Query

Query

[Oswald Wyler]

For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z),
with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x).  Let Q-alg denote
the category of algebras for the monad induced by this self-adjunction.
If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg
is an equivalence by results of M. Sobral.  If Z has two members, then Q-alg
is isomorphic to CaBool, the category of complete atomic Boolean algebras.
What is known about Q-alg if Z has more than two members (beyond the fact
that Q-alg and CaBool are equivalent)?

re: Query (Q-algebras)

[Vaughan Pratt]

The operations of Q-alg are the functions between the powers of the set Z,
forming the (Lawvere) theory T of Q-algebras, which are product-preserving
functors from T to Set.  (So if Z is 3 then there are 27 = 3^3 "Boolean"
operations in place of the familiar 4 = 2^2.)  With all powers Q-algebras are
equivalent to CABAs, with only finite powers and finite-product-preserving
functors they are equivalent to Boolean algebras.

A natural next question would be, what is obtained when T is taken to be
Set itself, in each of the cases when the functors T->Set are required to
preserve all limits, and just the discrete ones?

Vaughan Pratt

>From: Oswald Wyler <owyler@suscom-maine.net>
>For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z),
>with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x).  Let Q-alg denote
>the category of algebras for the monad induced by this self-adjunction.
>If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg
>is an equivalence by results of M. Sobral.  If Z has two members, then Q-alg
>is isomorphic to CaBool, the category of complete atomic Boolean algebras.
>What is known about Q-alg if Z has more than two members (beyond the fact
>that Q-alg and CaBool are equivalent)?

Re: Query (Q-algebras)

[Vaughan Pratt]

>(So if Z is 3 then there are 27 = 3^3 "Boolean" operations in place of
>the familiar 4 = 2^2.)

There should have been a "unary" in there of course.

Another question about these Q-algebras that Oswald Wyler was asking
about: what is a necessary and sufficient condition for a complete basis
for finitary Q-algebras (the theory of Boolean algebras rather than CABAs)
having any given Z?  For Z = 2 one answer (at least for the version of the
problem which only considers nonzeroary operations) is that for each of the
following properties the basis must contain a counterexample to that property.
Necessity follows because each property is preserved under composition;
sufficiency takes more work.

  * selfdual (e.g. xy+yz+zx = (x+y)(y+z)(z+x))
  * monotone
  * affine (expressible as the XOR of its arguments, optionally plus 1)
  * strict (maps the all-zeros input to zero)
  * costrict (maps the all-ones input to one)

(NAND violates all five at once.)  Is there a fixed number of such properties
that works for all finite cardinalities of Z, or must the number of properties
of this kind grow with Z?

Vaughan Pratt

Re: Query (Q-algebras)

[Ernie Manes]

Hi Vaughan,

Regarding these and similar questions, I suggest looking at the following
two gems:

A. L. Foster, Gerneralized "Boolean" theory of universal algebras, Part II.
Identities and subdirect sums of functionally complete algebras, Math. Zeit.
59, 1953, 191-199.

T.-K. Hu, Stone duality for primal algebra theory, Math. Zeit. 110, 1060,
180-198.

Ernie Manes


----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: <categories@mta.ca>
Sent: Thursday, May 08, 2003 3:05 PM
Subject: categories: Re: Query (Q-algebras)


>
>
> >(So if Z is 3 then there are 27 = 3^3 "Boolean" operations in place of
> >the familiar 4 = 2^2.)
>
> There should have been a "unary" in there of course.
>
> Another question about these Q-algebras that Oswald Wyler was asking
> about: what is a necessary and sufficient condition for a complete basis
> for finitary Q-algebras (the theory of Boolean algebras rather than CABAs)
> having any given Z?  For Z = 2 one answer (at least for the version of the
> problem which only considers nonzeroary operations) is that for each of
the
> following properties the basis must contain a counterexample to that
property.
> Necessity follows because each property is preserved under composition;
> sufficiency takes more work.
>
>   * selfdual (e.g. xy+yz+zx = (x+y)(y+z)(z+x))
>   * monotone
>   * affine (expressible as the XOR of its arguments, optionally plus 1)
>   * strict (maps the all-zeros input to zero)
>   * costrict (maps the all-ones input to one)
>
> (NAND violates all five at once.)  Is there a fixed number of such
properties
> that works for all finite cardinalities of Z, or must the number of
properties
> of this kind grow with Z?
>
> Vaughan Pratt
>
>