---
tag: TCD8SZW95Z
kind: statement
role: theorem
name: Foster's theorem on primal algebras
msc: [08A40, 08B26, "06E15"]
uses: [XD01Y7B6XW, AJNZJD8442, 0XADJKM699, ZV38HA5PAQ]
about: [XD01Y7B6XW]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_completeness
  - book: isbn:9780387905785
---

## Statement

Let $P$ be a [[primal algebra|XD01Y7B6XW]] and let $V(P)$ be the [[variety|AJNZJD8442]] it generates.
For a [[Boolean algebra|0XADJKM699]] $B$ with [[Stone space|ZV38HA5PAQ]] $X$, let $P[B]^*$ be the set of continuous maps from $X$ to $P$ with the discrete topology.
It is an algebra under the pointwise operations, called a Boolean power of $P$.
Then every algebra in $V(P)$ is isomorphic to $P[B]^*$ for some Boolean algebra $B$.

## 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 $P$.

## Explanations

### L1 human

The theorem says that all algebras built from a primal algebra $P$ are, in a precise sense, families of copies of $P$ indexed by the points of a Stone space.
So once one knows $P$ and Boolean algebras, one knows the whole variety.
This is why Boolean algebras are the simplest model of the general situation.

### L2 human

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.