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

Abstract

The cyclically reduced product of two words u, v, denoted u * 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 * and of the structure of the set of cyclically reduced words F (X) equipped with * . In particular we have found that a certain number of properties of the free group equipped with the reduced product can be generalized to ( F (X), * ).In this paper we continue this study by proving that a generalized version of the associative property holds for * in a special case. In a following paper we will prove that a more general version of the associative property holds for any case.

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

“…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. In the present paper we prove that a more general version of the associativity property holds for * in the general case.…”

“…Indeed there exist words v 1 , v 2 such that ρ(u) = v 1 v 2 and v = ρ(v 2 v 1 ). By (10) of Proposition 1.1 there exist words u 1 , u 2 such that u = u 1 u 2 and ρ(u…”

“…In order to prove that Lemma 3.1 implies Theorem 2.4 we make use of the main theorem of [10]. The latter states that a generalized version of the associative property holds for the triple of words u, u −1 , w. In the proof of Theorem 2.4 we use that result for the triple of words p, p −1 , γ(d * w)γ −1 where p is a cyclic permutation of u or of v and γ is some word.…”

“…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 [10] we have proved that if we take v = u −1 then the cyclically reduced product verifies a "twisted" version of the associative property, i.e., up to cyclic permutations of some of the terms. However for a general v this property does not hold true, but a more general one holds.…”