The family
{\mathcal{R}} X^*
of regular subsets of the free monoid
X^*
generated by a finite set X is the standard example of a
{}^*
-continuous Kleene algebra. Likewise, the family
{\mathcal{C}} X^*
of context-free subsets of
X^*
is the standard example of a
\mu
-continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator
\mu
. For arbitrary monoids M,
{\mathcal{C}} M
is the closure of
{\mathcal{R}}M
as a
\mu
-continuous Chomsky algebra, more briefly, the fixed-point closure of
{\mathcal{R}} M
. We provide an algebraic representation of
{\mathcal{C}} M
in a suitable product of
{\mathcal{R}} M
with
C_2'
, a quotient of the regular sets over an alphabet
\Delta_2
of two pairs of bracket symbols. Namely,
{\mathcal{C}}M
is isomorphic to the centralizer of
C_2'
in the product of
{\mathcal{R}} M
with
C_2'
, i.e. the set of those elements that commute with all elements of
C_2'
. This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets
X\subseteq M
by regular expressions over
X\cup\Delta_2
interpreted in the product of
{\mathcal{R}} M
and
C_2'
. More generally, for any
{}^*
-continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of
C_2'
in the product of K with
C_2'
.