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
following properties the basis must contain a counterexample to that
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
that works for all finite cardinalities of Z, or must the number of
the property. properties properties
of this kind grow with Z?
Vaughan Pratt