---
tag: 1D541B576S
kind: conjecture
name: Park's conjecture
msc: [08B05, 08B26]
uses: [YGYZAQ12MZ, DASF939439, TH2XQGGFWH, AJNZJD8442, 35G4HQG2FN, 9GZ7F6ETC6, QAQAS3JWXW]
about: [YGYZAQ12MZ]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
history:
  posed_by: "Robert Park"
  first_stated: "1976"
---

## Statement

Let $A$ be a finite algebra of finite signature, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.
Suppose that $V(A)$ has a finite residual bound.
This means that there is a natural number $n$ such that every [[subdirectly irreducible algebra|DASF939439]] in $V(A)$ has fewer than $n$ elements.
Then $A$ is [[finitely based|YGYZAQ12MZ]].

## Remarks

Park stated the conjecture in his doctoral dissertation at UCLA in 1976.
It is open in general.
It holds when $V(A)$ is [[congruence-distributive|9GZ7F6ETC6]], by [[Baker's finite basis theorem|35G4HQG2FN]], which in that case needs no residual bound.
McKenzie proved it in 1987 when $V(A)$ is [[congruence-modular|QAQAS3JWXW]].
Willard proved it in 2000 when $V(A)$ is congruence meet-semidistributive.
Kearnes, Szendrei and Willard extended it to varieties with a difference term.
A variety with a finite residual bound is in particular [[residually small|TH2XQGGFWH]].

## Explanations

### L1 human

Every algebra in the variety of $A$ is built from "irreducible" pieces.
Park's conjecture says that if these pieces are all small, below a fixed size, then finitely many laws are enough to describe everything true in $A$.
All results so far agree with the conjecture, but nobody has proved it in general.