(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