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)?