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Natural dualities, nilpotence and projective planes

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Abstract

We use an interpretation of projective planes to show the inherent nondualisability of some finite semigroups. The method is sufficiently flexible to demonstrate the nondualisability of (asymptotically) almost all finite semigroups as well as to give a fresh proof of the Quackenbush-Szabó result that any finite group with a nonabelian Sylow subgroup is nondualisable. A novel feature is that the ostensibly different notions of nilpotence for semigroups, nilpotence for groups, and the property of being nonorthodox for a completely 0-simple semigroup are unified by way of a single construction. We also give a semigroup example of two dualisable finite semigroups whose direct product is inherently nondualisable.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia

    Marcel Jackson

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  1. Marcel Jackson
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Correspondence to Marcel Jackson.

Additional information

Presented by R. Quackenbush.

Dedicated to Brian Davey on the occasion of his 65th birthday

Results in this article were obtained over a eleven year period during which the author was supported by Australian Postdoctoral Fellowship DP0342459, ARC Discovery Project DP1094578 and ARC Future Fellowship FT1201000666.

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Jackson, M. Natural dualities, nilpotence and projective planes. Algebra Univers. 74, 65–85 (2015). https://doi.org/10.1007/s00012-015-0340-5

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