Twisted associativity of the cyclically reduced product of words, part 1

The cyclically reduced product of two words $u,v$, denoted $u\ast v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have started a thorough study of $\ast$ and of the structure of the set of cyclically reduced words $\widehat{\mathcal{F}}(X)$ equipped with $\ast$. In particular we have found that a certain number of properties of the free group equipped with the reduced product can be generalized to $(\widehat{\mathcal{F}}(X),\ast )$. In this paper we continue this study by proving that a generalized version of the associative property holds for $\ast$ in a special case. In a following paper we will prove that a more general version of the associative property holds for any case.