FOM: Boolean algebra vs Boolean ring
Stephen G Simpson
simpson at math.psu.edu
Wed Mar 11 13:12:12 EST 1998
Vaughan Pratt writes:
> From: Sikorski, R., "Boolean Algebras", Springer-Verlag, 1960:
> >17.1 Every Boolean algebra is a Boolean ring...
> >Conversely, every Boolean ring is a Boolean algebra....
Vaughan, I think if you read Sikorski's comments carefully, you will
find that the remarks which you quote are not to be taken as the
*definitions* of Boolean algebra and Boolean ring. Here, when
Sikorski says "is", this is to be understood in a certain loose sense,
which is defined by the context. In particular, Sikorski is not
saying that a Boolean algebra is isomorphic to or identical with a
Boolean ring. I don't have my copy of Sikorski's book handy, but I'm
pretty sure that he makes it clear. Do you understand this point? Or
are you under the impression that Sikorski is saying that Boolean
algebras and Boolean rings are isomorphic? Or perhaps that they are
exactly the same concept, i.e. no distinctions whatsoever?
> Sikorski (op.cit.):
> >A Boolean algebra is a non-empty set with three operations AUB, A^B, -A
> >satisfying axioms (A1)-(A5) [omitted here]
> >A ring is said to be a Boolean ring provided it contains a unit 1 and
> >A.A = A for every A.
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.) Do you
agree? (If you don't, maybe you ought to state "your" definition of
when two structures are isomorphic.) So from this it follows that a
Boolean ring is not isomorphic to a Boolean algebra. Do you agree?
There is also the matter of the axioms, which you omitted, and various
other identities. According to Sikorski's definitions, in a Boolean
algebra we have A U A = A, but in a Boolean ring we have A + A = 0.
Do you agree? So there are distinctions that go beyond the actual
symbols in the signatures. Do you agree?
> From: Colin McLarty 3/10/98 7:48PM
> >Now Vaughan, like many other logicians and mathematicians, understands an
> >algebraic theory without distinguishing "basic" operations from "derived"
> >ones--that is primitive from defined operations. There is nothing arcane
> >about this, and on this basis Boolean algebras are exactly the same
> >things as Boolean rings.
Vaughan, since you quote this without comment, I assume you agree with
it. Never mind what McLarty means by it; the question is, what do you
mean by it? Does it mean that you don't distinguish basic operations
from derived ones? So you are saying that, according to Sikorski, U
is one of the operations of a Boolean ring, but not of an arbitrary
ring? Does this mean that, "for you", Boolean rings and arbitrary
rings have different signatures?
All of this is of course very standard material in mathematics, but
you seem to be using mathematical language in a somewhat unusual way
(at least differently from how it is used by the mathematicians that I
am familiar with), so I think we need to go into these details if we
are to have any basis for a rational discussion. Do you agree?
In trying to untangle this mess, it occurs to me that perhaps I need
to raise the following question. Is the concept *signature*
(i.e. similarity type) valid "for you"? It's my impression that most
mathematicians grasp this concept. The signature of a structure is
simply the set of names or symbols denoting the basic operations of
the structure. If you want, I think I can perhaps explain this to you
(not rigorously, however) in terms of structs in the C programming
language.
-- Steve
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