FOM: organizing; Boolean rings; signature

Vaughan Pratt pratt at cs.Stanford.EDU
Thu Mar 19 00:55:11 EST 1998

```From: Stephen G Simpson <simpson at math.psu.edu>
>Seriously, could you
>people please give me some feedback about the search engines on the
>FOM web page at http://www.math.psu.edu/simpson/fom/ ?

I've found them extremely helpful in finding old articles for
cross-referencing etc.  So far I've only used grep but this seems to
have done everything I needed.

>Basically, Pratt wants to
>dispense with the well-known notion of "signature" (i.e. a set of
>symbols with arities assigned) and replace it with an incredibly
>complicated set of alternative concepts which are most conveniently
>expressed in terms of category theory.

One consequence of having the notion of signature be independent of
theory is the one we've seen, that it seems inevitably to lead to more
than one variety of Boolean algebras, which one must then account for

McKenzie, McNulty and Taylor in "Algebras, Lattices, and Varieties"
explicitly invent a notion of equivalence of varieties to handle this.
Other authors mostly just gloss over the issue hoping the reader won't
notice this glitch in the notion of variety.

Those working within what they perceive as the one variety, such as
Sikorski writing about Boolean algebras, don't have this problem as
they don't need to modify the theory let alone remove it, and so think
in terms of a single variety, as Sikorski evidently does when he says
"Every Boolean algebra is a Boolean ring" where Mckenzie et al (and Steve)
would use the more careful "is equivalent to".

For those of us who believe in the essential unity of the concept of
Boolean algebras, having multiple varieties for what is intuitively a
single variety is a crock.  One wants independence of choice of basis in
the concept of variety of Boolean algebras.

My proposal to Steve for achieving basis independence was simply not
to permit signatures to exist in isolation from theories.  With this
approach it becomes possible to have signatures some operations of
which are e.g. composites of others.  This is possible because such
relationships can be encoded in the associated theory.  The point that
Steve had drawn my attention to is that in the absence of any theory
at all, the situation when considering a signature in isolation, all
symbols must be independent (unless one introduces some other structure
on signatures to encode such composition relationships, but this is
pointless when the theory itself can do this job).

When Steve challenged me as to whether this anti-isolation principle
for signatures had been worked out I said that I hadn't wanted to say so
because he wouldn't like it, but that this exactly described both Lawvere
algebraic theories and monads, both of which characterize varieties,
neither in a way that has a notion of signature independent of theory.

Naturally Steve views Lawvere theories, monads, and all other "categorical
dys-foundations" as "incredibly complex."  This should come as no suprise
to anyone who's read Steve's correspondence with Colin McLarty.  Steve is
convinced category theory will join communism in hell.

However I also said to Steve that doing universal algebra with theories
always accompanying the signature obviously does not require any
category theory and obviously need not be complicated.  Naturally Steve
is skeptical, so I guess I need to either find a non-category theoretic
account in the literature of this solution to the problem of multiple
varieties where there should only be one, aka basis-independence, or
write one myself.

Please, God, let the literature contain the former, there is no glory
in documenting the obvious.

Vaughan Pratt

```