  The paper explores properties of the {\L}ukasiewicz {\mu}-calculus, or
{\L}{\mu} for short, an extension of {\L}ukasiewicz logic with scalar
multiplication and least and greatest fixed-point operators (for monotone
formulas). We observe that {\L}{\mu} terms, with $n$ variables, define monotone
piecewise linear functions from $[0, 1]^n$ to $[0, 1]$. Two effective
procedures for calculating the output of {\L}{\mu} terms on rational inputs are
presented. We then consider the {\L}ukasiewicz modal {\mu}-calculus, which is
obtained by adding box and diamond modalities to {\L}{\mu}. Alternatively, it
can be viewed as a generalization of Kozen's modal {\mu}-calculus adapted to
probabilistic nondeterministic transition systems (PNTS's). We show how
properties expressible in the well-known logic PCTL can be encoded as
{\L}ukasiewicz modal {\mu}-calculus formulas. We also show that the algorithms
for computing values of {\L}ukasiewicz {\mu}-calculus terms provide automatic
(albeit impractical) methods for verifying {\L}ukasiewicz modal {\mu}-calculus
properties of finite rational PNTS's.
