Foster's theorem on primal algebras

Remarks

The result is due to Alfred L. Foster. The idea is that a primal algebra can express, for each element $a$, a term that tests equality with $a$, and a term that selects according to such tests. These terms let one split any algebra of $V(P)$ along a Boolean algebra of "decisions" and rebuild it from copies of $P$. For a finite $B$ with $k$ atoms, $X$ is a discrete set of $k$ points and $P[B]^* \cong P^k$. So the finite members of $V(P)$ are, up to isomorphism, the finite direct powers $P^k$. For $P$ the two-element Boolean algebra, the theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space, which is Stone's representation theorem. Primality is needed: for the two-element lattice, which is not primal, $V(P)$ contains the three-element chain, which is not a power of $P$.

Going deeper

For a student

The map $B \mapsto P[B]^*$ gives an equivalence between the category of Boolean algebras and the variety $V(P)$. This is the starting point of the theory of Boolean powers and of discriminator varieties. For quasi-primal algebras, which are more general, a similar description uses Boolean products of subalgebras of $P$. Foster's theorem makes the structure of Post algebras of each order explicit, since they form the variety generated by a primal algebra.