FOM: Boolean algebras/rings; isomorphism; categorical confusion
Till Mossakowski
till at Informatik.Uni-Bremen.DE
Fri Mar 13 09:36:30 EST 1998
Steve Simpson wrote:
>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?
Of course, "isomorphic" and "isomorphic after a signature change"
are distinct. I think often category theory helps to
clarify this distinction, because when you say "isomorphic",
you always have to be specific about "isomorphic in which category".
>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.
O.K.
I use a very similar notion, except that I would say it does not
make sense to speak of isomorphism or non-isomorphism if the
structures do not have the same signature.
The confusion does not come from category theory, but from
the imprecise formulation of the natural language sentence
"Boolean algebras are isomorphic to Boolean rings".
If I understand your interpretation right:
forall Boolean algebras A. forall Boolean rings R. A is isomorphic to R
With your notion of isomorphism, this is trivially false,
and no one would state the sentence to be true with this
interpretation in mind.
(With mine notion of isomorphism, it is meaningless, because
the signatures are different.)
Category theory people would interpret the sentence more globally, like:
"All Boolean algebras" and "all Boolean rings" (together with
the standard homomorphism) are isomorphic categories
(Of course, "all" in this interpretation is *not*
universal quantification, but comprehension.)
Or, even stronger and more syntactically:
The clones or monads induces by the presentations of Boolean algebras
and Boolean rings are isomorphic.
You rejected this global view.
But its importance is the following:
There are many different approaches to propositional logic, differing
in the choice of basic logical connectives. But these differences
are inessential, because the derived connectives are the same.
Likewise, there are two standard definition of groups:
one is "monoid plus an inverse operation",
the other is "monoid, such that there exists an inverse for each element".
Usually people do not care about this difference, because
the corresponding categories of groups are isomorphic
(this example goes beyond derived operations).
Till
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