  We investigate the canonicity of inequalities of the intuitionistic
mu-calculus. The notion of canonicity in the presence of fixed point operators
is not entirely straightforward. In the algebraic setting of canonical
extensions we examine both the usual notion of canonicity and what we will call
tame canonicity. This latter concept has previously been investigated for the
classical mu-calculus by Bezhanishvili and Hodkinson. Our approach is in the
spirit of Sahlqvist theory. That is, we identify syntactically-defined classes
of inequalities, namely the restricted inductive and tame inductive
inequalities, which are, respectively, canonical or tame canonical. Our
approach is to use an algorithm which processes inequalities with the aim of
eliminating propositional variables. The algorithm we introduce is closely
related to the algorithms ALBA and mu-ALBA studied by Conradie, Palmigiano, et
al. It is based on a calculus of rewrite rules, the soundness of which rests
upon the way in which algebras embed into their canonical extensions and the
order-theoretic properties of the latter. We show that the algorithm succeeds
on every restricted inductive inequality by means of a so-called proper run,
and that this is sufficient to guarantee their canonicity. Likewise, we are
able to show that the algorithm succeeds on every tame inductive inequality by
means of a so-called tame run. In turn, this guarantees their tame canonicity.
