  The cyclically reduced product of two words is the cyclically reduced form of
the concatenation of the two words. While the reduced form of such a
concatenation (which is the product of the free group) verifies many basic
properties like for example associativity, the same is not true for the
cyclically reduced product which has been very little studied in the
literature. Recently Sergei Ivanov has proved that the Andrews-Curtis
conjecture (stated in 1965 and still not solved) is equivalent to a formulation
where the reduced product is replaced by the cyclically reduced product (and
the conjugations replaced by cyclic permutations). In this paper we study
properties of the cyclically reduced product $*$ and of the set of cyclically
reduced words $\hat{\mathcal{F}(X)}$ equipped with $*$. In particular we find
that even if $*$ is not commutative nor verifies the Latin square property,
weaker versions of these properties hold true. We also show that
$\hat{\mathcal{F}(X)}$ equipped with $*$ and with cyclic permutations enjoys
similar properties as the free group equipped with the reduced product and
conjugations.
