FOM: Boolean algebra vs Boolean ring
Jaap van Oosten
jvoosten at math.ruu.nl
Thu Mar 12 06:35:24 EST 1998
> OK, good. So from this we see that a Boolean ring is a special kind
> ring. The signature (i.e. similarity type) of a ring, in particular a
> Boolean ring, is {+,-,0,.,1}. Do you agree? In any case, you have
> acknowledged that the signature of rings contains {.}. Is that
> correct? Now, according to to Sikorski as quoted by you, the
> signature of Boolean algebras is {U,^,-}. Is that correct? So a
> Boolean algebra doesn't have the same signature as a Boolean ring. Do
> you agree? And part of the definition of two structures being
> isomorphic is that they have the same signature. (See for instance my
> lecture notes on model theory, at
> http://www.math.psu.edu/simpson/courses/math563/. But I think that
> this is the same concept of isomorphism that you will find in any
> algebra texbook, e.g. van der Waerden, Jacobson, or Lang.)
Algebra books define what an isomorphism of groups is, an isomorphism
of rings, etc.; you will never find in an algebra book the statement that
the group of integers is not isomorphic to the field of rationals. If two
structures have different types you can't talk about isomorphism, so you
can't say that they are not isomorphic either. You created the confusion yourself
with the off-base comment that Boolean algebras are not isomorphic to
Boolean rings, and please reserve your patronizing language ("Do you agree?"
"Is that correct?") to your children.
Isn't it about time people stop writing pages and pages about this kind of
trivia?
Jaap van Oosten
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