{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T10:30:34Z","timestamp":1773225034641,"version":"3.50.1"},"reference-count":0,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[1993,9]]},"abstract":"<jats:p> Nous donnons une g\u00e9n\u00e9ralisation de la d\u00e9composition de M. Hall des \u00e9l\u00e9ments du groupe libre en produits d\u00e9croissant de commutateurs de Hall. Nous g\u00e9n\u00e9ralisons les identit\u00e9s de Th\u00e9rien, qui expriment les exposants de la d\u00e9composition comme des sommes \u00e0 coefficients entiers positifs de fonctions sous-mots. Nous \u00e9tudions l\u2019alg\u00e8bre des fonctions sous-mots et nous montrons que cette alg\u00e8bre est librement engendr\u00e9e par les fonctions qui donnent ces exposants; nous montrons aussi la continuit\u00e9 de ces fonctions pour la topologie de Hall sur le groupe libre. De plus, nous donnons de nouvelles preuves de r\u00e9sultats connus, entre autres les th\u00e9or\u00e8mes de Magnus et Witt qui caract\u00e9risent les \u00e9l\u00e9ments de la s\u00e9rie centrale descendante du grouple libre. <\/jats:p><jats:p> <\/jats:p><jats:p> We give the generalization of M. Hall\u2019s expansion of each element of the free group as a decreasing product of Hall commutators. We also prove the generalization of Therien\u2019s identities expressing the Hall exponents as nonnegative linear combinations of subword functions. We study the algebra of subword functions and show that it is freely generated by the Hall exponents functions; we also prove the continuity of these functions for the Hall topology on the free group. Besides these results, we give new proofs of known results, especially of the theorem of Magnus and Witt on the lower central series of the free group. <\/jats:p>","DOI":"10.1142\/s0218196793000196","type":"journal-article","created":{"date-parts":[[2004,11,22]],"date-time":"2004-11-22T22:29:30Z","timestamp":1101162570000},"page":"275-294","source":"Crossref","is-referenced-by-count":2,"title":["COMPUTING HALL EXPONENTS IN THE FREE GROUP"],"prefix":"10.1142","volume":"03","author":[{"given":"GUY","family":"MELAN\u00c7ON","sequence":"first","affiliation":[{"name":"LaBRI, Universit\u00e9 de Bordeaux I, 351, Cours de la Lib\u00e9ration, 33405 Talence Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CHRISTOPHE","family":"REUTENAUER","sequence":"additional","affiliation":[{"name":"LaCIM, D\u00e9partement de math. et info., Universit\u00e9 du Qu\u00e9bec a Montr\u00e9al, C.P. 8888, Succ. A, Montr\u00e9al (Qu\u00e9bec), Canada H3C 3P8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196793000196","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T09:26:16Z","timestamp":1565169976000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196793000196"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,9]]},"references-count":0,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[1993,9]]}},"alternative-id":["10.1142\/S0218196793000196"],"URL":"https:\/\/doi.org\/10.1142\/s0218196793000196","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,9]]}}}