---
tag: Z6TKYRZ9WX
kind: concept
name: Inherently nonfinitely based algebra
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

A finite algebra is inherently nonfinitely based when no locally finite variety that contains it is [[finitely based|YGYZAQ12MZ]].
This is a strong and robust way of failing to be finitely based: it passes to every finite algebra whose variety contains the given one.
It gives a convenient tool to show that many finite algebras, such as some finite semigroups, are not finitely based.

## Explanations

### L1 human

Some finite algebras cannot be described by finitely many laws.
An inherently nonfinitely based algebra has a stronger defect.
Put it inside any larger, still well-behaved family of algebras, and that family cannot be described by finitely many laws either.
So the defect cannot be repaired by enlarging the setting.
A finite group is never like this, because the variety it generates is locally finite and finitely based.