{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T11:18:47Z","timestamp":1758280727012},"reference-count":12,"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":[[2001,6]]},"abstract":"<jats:p> Let h<jats:sub>1<\/jats:sub>, h<jats:sub>2<\/jats:sub>,\u2026 be a sequence of elements in a free group and let H be the subgroup they generate. Let H\u2032 be the subgroup generated by w<jats:sub>1<\/jats:sub>, w<jats:sub>2<\/jats:sub>, \u2026, where each w<jats:sub>i<\/jats:sub> is a word in h<jats:sub>i<\/jats:sub> and possibly other h<jats:sub>j<\/jats:sub>, such that the associated directed graph has the finite paths property. We show that rank H\u2032\u2265 rank H. As a corollary we get that [Formula: see text], where [Formula: see text] is the subgroup generated by the roots of the elements in H. If H<jats:sub>0<\/jats:sub> is finitely generated and the sequence of subgroups H<jats:sub>0<\/jats:sub>, H<jats:sub>1<\/jats:sub>, H<jats:sub>2<\/jats:sub>, \u2026 satisfies [Formula: see text] then the sequence stabilizes, i.e. for some m, H<jats:sub>i<\/jats:sub>=H<jats:sub>i+1<\/jats:sub> for every i\u2265 m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in \"Lyndon equation\" [Formula: see text] the exponents a<jats:sub>i<\/jats:sub> satisfy gcd (a<jats:sub>1<\/jats:sub>,\u2026,a<jats:sub>n<\/jats:sub>)\u22601 then the inner rank is \u230a n\/2\u230b. The proofs are mostly elementary. <\/jats:p>","DOI":"10.1142\/s0218196701000565","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T10:58:59Z","timestamp":1027767539000},"page":"375-390","source":"Crossref","is-referenced-by-count":3,"title":["ON RANK, ROOT AND EQUATIONS IN FREE GROUPS"],"prefix":"10.1142","volume":"11","author":[{"given":"AMNON","family":"ROSENMANN","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"p_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02546390"},{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-35.1.30"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(65)90008-6"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1997.7183"},{"key":"p_6","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028998143"},{"key":"p_7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160260520"},{"key":"p_9","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028998766"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(94)00120-8"},{"key":"p_14","doi-asserted-by":"publisher","DOI":"10.2307\/1970329"},{"key":"p_15","first-page":"2435","volume":"248","author":"Sch\u00fctzenberger M. P.","year":"1959","journal-title":"C. R. Acad. Sci. Paris"},{"key":"p_16","first-page":"532","volume":"6","author":"Stallings J.","year":"1959","journal-title":"Notices Amer. Math. Soc."},{"key":"p_17","first-page":"87","volume":"18","author":"Steinberg A.","year":"1959","journal-title":"Michigan Math. J."}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196701000565","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:55:21Z","timestamp":1565128521000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196701000565"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6]]},"references-count":12,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2001,6]]}},"alternative-id":["10.1142\/S0218196701000565"],"URL":"https:\/\/doi.org\/10.1142\/s0218196701000565","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,6]]}}}