FOM: rings vs. algebras
Vaughan Pratt
pratt at cs.Stanford.EDU
Sun Mar 15 17:23:18 EST 1998
From: kanovei at wminf2.math.uni-wuppertal.de (Kanovei)
>VP>every Boolean ring *is* a Boolean algebra
>Let me stick on this claim. As it is clear that formally BR and BA are
>objects of different signature, the only way I see to understand the
>claim is that there is a definition of the signature of BR in terms
>of the signature of BA, and a definition in the opposite direction,
>which convert each BA in BR and vice versa, and (most likely) give the
>identity in composition (modulo isomorphism).
>
>Please continue to comment.
Ok, I will. For the second time in a row I am in nearly complete
agreement with you. Just delete the "modulo isomorphism" bit because
it isn't needed, we're talking about algebras with the same carrier and
the same operations, which makes them the same algebra.
Furthermore when Simpson gets his hands on Sikorski on Monday he will
see that Sikorski elaborates his "Every Boolean algebra is a Boolean
ring...Conversely every Boolean ring is a Boolean algebra" with exactly
the sort of conversion you have in mind. I'd quote Sikorski in full here
except that this would deprive Steve of the pleasure of predicting what
Sikorski's conversion is going to be before he looks it up. I'll quote
just the last bit---"In both cases the algebraic zero and unit coincide
with the Boolean zero and unit respectively"---to rule out the De Morgan
dual of this conversion that swaps 0 and 1.
Sikorski attributes his theorem 17.1 (the above) to Stone. However
Stone found it only in 1935, by his own admission when he acknowledged
that his discovery forced a major rewrite of his Stone duality paper.
But Zhegalkin had already published it in 1927, apparently unbeknownst
to Stone, and would therefore seem to have a handy margin of priority.
I had assumed that this conversion was well known to all FOM readers,
or I would have been more explicit about it from the beginning, for
which I apologize. Logicians don't have a corner on boolean logic,
it is the bread and butter of digital engineers, who tend to take its
theory as much for granted as analog engineers take calculus.
Vaughan Pratt
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