FOM: Boolean algebra vs Boolean ring
Till Mossakowski
till at Informatik.Uni-Bremen.DE
Thu Mar 12 14:02:25 EST 1998
Steve Simpson wrote:
>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?
The only definition of isomorphism that applies here seems
to be isomporphism in the category of categories,
since "all Boolean algebras" and "all Boolean rings"
are categories. And these two categories are isomorphic.
>You can talk about things like this if you have a comprehensive
>foundational scheme such as set-theoretic foundations, because
>everything is a set or whatever. Perhaps you are thinking of
>something else, e.g. category-theoretic dys-foundations. In catogory
>theory, "isomorphism" is only defined within each category, not
>generally.
What would such an isomorphism be in set-theoretic terms?
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).
More generally, if you have different presentations P
and P', and a morphism between them, this induces
a functor from Mod(P') to Mod(P), which can be used to compare
algebras (or models) of different signatures.
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).
Till Mossakowski
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