Math 2.0

General algebraic systems Definition

Inherently nonfinitely based algebra

Builds on Finitely based algebraLocally finite varietyVariety of algebrasOates–Powell theorem

Definition

Let AA be a finite algebra of a finite signature with only function symbols. The algebra AA is inherently nonfinitely based if every locally finite variety that contains AA is not finitely based.

Remarks

The variety generated by a finite algebra is locally finite, so an inherently nonfinitely based finite algebra is not finitely based. Let AA be inherently nonfinitely based and let BB be a finite algebra whose variety contains AA. Then BB is inherently nonfinitely based as well, since every locally finite variety containing BB contains AA.

Example: the six-element Brandt monoid B21B_2^1, as a semigroup, is inherently nonfinitely based, by a theorem of Sapir.

Non-example: no finite group GG is inherently nonfinitely based. The variety generated by GG is locally finite and contains GG, and it is finitely based by the Oates–Powell theorem.

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Finitely based algebra, concept, no evidence levelFinitely based algebraconceptGroup, concept, no evidence levelGroupconceptIdentity, concept, no evidence levelIdentityconceptLocally finite variety, concept, no evidence levelLocally finite varietyconceptVariety of algebras, concept, no evidence levelVariety of algebrasconceptOates–Powell theorem, statement, no evidence levelOates–Powell theoremstatementInherently nonfinitely based algebra, definition, no evidence levelInherently nonfinitel…definition
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. Finitely based algebra concept
  2. Group concept
  3. Identity concept
  4. Locally finite variety concept
  5. Variety of algebras concept
  6. Oates–Powell theorem statement
  7. Inherently nonfinitely based algebra definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Variety_(universal_algebra)
About this page
TagJ9BZ6ZWGH6
URIm2://J9BZ6ZWGH6
Kinddefinition
WritingW0 stub
MSC08B05
DefinesInherently nonfinitely based algebra
UsesFinitely based algebra, Locally finite variety, Variety of algebras, Oates–Powell theorem

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