FOM: isomorphism

Jaap van Oosten jvoosten at
Thu Mar 12 13:50:37 EST 1998

> From: Stephen G Simpson <simpson at>

> 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

More information about the FOM mailing list