  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.
