{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T23:40:58Z","timestamp":1716853258846},"reference-count":45,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2014,8]]},"abstract":"<jats:p>A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.<\/jats:p>","DOI":"10.1142\/s021819671450026x","type":"journal-article","created":{"date-parts":[[2014,6,23]],"date-time":"2014-06-23T02:13:38Z","timestamp":1403489618000},"page":"609-653","source":"Crossref","is-referenced-by-count":1,"title":["Markov semigroups, monoids and groups"],"prefix":"10.1142","volume":"24","author":[{"given":"Alan J.","family":"Cain","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica, Universidade do Porto, Rua do Campo Alegre 687, 4169\u2013007 Porto, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Victor","family":"Maltcev","sequence":"additional","affiliation":[{"name":"Mathematics Department, Technion \u2013 Israel Institute of Technology, Haifa 32000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,9,3]]},"reference":[{"key":"rf1","unstructured":"J. M.\u00a0Alonso, Group Theory from a Geometrical Viewpoint, ed. H.\u00a0Short (World Scientific Publishing, River Edge, NJ, 1991)\u00a0pp. 3\u201363."},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9771-7"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2008.10.006"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-012-9406-2"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196712500610"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-013-9531-6"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2009.02.005"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700001816"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(99)00151-6"},{"key":"rf11","series-title":"Mathematical Surveys","volume-title":"The Algebraic Theory of Semigroups","volume":"2","author":"Clifford A. H.","year":"1967"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700002354"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90037-X"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004103007497"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004199003722"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.03.026"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1201\/9781439865699"},{"key":"rf19","doi-asserted-by":"crossref","unstructured":"\u00c9.\u00a0Ghys and P.\u00a0de la Harpe, Sur les groupes hyperboliques d'apr\u00e9s Mikhael Gromov (Birkh\u00e4usa, 1996)\u00a0pp. 165\u2013187.","DOI":"10.1007\/978-1-4684-9167-8_9"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.07.008"},{"key":"rf21","first-page":"428","volume":"43","author":"Grigorchuk R. I.","year":"1988","journal-title":"Mat. Zametki"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02471762"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02573555"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9586-7_3"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(95)00165-4"},{"key":"rf26","doi-asserted-by":"publisher","DOI":"10.1142\/9789812776884_0016"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196702000833"},{"key":"rf28","volume-title":"Introduction to Automata Theory, Languages, and Computation","author":"Hopcroft J. E.","year":"1979"},{"key":"rf29","series-title":"London Mathematical Society Monographs (New Series)","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198511946.001.0001","volume-title":"Fundamentals of Semigroup Theory","volume":"12","author":"Howie J. M.","year":"1995"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2005.11.028"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(74)90146-X"},{"key":"rf32","first-page":"107","volume":"4","author":"Malcev A. I.","year":"1953","journal-title":"Ivanov. Gos. Ped. Inst. U\u010den. Zap. Fiz.-Mat. Nauki"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1142\/S021819670900497X"},{"key":"rf35","first-page":"19","volume":"77","author":"Markov A.","year":"1951","journal-title":"Dok. Akad. Nauk SSSR (N.S.)"},{"key":"rf36","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/117"},{"key":"rf37","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700034171"},{"key":"rf39","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1998.0230"},{"key":"rf40","series-title":"Hamburger Mathematische Einzelschriften","volume-title":"Theorie der Endlich Erzeugbaren Kommutativen Halbgruppen","volume":"41","author":"R\u00e9dei L.","year":"1963"},{"key":"rf41","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-98-02074-1"},{"key":"rf42","volume-title":"Finitely Generated Commutative Monoids","author":"Rosales J. C.","year":"1999"},{"key":"rf43","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1974-0342955-9"},{"key":"rf44","doi-asserted-by":"publisher","DOI":"10.1112\/S0024611598000124"},{"key":"rf45","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1997.7392"},{"key":"rf46","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196705002761"},{"key":"rf47","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511574702"},{"key":"rf48","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(94)90175-9"},{"key":"rf49","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050090"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021819671450026X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T23:12:41Z","timestamp":1716851561000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021819671450026X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8]]},"references-count":45,"journal-issue":{"issue":"05","published-online":{"date-parts":[[2014,9,3]]},"published-print":{"date-parts":[[2014,8]]}},"alternative-id":["10.1142\/S021819671450026X"],"URL":"https:\/\/doi.org\/10.1142\/s021819671450026x","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,8]]}}}