About the cyclically reduced product of words

Abstract

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 S. V. 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 F (X) equipped with * . In particular we find that even if * is not commutative nor verifies the Latin square property, generalized versions of these properties hold true.We also show that F (X) equipped with * and with cyclic permutations enjoys similar properties as the free group equipped with the reduced product and conjugations.

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

“…However ( F (X), * ) has interesting properties. Indeed we have proved in [12] the following facts concerning F (X). There is a unique identity element and each element has a unique inverse.…”

“…This was hinted at in the papers [10], [11], [4], [5]. In [12] we have explored further this fact; in particular we have proved that for words u and v the cyclically reduced product u * v is a cyclic permutation of v * u and the identity among relations that follows from this fact is a generalization of the identity among relations that follows from the fact that in the free group u • v is a conjugate of v • u.…”

Twisted associativity of the cyclically reduced product of words, part 2

“…Proof See Corollary 4.4 of [9]. Proposition 1.9 Let u and w be words and let us set f := u −1 * w and g := w * u −1 .…”

“…This section deals with identities among relations. Some of the material in this section can also be found in Appendix A of [9] and Appendix A of [10], but we have included here in order to make the paper self-contained.…”

“…In [9] we have started the study of properties of * and of F (X) equipped with * ; in particular we have proved that * verifies generalizations of properties of the reduced product. In [10] we have proved that * verifies a generalized version of the associativity property in a special case.…”