Counting Semigroups

Abstract

This paper proves that almost all finite nilpotent semigroups of index 3 are rigid and noncommutative, and gives estimates of the number of these semigroups. The number of finite almost N ₃ semigroups of types B, C, and D is also shown to be negligible relative to the number of nilpotent semigroups of index 3. Corresponding results hold for commutative semigroups.

Key Words:

• Almost N₃ semigroup
• Antiautomorphism
• Automorphism
• Automorphism conjecture
• Kleitman-Rothschild-Spencer conjecture
• N₃ semigroup
• Nilpotent semigroup
• Number of semigroups of order n
• Rigid semigroup

2010 Mathematics Subject Classification:

• 20M10
• 20M14
• 20M15

Notes

Communicated by V. Gould.