{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:15:33Z","timestamp":1760235333923,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,11]],"date-time":"2021-08-11T00:00:00Z","timestamp":1628640000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005203","name":"OeAD-GmbH","doi-asserted-by":"publisher","award":["KONTAKT CZ 02\/2019"],"award-info":[{"award-number":["KONTAKT CZ 02\/2019"]}],"id":[{"id":"10.13039\/501100005203","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal\u2019cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.<\/jats:p>","DOI":"10.3390\/sym13081471","type":"journal-article","created":{"date-parts":[[2021,8,11]],"date-time":"2021-08-11T08:35:52Z","timestamp":1628670952000},"page":"1471","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Centralising Monoids with Low-Arity Witnesses on a Four-Element Set"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0050-8085","authenticated-orcid":false,"given":"Mike","family":"Behrisch","sequence":"first","affiliation":[{"name":"Institute of Discrete Mathematics and Geometry, Technische Universit\u00e4t Wien, Wiedner Hauptstr. 8\u201310, 1040 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9677-9087","authenticated-orcid":false,"given":"Edith","family":"Vargas-Garc\u00eda","sequence":"additional","affiliation":[{"name":"Departamento Acad\u00e9mico de Matem\u00e1ticas, ITAM, R\u00edo Hondo No. 1, Col. Tizap\u00e1n San Angel, Del. \u00c1lvaro Obreg\u00f3n, Ciudad de M\u00e9xico C.P. 01080, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,11]]},"reference":[{"key":"ref_1","unstructured":"Cohn, P.M. (1965). Universal Algebra, Harper & Row Publishers."},{"key":"ref_2","unstructured":"Kuznecov, A.V. (1979). \u041e \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430\u0445 \u0434\u043b\u044f o\u0431\u043d\u0430\u0440\u0443\u0436\u0435\u043d\u0438\u044f \u043d\u0435\u0432\u044b\u0432o\u0434\u0438\u043co\u0441\u0442\u0438 \u0438\u043b\u0438 \u043d\u0435\u0432\u044b\u0440\u0430\u0437\u0438\u043co\u0441\u0442\u0438 [Means for detection of nondeducibility and inexpressibility]. \u041bo\u0433\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\u044b\u0432o\u0434 [Logical Inference] (Moscow, 1974), Nauka. 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