{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:45:29Z","timestamp":1740109529981,"version":"3.37.3"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005855","name":"Universidade Nova de Lisboa","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005855","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Semigroup Forum"],"published-print":{"date-parts":[[2024,8]]},"abstract":"Abstract<\/jats:title>Cameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup $$\\mathcal {T}(X)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on a finite set X<\/jats:italic> and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X<\/jats:italic> is finite, the maximum order of a commutative nilpotent subsemigroup of $$\\mathcal {T}(X)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is equal to the maximum order of a null subsemigroup of $$\\mathcal {T}(X)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and we prove that the largest commutative nilpotent subsemigroups of $$\\mathcal {T}(X)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are the null semigroups previously characterized by Cameron et al.<\/jats:p>","DOI":"10.1007\/s00233-024-10444-8","type":"journal-article","created":{"date-parts":[[2024,7,2]],"date-time":"2024-07-02T05:25:09Z","timestamp":1719897909000},"page":"60-75","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Commutative nilpotent transformation semigroups"],"prefix":"10.1007","volume":"109","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0706-1354","authenticated-orcid":false,"given":"Alan J.","family":"Cain","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1186-6216","authenticated-orcid":false,"given":"Ant\u00f3nio","family":"Malheiro","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0001-8404-8305","authenticated-orcid":false,"given":"T\u00e2nia","family":"Paulista","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,1]]},"reference":[{"key":"10444_CR1","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s11856-015-1173-9","volume":"207","author":"J Ara\u00fajo","year":"2015","unstructured":"Ara\u00fajo, J., Bentz, W., Konieczny, J.: The commuting graph of the symmetric inverse semigroup. 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