  We propose a new axiomatisation of the alpha-equivalence relation for nominal
terms, based on a primitive notion of fixed-point constraint. We show that the
standard freshness relation between atoms and terms can be derived from the
more primitive notion of permutation fixed-point, and use this result to prove
the correctness of the new $\alpha$-equivalence axiomatisation. This gives rise
to a new notion of nominal unification, where solutions for unification
problems are pairs of a fixed-point context and a substitution. Although it may
seem less natural than the standard notion of nominal unifier based on
freshness constraints, the notion of unifier based on fixed-point constraints
behaves better when equational theories are considered: for example, nominal
unification remains finitary in the presence of commutativity, whereas it
becomes infinitary when unifiers are expressed using freshness contexts. We
provide a definition of $\alpha$-equivalence modulo equational theories that
take into account A, C and AC theories. Based on this notion of equivalence, we
show that C-unification is finitary and we provide a sound and complete
C-unification algorithm, as a first step towards the development of nominal
unification modulo AC and other equational theories with permutative
properties.
