What are the algebras of the double powerset monad?
Solution 1:
The signature of complete atomic boolean algebras is the class of all functions $\{0,1\}^\kappa \to \{0,1\}$, where $\kappa$ can be any set. Up to canonical isomorphism, for any algebra $a:\mathscr{P}^2X\to X$ and any operator $P:\{0,1\}^\kappa \to \{0,1\}$, we have: $$X^Y \stackrel{(\eta_X)^\kappa}\longrightarrow \{0,1\}^{\mathscr PX\times \kappa} \stackrel{P^{\mathscr PX}}\longrightarrow \{0,1\}^{\mathscr P X} \stackrel a \longrightarrow X$$ This way every $\#\kappa$-ary operation on $\{0,1\}$ in implemented on $X$.