---
tag: G24036DJ5K
kind: definition
name: Post algebra
msc: [06D25, 03G20, 08A40]
defines: RD2A5Z7E2B
uses: [AJNZJD8442, XD01Y7B6XW, DWYCKAKXZD]
sources:
  - url: https://en.wikipedia.org/wiki/Many-valued_logic
  - book: isbn:9780387905785
---

## Definition

Let $n \ge 2$ and let $P_n = \set{0, 1, \dots, n - 1}$.
The *$n$-element Post algebra* is $\mathbf{P}_n = (P_n, \vee, \wedge, ', 0, n - 1)$.
Here $x \vee y = \max(x, y)$, $x \wedge y = \min(x, y)$, and $x' = x + 1$ for $x < n - 1$ while $(n - 1)' = 0$.
A *Post algebra of order $n$* is an algebra in the [[variety|AJNZJD8442]] generated by $\mathbf{P}_n$.

## Remarks

Example: $\mathbf{P}_n^2$ is a Post algebra of order $n$, since a variety is closed under direct products.
Non-example: the three-element chain $(\set{0, 1, 2}, \max, \min, ', 0, 2)$ with $x' = 2 - x$ is not a Post algebra of order $3$.
In $\mathbf{P}_3$ the identity $x''' \approx x$ holds, but here $0''' = 2 \ne 0$.
The algebra $\mathbf{P}_n$ is [[primal|XD01Y7B6XW]], because $\max$ and the cyclic negation already generate every operation on $P_n$, a result of Post.
For $n = 2$, $\mathbf{P}_2$ is the two-element Boolean algebra, so Post algebras of order $2$ are Boolean algebras.
Other texts present Post algebras through lattice axioms, as bounded distributive lattices with a chain of constants $e_0 < e_1 < \cdots < e_{n - 1}$.
Post algebras give algebraic semantics for Post's [[many-valued logic|DWYCKAKXZD]].