Math 2.0

General algebraic systems Terms, identities and equational logic Conjecture

Park's conjecture

Builds on Finitely based algebraSubdirectly irreducible algebraResidually small varietyVariety of algebrasBaker's finite basis theoremCongruence-distributive varietyCongruence-modular variety

Statement

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

Remarks

Park stated the conjecture in his doctoral dissertation at UCLA in 1976. It is open in general. It holds when V(A)V(A) is congruence-distributive, by Baker's finite basis theorem, which in that case needs no residual bound. McKenzie proved it in 1987 when V(A)V(A) is congruence-modular. Willard proved it in 2000 when V(A)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.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Congruence-distributive variety, concept, no evidence levelCongruence-distributi…conceptCongruence-modular variety, concept, no evidence levelCongruence-modular va…conceptFinitely based algebra, concept, no evidence levelFinitely based algebraconceptResidually small variety, concept, no evidence levelResidually small vari…conceptSubdirectly irreducible algebra, concept, no evidence levelSubdirectly irreducib…conceptVariety of algebras, concept, no evidence levelVariety of algebrasconceptBaker's finite basis theorem, statement, no evidence levelBaker's finite basis …statementPark's conjecture, conjecture, no evidence levelPark's conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Congruence-distributive variety concept
  2. Congruence-modular variety concept
  3. Finitely based algebra concept
  4. Residually small variety concept
  5. Subdirectly irreducible algebra concept
  6. Variety of algebras concept
  7. Baker's finite basis theorem statement
  8. Park's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Variety_(universal_algebra)
About this page
Tag1D541B576S
URIm2://1D541B576S
Kindconjecture
WritingW0 stub
MSC08B05, 08B26
StatusThis copy of the site has no build data. Run m2 status 1D541B576S.
AboutFinitely based algebra
UsesFinitely based algebra, Subdirectly irreducible algebra, Residually small variety, Variety of algebras, Baker's finite basis theorem, Congruence-distributive variety, Congruence-modular variety
Changed2 November 2026

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