we could instead use the set of ideals, but aside
from the fact that that will include non-prime ideals, the topology is the
opposite of the Zariski topology.

Michael


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Re: Does this topology have a name?

[Dana Scott]

On Jan 1, 2011, at 6:59 AM, Michael Barr wrote:

> Let A be a model of a finitary equational theory and let X be the set of
> congruences on A.  For a,b in A, let M(a,b) = {E} such that E is a
> congruence on A and aEb.  Does this topology have a name?  It turns out
> that this topology is coherent which means, among other things, that if we
> make the M(a,b) clopen, the result is a Stone space.

Consider the powerset space P(A x A) = 2^(A x A).  The product topology
makes it a Stone space.  This is elementary.

Now, the space X of congruences is defined by logical formulae with
only universal quantifiers and atomic formulae xEy for variables
ranging over A.  That makes X a CLOSED subspace of P(A x A).

This is so easy, it hardly needs a name.  And it works even if A has
infinitely many operations.  That there is an equational theory in the
background seems neither here nor there to get a Stone space of congruences.

HAPPY NEW YEAR!




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Re: Does this topology have a name?

[Michael Barr]

Yes, the fact that when these sets are taken as clopens gives a Stone
space is easy.  But I want to know what to call the weaker topology in
which you take these sets as a basis of opens.

Happy New Year to you!

Michael


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