FOM: isomorphism
Jaap van Oosten
jvoosten at math.ruu.nl
Thu Mar 12 13:50:37 EST 1998
> From: Stephen G Simpson <simpson at math.psu.edu>
> Jaap van Oosten writes:
> > 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.
>
> OK, I take your point. Your point is that classical algebra books
> (Lang, etc.) don't define the concept of isomorphism in general; they
> only define it piece-meal for specific structures such as groups,
> rings, etc. But what about universal algebra books, e.g. books by
> P.M.Cohen, Graetzer, etc.? In there I think they define isomorphism
> in general, and part of the definition is that isomorphic structures
> have the same signature. Sorry, I don't have any of those books handy
> right now to pull out a reference, but -- dare I ask -- do you agree?
So now you shift from algebra to universal algebra. Without knowing what
you're talking about. 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.
Get the picture?
Jaap van Oosten
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