Certain Relations on a Lattice of Varieties of Completely Regular Semigroups

Abstract

In a previous work we constructed a sublattice Σ of the lattice ℒ(𝒞ℛ) of varieties of completely regular semigroups. The lattice Σ is generated by the first few varieties of those we dubbed canonical in another communication. The latter arise as upper ends of the intervals which are classes of the congruence induced on ℒ(𝒞ℛ) by intersections with the lattice ℬ of band varieties.

We determine the T-, L-, and C-classes of all members of Σ in terms of intersections of canonical varieties. This provides the varieties so obtained with a basis, some of them consisting of a single identity. While the description of T- and L-classes are relatively easy to determine, C-classes are not and the argument is split into several lemmas. We also discuss the structure theory of completely regular semigroups in relation to Σ. The theory is illustrated by five diagrams.

Key Words:

• C-classes
• Canonical variety
• Completely regular semigroup
• L-classes
• Lattice
• Structure theorem
• T-classes

2010 Mathematics Subject Classification:

• 20M07
• 20M10

ACKNOWLEDGMENT

The assistance of Edmond W. H. Lee and the referee is deeply appreciated.

Notes

Communicated by V. Gould.