  Nominal algebra includes $\alpha$-equality and freshness constraints on
nominal terms endowed with a nominal set semantics that facilitates reasoning
about languages with binders. Nominal unification is decidable and unitary,
however, its extension with equational axioms such as Commutativity (which is
finitary) is no longer finitary unless permutation fixed-point constraints are
used. In this paper, we extend the notion of nominal algebra by introducing
fixed-point constraints and provide a sound semantics using strong nominal
sets. We show, by providing a counter-example, that the class of nominal sets
is not a sound denotation for this extended nominal algebra. To recover
soundness we propose two different formulations of nominal algebra, one
obtained by restricting to a class of fixed-point contexts that are in direct
correspondence with freshness contexts and another obtained by using a
different set of derivation rules.
