  We introduce a new measure on regular languages: their nondeterministic
syntactic complexity. It is the least degree of any extension of the `canonical
boolean representation' of the syntactic monoid. Equivalently, it is the least
number of states of any subatomic nondeterministic acceptor. It turns out that
essentially all previous structural work on nondeterministic state-minimality
computes this measure. Our approach rests on an algebraic interpretation of
nondeterministic finite automata as deterministic finite automata endowed with
semilattice structure. Crucially, the latter form a self-dual category.
