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.
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.
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 proved that * verifies generalizations of properties of the product in the free group. In another previous paper 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.
No abstract
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.