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.
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.
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
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.
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.
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.
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 ************************************************************
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
participants (6)
-
F W Lawvere -
jdolan＠math.ucr.edu -
Michael Barr -
Michael Barr -
Peter Freyd -
Prof. Peter Johnstone