Re: equivalent varieties

[Michael Barr <barr@math.mcgill.ca>]

Two comments on Peter's posting.  First the particular example he mentions
was apparently first discovered by a French mathematician named Batbedat.
Second, there is an example of an infinitary theory whose category of
algebras is equivalent to the category of sets!  Simply take the
underlying functor to sets represented by an infinite set and prove it is
tripleable using Beck's PTT (very easy).  The theory has as n-ary
operations all functions X --> X^n where X is the representing set.

On Fri, 25 Apr 2003, Peter Freyd wrote:

> Varieties of algebras when viewed as categories can be unexpectedly
> equivalent. For a reason explained at the end, I was looking at
> varieties of unital rings satisfying the equations  p = 0  and
> x^p = x, one such variety for each prime integer  p.
>
> The equivalence type of these categories is independent of  p. The
> easiest way of establishing that is to show that each is equivalent to
> the category of Boolean algebras (a well-known fact when  p = 2) and
> all the equivalences can by established by just one functor. Given a
> unital ring, R, define  B(R)  to be the boolean algebra of its central
> idempotents where the meet of  a  and  b  is  ab  and the join is
> a + b - ab. Then the restriction of  B  to the  p'th variety described
> above is always an equivalence of categories.
>
> The fastidious will note (one would certainly hope) that  B  is not a
> functor in general (homomorphisms don't preserve centrality). But in a
> ring "without nilpotents" (that is, in which  x^2 = 0  implies  x = 0)
> all idempotents are central. The equations  x^p = x, of course, imply
> the absence of nilpotents.
>
> (Given  p  the inverse functor to  B  can be described as follows: for
> a Boolean algebra  C  consider the set of  "p-labeled partitions of
> unity", that is, the set of functions  f:Z_p  ->  C  whose values are
> pairwise disjoint and have unity as their join. Given two such, f  and
> g, define their sum by setting  (f+g)i  to be the join of the set
> { fj ^ gk | j+k = i }  and their product by setting  (fg)i  to be the
> join of  { fj ^ gk | jk = i }.)
>
> I was looking for examples of equational theories with unique maximal
> consistent equational extensions. The best known example is the theory
> of lattices: every equation consistent with the theory of lattices is
> a consequence of distributivity. (Inconsistent in the equational
> setting means that all equations can be proved, or equivalently, the
> one equation  x = y  can be proved.) That is, the unique maximal
> consistent extension of the theory of lattices is the theory of
> distributive lattices (fortunately this is independent of your choice
> of whether top and/or bottom are considered to be part of the theory
> of lattices). A less-well-known example is the theory of Heyting
> algebras: every equation consistent with the theory of Heyting
> algebras is a consequence of the law of double-negation:
> (x -> 0) -> 0 = x. That is, the unique maximal consistent extension of
> the theory of Heyting algebras is the theory of Boolean algebras.
>
> This search for examples was sparked by what I consider a great
> example -- not to be described here -- in "algebraic real analysis".
> The only other examples I've found are the theories of unital rings of
> characteristic  p, one such example for each prime  p. To shift to the
> traditional language here, any polynomial identity consistent with
> characteristic  p  is a consequence of characteristic  p  and the
> identity  x^p = x.  A lot of examples. But, then again, maybe just one
> example.
>
>
>