{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:52:42Z","timestamp":1747173162152,"version":"3.40.5"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,5,18]],"date-time":"2021-05-18T00:00:00Z","timestamp":1621296000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Bull. Aust. Math. Soc."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:italic>V<\/jats:italic>be an infinite-dimensional vector space over a field<jats:italic>F<\/jats:italic>and let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline1.png\"\/><jats:tex-math>$I(V)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be the inverse semigroup of all injective partial linear transformations on<jats:italic>V<\/jats:italic>. Given<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline2.png\"\/><jats:tex-math>$\\alpha \\in I(V)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we denote the domain and the range of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline3.png\"\/><jats:tex-math>$\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>by<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline4.png\"\/><jats:tex-math>${\\mathop {\\textrm {dom}}}\\,\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline5.png\"\/><jats:tex-math>${\\mathop {\\textrm {im}}}\\,\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and we call the cardinals<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline6.png\"\/><jats:tex-math>$g(\\alpha )={\\mathop {\\textrm {codim}}}\\,{\\mathop {\\textrm {dom}}}\\,\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline7.png\"\/><jats:tex-math>$d(\\alpha )={\\mathop {\\textrm {codim}}}\\,{\\mathop {\\textrm {im}}}\\,\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>the \u2018gap\u2019 and the \u2018defect\u2019 of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline8.png\"\/><jats:tex-math>$\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We study the semigroup<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline9.png\"\/><jats:tex-math>$A(V)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of all injective partial linear transformations with equal gap and defect and characterise Green\u2019s relations and ideals in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972721000344_inline10.png\"\/><jats:tex-math>$A(V)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This is analogous to work by Sanwong and Sullivan [\u2018Injective transformations with equal gap and defect\u2019,<jats:italic>Bull. Aust. Math. Soc.<\/jats:italic><jats:bold>79<\/jats:bold>(2009), 327\u2013336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.<\/jats:p>","DOI":"10.1017\/s0004972721000344","type":"journal-article","created":{"date-parts":[[2021,5,18]],"date-time":"2021-05-18T13:03:10Z","timestamp":1621342990000},"page":"106-116","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["INJECTIVE LINEAR TRANSFORMATIONS WITH EQUAL GAP AND DEFECT"],"prefix":"10.1017","volume":"105","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3579-0749","authenticated-orcid":false,"given":"C.","family":"MENDES ARA\u00daJO","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0618-2223","authenticated-orcid":false,"given":"S.","family":"MENDES-GON\u00c7ALVES","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,5,18]]},"reference":[{"key":"S0004972721000344_r7","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972708001330"},{"key":"S0004972721000344_r2","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198511946.001.0001","volume-title":"Fundamentals of Semigroup Theory","author":"Howie","year":"1995"},{"key":"S0004972721000344_r1","volume-title":"The Algebraic Theory of Semigroups","volume":"1","author":"Clifford","year":"1961"},{"key":"S0004972721000344_r4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1986-0840614-0"},{"key":"S0004972721000344_r3","doi-asserted-by":"publisher","DOI":"10.1080\/00927870500442013"},{"key":"S0004972721000344_r5","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-012-9381-7"},{"key":"S0004972721000344_r6","doi-asserted-by":"publisher","DOI":"10.1142\/S1005386717000220"}],"container-title":["Bulletin of the Australian Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0004972721000344","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T21:24:31Z","timestamp":1699046671000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0004972721000344\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,18]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["S0004972721000344"],"URL":"https:\/\/doi.org\/10.1017\/s0004972721000344","relation":{},"ISSN":["0004-9727","1755-1633"],"issn-type":[{"type":"print","value":"0004-9727"},{"type":"electronic","value":"1755-1633"}],"subject":[],"published":{"date-parts":[[2021,5,18]]},"assertion":[{"value":"\u00a9 2021 Australian Mathematical Publishing Association Inc.","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}