Natural dualities, nilpotence and projective planes

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.2000 Mathematics Subject Classification. Primary: 08C20. Secondary: 20M07, 20M17.

Dualizability of automatic algebras

“…Remark 2.6. The automatic algebra F 0 is a 3-nilpotent semigroup, and is therefore also covered by M. Jackson's general result [11] that all finite proper 3-nilpotent semigroups are inherently non-dualizable.…”

Bohr Compactifications of Algebras and Structures

“…There is only fragmentary information on dualisability of semigroups, which form a large and diverse class, with only certain subclasses (for example bands) analysed in depth. (See [36] for a detailed discussion of dualisability for both groups and semigroups. )…”