{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T02:53:10Z","timestamp":1687229590792},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2010,6]]},"abstract":"<jats:p> We study the monoid generalization M<jats:sub>k, 1<\/jats:sub> of the Thompson\u2013Higman groups, and we characterize the [Formula: see text]- and the [Formula: see text]-order of M<jats:sub>k, 1<\/jats:sub>. Although M<jats:sub>k, 1<\/jats:sub> has only one nonzero [Formula: see text]-class and k-1 nonzero [Formula: see text]-classes, the [Formula: see text]- and the [Formula: see text]-order are complicated; in particular, [Formula: see text] is dense (even within an [Formula: see text]-class), and [Formula: see text] is dense (even within an [Formula: see text]-class). <\/jats:p><jats:p> We study the computational complexity of the [Formula: see text]- and the [Formula: see text]-order. When inputs are given by words over a finite generating set of M<jats:sub>k, 1<\/jats:sub>, the [Formula: see text]- and the [Formula: see text]-order decision problems are in P. However, over a \"circuit-like\" generating set the [Formula: see text]-order decision problem of M<jats:sub>k, 1<\/jats:sub> is [Formula: see text]-complete, whereas the [Formula: see text]-order decision problem is coNP-complete. Similarly, for acyclic circuits the surjectiveness problem is [Formula: see text]-complete, whereas the injectiveness problem is coNP-complete. <\/jats:p>","DOI":"10.1142\/s0218196710005741","type":"journal-article","created":{"date-parts":[[2010,7,9]],"date-time":"2010-07-09T11:00:35Z","timestamp":1278673235000},"page":"489-524","source":"Crossref","is-referenced-by-count":8,"title":["THE ${\\mathcal R}$- AND ${\\mathcal L}$-ORDERS OF THE THOMPSON\u2013HIGMAN MONOID M<sub>k, 1<\/sub> AND THEIR COMPLEXITY"],"prefix":"10.1142","volume":"20","author":[{"given":"JEAN-CAMILLE","family":"BIRGET","sequence":"first","affiliation":[{"name":"Department of Computer Science, Rutgers University at Camden, Camden, NJ 08102, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,4,30]]},"reference":[{"key":"rf2","author":"Birget J. C.","journal-title":"Int. J. Algebra Computation"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2008.06.012"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.05.035"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196708004457"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196706002822"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196704001980"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610799007905"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698834"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388519"},{"key":"rf11","first-page":"215","volume":"42","author":"Cannon J. W.","journal-title":"L'Enseignement Math\u00e9matique"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/007.1"},{"key":"rf13","volume-title":"Semigroups, An Introduction to the Structure Theory","author":"Grillet P. A.","year":"1995"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032916"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04880-1"},{"key":"rf16","volume-title":"Notes on Pure Mathematics 8","author":"Higman G.","year":"1974"},{"key":"rf17","unstructured":"R.\u00a0McKenzie and R. J.\u00a0Thompson, Word Problems, eds. W. W.\u00a0Boone, F. B.\u00a0Cannonito and R. C.\u00a0Lyndon (North-Holland, 1973)\u00a0pp. 457\u2013478."},{"key":"rf18","volume-title":"Computational Complexity","author":"Papadimitriou C. H.","year":"1994"},{"key":"rf19","volume-title":"Infinite Words","author":"Perrin D.","year":"2004"},{"key":"rf20","volume-title":"Models of Computation","author":"Savage J. E.","year":"1998"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(84)90172-8"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71348-X"},{"key":"rf24","volume-title":"Handbook of Theoretical Computer Science","author":"van Leeuwen J.","year":"1990"},{"key":"rf25","volume-title":"The Complexity of Boolean Functions","author":"Wegener I.","year":"1987"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196710005741","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T03:25:48Z","timestamp":1565148348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196710005741"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6]]},"references-count":23,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2012,4,30]]},"published-print":{"date-parts":[[2010,6]]}},"alternative-id":["10.1142\/S0218196710005741"],"URL":"https:\/\/doi.org\/10.1142\/s0218196710005741","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,6]]}}}