---
tag: J9BZ6ZWGH6
kind: definition
name: Inherently nonfinitely based algebra
msc: [08B05]
defines: Z6TKYRZ9WX
uses: [YGYZAQ12MZ, 8QPW65PZVW, AJNZJD8442, 5RFPT46DXX]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Definition

Let $A$ be a finite algebra of a finite signature with only function symbols.
The algebra $A$ is *inherently nonfinitely based* if every [[locally finite variety|8QPW65PZVW]] that contains $A$ is not [[finitely based|YGYZAQ12MZ]].

## Remarks

The [[variety|AJNZJD8442]] generated by a finite algebra is locally finite, so an inherently nonfinitely based finite algebra is not finitely based.
Let $A$ be inherently nonfinitely based and let $B$ be a finite algebra whose variety contains $A$.
Then $B$ is inherently nonfinitely based as well, since every locally finite variety containing $B$ contains $A$.

Example: the six-element Brandt monoid $B_2^1$, as a semigroup, is inherently nonfinitely based, by a theorem of Sapir.

Non-example: no finite group $G$ is inherently nonfinitely based.
The variety generated by $G$ is locally finite and contains $G$, and it is finitely based by the [[Oates–Powell theorem|5RFPT46DXX]].