{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T05:19:48Z","timestamp":1775452788817,"version":"3.50.1"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:p> This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case. <\/jats:p>","DOI":"10.1142\/s0218196721500570","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T04:33:35Z","timestamp":1631853215000},"page":"1587-1599","source":"Crossref","is-referenced-by-count":7,"title":["Relatively free dimonoids and bar-units"],"prefix":"10.1142","volume":"31","author":[{"given":"Anatolii V.","family":"Zhuchok","sequence":"first","affiliation":[{"name":"Department of Algebra and System Analysis, Luhansk Taras Shevchenko National University, Gogol Square, 1, Starobilsk 92703, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,9,15]]},"reference":[{"key":"S0218196721500570BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196710005753"},{"key":"S0218196721500570BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2005.01.014"},{"key":"S0218196721500570BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45328-8_3"},{"issue":"4","key":"S0218196721500570BIB004","first-page":"99","volume":"17","author":"Kinyon M. K.","year":"2007","journal-title":"J. Lie Theory"},{"key":"S0218196721500570BIB005","doi-asserted-by":"publisher","DOI":"10.1080\/03081087.2012.686108"},{"key":"S0218196721500570BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45328-8_2"},{"key":"S0218196721500570BIB008","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/346\/06296"},{"key":"S0218196721500570BIB009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-03-03387-7"},{"key":"S0218196721500570BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-004-0169-2"},{"key":"S0218196721500570BIB011","doi-asserted-by":"publisher","DOI":"10.2478\/BF02476006"},{"key":"S0218196721500570BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/s11202-009-0118-0"},{"key":"S0218196721500570BIB013","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2015.1065841"},{"issue":"1","key":"S0218196721500570BIB014","first-page":"187","volume":"19","author":"Schein B. M.","year":"1964","journal-title":"Uspekhi Mat. Nauk."},{"issue":"44","key":"S0218196721500570BIB015","first-page":"168","volume":"1","author":"Schein B. M.","year":"1965","journal-title":"Izv. Vyssh. Uchebn. Zaved. Mat."},{"key":"S0218196721500570BIB016","doi-asserted-by":"publisher","DOI":"10.1515\/math-2019-0010"},{"key":"S0218196721500570BIB017","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-011-9144-7"},{"issue":"2","key":"S0218196721500570BIB018","first-page":"299","volume":"16","author":"Zhuchok A. V.","year":"2013","journal-title":"Algebra Discrete Math."},{"issue":"2","key":"S0218196721500570BIB019","first-page":"273","volume":"21","author":"Zhuchok A. V.","year":"2013","journal-title":"Quasigroups Related Syst."},{"key":"S0218196721500570BIB020","doi-asserted-by":"publisher","DOI":"10.1134\/S0037446615050055"},{"issue":"2","key":"S0218196721500570BIB021","first-page":"330","volume":"20","author":"Zhuchok Y. V.","year":"2015","journal-title":"Algebra Discrete Math."},{"key":"S0218196721500570BIB022","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2016.1222404"},{"key":"S0218196721500570BIB023","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2017.1287274"},{"key":"S0218196721500570BIB024","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-019-09995-y"},{"key":"S0218196721500570BIB025","doi-asserted-by":"publisher","DOI":"10.1142\/S1793557115500898"},{"issue":"2","key":"S0218196721500570BIB026","first-page":"304","volume":"22","author":"Zhuchok A. V.","year":"2016","journal-title":"Algebra Discrete Math."},{"key":"S0218196721500570BIB027","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2020.1802472"},{"key":"S0218196721500570BIB028","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-015-9743-z"},{"key":"S0218196721500570BIB029","doi-asserted-by":"publisher","DOI":"10.1007\/s40863-016-0038-4"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196721500570","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,29]],"date-time":"2021-12-29T06:45:25Z","timestamp":1640760325000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196721500570"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,15]]},"references-count":28,"journal-issue":{"issue":"08","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["10.1142\/S0218196721500570"],"URL":"https:\/\/doi.org\/10.1142\/s0218196721500570","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,15]]}}}