Natural dualities, nilpotence and projective planes

TL;DR

This paper explores the nondualisability of finite semigroups using projective planes, unifies various notions of nilpotence, and provides new proofs and examples related to dualisability properties.

Contribution

It introduces a flexible method to demonstrate nondualisability of finite semigroups and unifies different nilpotence concepts through a novel construction.

Findings

Most finite semigroups are nondualisable.

Finite groups with nonabelian Sylow subgroups are nondualisable.

Two dualisable semigroups can have a nondualisable product.

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-Szab\'o 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.