FOM: rings vs. algebras
Vaughan Pratt
pratt at cs.Stanford.EDU
Sun Mar 15 13:44:14 EST 1998
[I'm posting this to fom because it contains an actual point for a change.]
From: kanovei at wminf2.math.uni-wuppertal.de (Kanovei)
>It perhaps would be useful if someone among those
>thinking that BR and BA *are* isomorphic, PRESENTS
>a CONCRETE, well defined isomorphism
The concept of isomorphism has nothing to do with the question. Only you
and Steve seem to think it does. The textbooks that make this connection
say that every Boolean ring *is* a Boolean algebra. Let us know if you
ever find a textbook claiming that every Boolean ring is isomorphic to a
Boolean algebra.
>(well defined in terms of ordinary algebra, not using
>special categorical notions which many do not understand
My promised "actual point" is that the category theoretic truth that
every identity is an isomorphism appears not to hold in universal algebra.
On the one hand we have Sikorski's statement that every Boolean ring
is a Boolean algebra and conversely (which I knew long before I'd ever
heard of Sikorski's book, which is simply the first book I laid hands on
to look up what textbooks have to say on this question). I'd be amazed
if there weren't other textbooks that said this (I'm too lazy to search).
On the other we have the quite correct point that Simpson and Kanovei are
making, that the official definitions of signature and isomorphism make
Boolean rings with their standard signature not isomorphic to Boolean
algebras with their standard signature (or at least with the signature
that goes with the definition of "Boolean algebra" as a complemented
distributive lattice).
Together we have placed these two undeniable facts on a collision
course which would appear to sink a particularly elementary assumption
of category theory.
Personally I like the idea that every identity is an isomorphism. The way
I (and therefore all the sensible people in the world :) want to resolve
this conflict is by working with clones, i.e. signatures that are closed
under composition. *No mathematical harm results thereby.* This is a
theorem: algebra suffers no loss of generality whatsoever by doing so.
For if f:A->B is a homomorphism (or an isomorphism or an automorphism
or a hemidemisemifemtomorphism) with respect to a given signature,
it remains one with respect to the clone generated by that signature.
Given that signatures not closed under composition lead to the above
inconsistency, I think we have here an open and shut case for clones
instead of signatures as the basis for the definition of "homomorphism",
as well as for direct product, subalgebra, congruence, quotient algebra,
etc., all being stable under composition. Make a note in your textbooks
to that effect.
Vaughan Pratt
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