FOM: Boolean algebras/rings; isomorphism; categorical confusion
Stephen G Simpson
simpson at math.psu.edu
Thu Mar 12 15:53:52 EST 1998
Jaap van Oosten writes:
> I looked it up for you: Graetzer (p.34) defines the notion of
> isomorphism for algebras of the same similarity class, not for
> algebras of possibly different classes. O! But what about model
> theory? Well, also Chang & Keisler define isomorphisms only for
> models of the same language.
Thanks for looking it up for me. But I don't get your point. Isn't
your reading of Graetzer, Chang/Keisler etc equivalent to, or at least
consistent with, the view that I expressed earlier? I.e. that
algebras of different similarity class, or models of different
languages, are defined to be non-isomorphic? Or, at least, not
defined to be isomorphic?
In any case, I think some of your fellow category theorists don't
agree with Graetzer, Chang/Keisler, etc. See below. Mossakowski
wants to say that structures of different signatures *are* isomorphic,
under certain circumstances. (Isomorphism of categories? Existence
of a functor?) I think he is blurring the concept of isomorphism, but
....
> Get the picture?
What are you so upset about? Please answer this, I'm genuinely
concerned.
Mossakowski writes:
> >I still want to claim that Boolean algebras are not isomorphic to
> >Boolean rings. Do you agree with this now? Try to put yourself in
> >the appropriate frame of mind.
>
> But what notion of isomorphism are you using within this statement?
I am using the following notion. Two structures (A,Phi) and (B,Psi)
are said to be isomorphic if dom(Phi)=dom(Psi)=s, i.e. s is the common
signature, and if there is a 1-1 onto mapping i:A->B such that for all
n-ary relation symbols R in s and all a1,...,an in A,
Phi(R)(a1,...,an) iff Psi(R)(i(a1),...,i(an)), and for all n-ary
operation symbols o in s,
i(Phi(o)(a1,...,an))=Psi(o)(i(a1),...,i(an)).
I think this definition of isomorphism agrees with what is in most
math books including books on algebra, universal algebra, and model
theory. It may also agree with some category theory books, but
probably not all of them.
> The only definition of isomorphism that applies here seems
> to be isomporphism in the category of categories,
No, that's not what I had in mind!
> since "all Boolean algebras" and "all Boolean rings"
> are categories. And these two categories are isomorphic.
No, you misunderstood me. "All Boolean algebras" does not refer to a
category. "All" is a universal quantifier. It is a standard
construct in the predicate calculus, going back to Frege. I assume
you are familiar with it. If not, please let me know and I'll try to
explain it, or you can read up on it in textbooks of mathematical
logic. It is sometimes denoted by an upside-down A.
> What would such an isomorphism be in set-theoretic terms?
The definition of isomorphism that I gave above is easily and
standardly formalized in set-theoretic terms, using sets of ordered
n-tuples, etc.
> If you want to compare a boolean algebra A and a boolean ring R,
> the only way seems to be to ask whether F(A) is isomorphic
> to R (where F the isomorphism functor from the category of
> boolean algebras to the category of boolean rings).
No, there are other ways to compare them. According to the standard
definitions of Boolean algebra and Boolean ring that Pratt quoted from
Sikorski, and using my definition of isomorphism as above (which I
maintain is well recognized in the mathematical literature), A and R
are not isomorphic, simply because they don't have the same signature.
By the way, it's perhaps interesting to note that some reference works
on Boolean algebras define them to have *the same* signature as rings,
i.e. {+,.,-,0,1}, rather than the more standard {U,^,-,0,1}. But +
denotes join, so one has a+a=a, not a+a=0. In this way of doing
things, Boolean algebras are never isomorphic to Boolean rings *even
though they have the same signature*. I happen to like this way of
doing things, because it's typographically convenient; see for
instance Jech's set theory textbook. But this notational convention
could be confusing in a side-by-side comparison of Boolean algebras
and Boolean rings, such as I am attempting here. (Only attempting,
because category-theoretic confusion is getting in the way.)
> By the way, these changes of signatures are quite important
> in theoretical computer science. It's a pity that they are only
> rudimentary recognized in mathematical logic (generally, only reducts
> induced by signature inclusions are considered).
No, you're mistaken on this. Various kinds of signature changes are
very well recognized and understood by mathematical logicians. Also,
they routinely consider more general notions: various kinds of
relative interpretability between theories. However, mathematical
logicians are also very clear on the distinction between "isomorphic"
and what we might call "isomorphic after a signature change"; this
distinction seems to be lost on some category theorists. Do you and
van Oosten grasp this distinction?
-- Steve
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