Combinatorial Rees–Sushkevich varieties that are Cross, finitely generated, or small

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Abstract

A variety is said to be a ReesSushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set F of finitely generated varieties constitutes an incomplete sublattice and the set S of small varieties constitutes a strict incomplete sublattice of F. Consequently, a combinatorial Rees–Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set Σ of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity O(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ.
Original languageAmerican English
Pages (from-to)64–84
JournalBulletin of the Australian Mathematical Society
Volume81
Issue number1
DOIs
StatePublished - Feb 2010

ASJC Scopus Subject Areas

  • General Mathematics

Keywords

  • Semigroup
  • 0-Simple
  • Variety
  • Rees–Sushkevich
  • Cross
  • Finitely based
  • Finitely generated
  • Small

Disciplines

  • Mathematics

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