---
tag: RD2A5Z7E2B
kind: concept
name: Post algebra
msc: [06D25, 03G20, 08A40]
sources:
  - url: https://en.wikipedia.org/wiki/Many-valued_logic
---

## Summary

Post algebras are the algebras of Post's $n$-valued logic, in the same way as Boolean algebras are the algebras of classical two-valued logic.
For each $n \ge 2$, the Post algebras of order $n$ form the variety generated by an $n$-element chain with a cyclic negation.
This generating algebra is [[primal|XD01Y7B6XW]], so Post algebras have a Boolean-like structure theory.
For $n = 2$ they are exactly the Boolean algebras.

## Explanations

### L1 human

Classical logic has two truth values, and Boolean algebras capture its laws.
Emil Post studied logics with $n$ truth values $0, 1, \dots, n - 1$, ordered like a ladder, with a negation that moves every value one step up and the top value back to $0$.
Post algebras collect all algebras that obey the same laws as this $n$-valued system.
With three values, for example, the negation sends $0 \mapsto 1 \mapsto 2 \mapsto 0$.