equivalent varieties

[Peter Freyd <pjf@saul.cis.upenn.edu>]

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.