Re: Query (Q-algebras)

["Ernie Manes" <manes@mtdata.com>]

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
>
>