{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T13:03:52Z","timestamp":1648991032795},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[1997,8,1]],"date-time":"1997-08-01T00:00:00Z","timestamp":870393600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[1997,8]]},"abstract":"<jats:p>For a finite set <jats:italic>D<\/jats:italic> of nodes let \n<jats:italic>E<\/jats:italic><jats:sub>2<\/jats:sub>(<jats:italic>D<\/jats:italic>)={(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>)[mid     ]\n<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>\u2208<jats:italic>D<\/jats:italic>, <jats:italic>x<\/jats:italic>\u2260<jats:italic>y<\/jats:italic>}. \nWe define an inversive \n\u03942-structure <jats:italic>g<\/jats:italic> as a function \n<jats:italic>g<\/jats:italic>[ratio   ]<jats:italic>E<\/jats:italic><jats:sub>2<\/jats:sub>(<jats:italic>D<\/jats:italic>)\u2192\u0394 \ninto a given group \u0394 satisfying the property \n<jats:italic>g<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>)=\n<jats:italic>g<\/jats:italic>(<jats:italic>y<\/jats:italic>, <jats:italic>x<\/jats:italic>)<jats:sup>\u22121<\/jats:sup> for all \n(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>)\u2208<jats:italic>E<\/jats:italic><jats:sub>2<\/jats:sub>(<jats:italic>D<\/jats:italic>). \nFor each function (selector) \n\u03c3[ratio   ]<jats:italic>D<\/jats:italic>\u2192\u0394 there is a corresponding inversive \n\u03942-structure <jats:italic>g<\/jats:italic><jats:sup>\u03c3<\/jats:sup> defined by \n<jats:italic>g<\/jats:italic><jats:sup>\u03c3<\/jats:sup>(<jats:italic>x<\/jats:italic>, \n<jats:italic>y<\/jats:italic>)=\u03c3(<jats:italic>x<\/jats:italic>)\u00b7<jats:italic>g<\/jats:italic>\n(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>)\u00b7\u03c3(<jats:italic>y<\/jats:italic>)<jats:sup>\u22121<\/jats:sup>.\n A function \n\u03b7 mapping each <jats:italic>g<\/jats:italic> into the group \u0394 is called an invariant \nif \u03b7(<jats:italic>g<\/jats:italic><jats:sup>\u03c3<\/jats:sup>)=\u03b7(<jats:italic>g<\/jats:italic>) for all <jats:italic>g<\/jats:italic> \nand \u03c3. We study the group of free invariants \u03b7 of inversive \n\u03942-structures, where \u03b7 is defined by a word from the free monoid \nwith involution generated by the set <jats:italic>E<\/jats:italic><jats:sub>2<\/jats:sub>(<jats:italic>D<\/jats:italic>).\n In \nparticular, if \u0394 is abelian, the group of free invariants is generated \nby triangle words of the form \n(<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>)(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \n<jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub>)(<jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>).<\/jats:p>","DOI":"10.1017\/s0960129597002260","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T13:36:03Z","timestamp":1027776963000},"page":"303-327","source":"Crossref","is-referenced-by-count":1,"title":["Invariants of inversive 2-structures on groups of labels"],"prefix":"10.1017","volume":"7","author":[{"given":"A.","family":"EHRENFEUCHT","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"HARJU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G.","family":"ROZENBERG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[1997,8,1]]},"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129597002260","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T20:28:23Z","timestamp":1557606503000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129597002260\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,8]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1997,8]]}},"alternative-id":["S0960129597002260"],"URL":"https:\/\/doi.org\/10.1017\/s0960129597002260","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,8]]}}}