Finite Nilpotent Semigroups of Small Coclass

Abstract

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this article a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.

Key Words:

• Coclass
• Classification
• Enumeration
• Nilpotent
• Presentation
• Semigroup

2010 Mathematics Subject Classification:

• 20M10

ACKNOWLEDGMENTS

The author thanks Bettina Eick, J. D. Mitchell, and an anonymous reviewer for comments on earlier versions of the article.

The author was supported by the doctoral program of the University of St Andrews and the project PTDC/MAT/101993/2008 of Centro de Álgebra da Universidade de Lisboa, financed by FCT and FEDER.

Notes

The remaining products are deduced as follows: vu ^k  = (vu ²)u ^k−2; u ^k v = u ^k−1(uv);

Communicated by M. Kambites.