equivalent varieties

equivalent varieties

[Peter Freyd]

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.

Re: equivalent varieties

[Michael Barr]

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.
>
>
>

Re: equivalent varieties

[Prof. Peter Johnstone]

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 }.)
>
The equivalence of these varieties for all p is well known. It's best
understood by seeing that they are all dual to the category of Stone
spaces: given a Stone space, the ring of continuous Z_p-valued functions
on it (where Z_p is given the discrete topology) is a ring satisfying
p1=0 and x^p=x; conversely, given such a ring, its prime (=maximal)
ideal spectrum is a Stone space.

Not having my copy of "Stone Spaces" to hand as I write this, I can't
remember whether this fact was in the book. But it certainly should have
been.

Peter Johnstone

Re: equivalent varieties

[Prof. Peter Johnstone]

On Sat, 26 Apr 2003, Prof. Peter Johnstone wrote:

> The equivalence of these varieties for all p is well known. It's best
> understood by seeing that they are all dual to the category of Stone
> spaces: given a Stone space, the ring of continuous Z_p-valued functions
> on it (where Z_p is given the discrete topology) is a ring satisfying
> p1=0 and x^p=x; conversely, given such a ring, its prime (=maximal)
> ideal spectrum is a Stone space.
>
> Not having my copy of "Stone Spaces" to hand as I write this, I can't
> remember whether this fact was in the book. But it certainly should have
> been.
>
Yes, it is there -- Exercise V 2.6, page 186.

Peter Johnstone

Re: equivalent varieties

[F W Lawvere]

An invariant way to see the particular equivalence discussed is to
note that the topos of presheaves on the category of finite sets
is the classifying topos for p-algebras for any given p > 1.
That is because any non-empty set is a retract of a finite power of
p and because left-exactness is equivalent to preserving finite
products in this particular case.

This representation suggests a different interpretation from the
usual "truth of properties" point of view concerning the essential
content of Boolean algebra. Namely, it concerns finite partitions
of a hypothetical whole and shuffling of these induced by arbitrary
maps between the index sets for the partitions, nothing more.

Coordinatizing the above shuffling of partitions using p = 3
has some advantages over p = 2, namely, the unary operations of the
theory suffice to characterize ultrafilters, i.e. to insure that
perceived points of a finite set are actually there; more formally,
the contravariant functor represented by 3 from finite sets to M-sets is
full where M is the 27-element monoid of these unary operations.



************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************

Re: equivalent varieties

[jdolan]

i wrote:

|for any finite k, the category of
|sets of cardinality a finite power of k has splitting-idempotents
|completion the category of finite sets.

non-empty, i guess.

Re: equivalent varieties

[jdolan]

peter johnstone wrote:

|The equivalence of these varieties for all p is well known. It's best
|understood by seeing that they are all dual to the category of Stone


i think of these equivalences as sort-of "morita equivalences" between
lawvere-style algebraic theories.  for any finite k, the category of
sets of cardinality a finite power of k has splitting-idempotents
completion the category of finite sets.

re: Equivalent varieties

[Michael Barr]

To expand on what I said about equivalent varieties, a French
mathematician named Batbedat showed many years ago that for any prime p,
the category of p-rings (a p-ring satisfies px = 0 and x^p = x) is
equivalent to the category of 2-rings, that is boolean rings.