{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T23:10:17Z","timestamp":1712358617908},"reference-count":27,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01n02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2011,2]]},"abstract":"<jats:p>The Thompson\u2013Higman groups G<jats:sub>k,i<\/jats:sub>have a natural generalization to monoids, called M<jats:sub>k,i<\/jats:sub>, and inverse monoids, called Inv<jats:sub>k,i<\/jats:sub>. We study some structural features of M<jats:sub>k,i<\/jats:sub>and Inv<jats:sub>k,i<\/jats:sub>and investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity.<\/jats:p><jats:p>The maximal subgroups of M<jats:sub>k,1<\/jats:sub>and Inv<jats:sub>k,1<\/jats:sub>are isomorphic to the groups G<jats:sub>k,j<\/jats:sub>(1 \u2264 j \u2264 k - 1); so we rediscover all the Thompson\u2013Higman groups within M<jats:sub>k,1<\/jats:sub>.<\/jats:p><jats:p>Deciding the Green relations [Formula: see text] and [Formula: see text] of M<jats:sub>k,1<\/jats:sub>, when the inputs are words over a finite generating set of M<jats:sub>k,1<\/jats:sub>, is in P.<\/jats:p><jats:p>When a circuit-like generating set is used for M<jats:sub>k,1<\/jats:sub>then deciding [Formula: see text] is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for [Formula: see text] is xNPsearch-complete, whereas the multiplier search problems of [Formula: see text] and [Formula: see text] are not in xNPsearch unless NP = coNP. The class of search problems xNPsearch is introduced as a slight generalization of NPsearch.<\/jats:p><jats:p>Deciding [Formula: see text] for M<jats:sub>k,1<\/jats:sub>when the inputs are words over a circuit-like generating set, is \u2295<jats:sub>k-1<\/jats:sub>\u2022 NP -complete; for any h \u2265 2, \u2295<jats:sub>h<\/jats:sub>\u2022NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # \u2022 NP-complete), and the image size modulo h problem (which is \u2295<jats:sub>h<\/jats:sub>\u2022NP-complete). For Inv<jats:sub>k,1<\/jats:sub>over a circuit-like generating set, deciding [Formula: see text] is \u2295<jats:sub>k-1<\/jats:sub>P-complete. It is interesting that the little known complexity classes coDP and \u2295<jats:sub>k-1<\/jats:sub>\u2022NP play a central role in M<jats:sub>k,1<\/jats:sub>.<\/jats:p>","DOI":"10.1142\/s0218196711006066","type":"journal-article","created":{"date-parts":[[2011,4,6]],"date-time":"2011-04-06T12:14:25Z","timestamp":1302092065000},"page":"1-34","source":"Crossref","is-referenced-by-count":6,"title":["THE THOMPSON\u2013HIGMAN MONOIDS M<sub>k,i<\/sub>: THE ${\\mathcal J}$-ORDER, THE ${\\mathcal D}$-RELATION, AND THEIR COMPLEXITY"],"prefix":"10.1142","volume":"21","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":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"2","volume":"103","author":"Beigel R.","journal-title":"Theoretical Computer Science"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196710005741"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2008.06.012"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.05.035"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196708004457"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196706002822"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196704001980"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02090768"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539790178069"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/007.1"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032916"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.03.012"},{"key":"rf15","volume-title":"Computers and Intractability, a Guide to the Theory of NP-Completeness","author":"Garey M.","year":"1979"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90165-9"},{"key":"rf17","volume-title":"Semigroups, An Introduction to the Structure Theory","author":"Grillet P. 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