{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T21:58:49Z","timestamp":1776290329469,"version":"3.50.1"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"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":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We give a presentation for the monoid <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr{I}\\mathscr{O}_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of all order-preserving transformations of an <jats:italic>n<\/jats:italic>-chain whose ranges are intervals. We also consider the submonoid <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr{I}\\mathscr{O}_n^-$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:msubsup>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                    <\/mml:msubsup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr{I}\\mathscr{O}_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.<\/jats:p>","DOI":"10.1007\/s00233-024-10466-2","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T17:02:16Z","timestamp":1724086936000},"page":"336-346","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the monoid of order-preserving transformations of a finite chain whose ranges are intervals"],"prefix":"10.1007","volume":"109","author":[{"given":"V\u00edtor H.","family":"Fernandes","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"key":"10466_CR1","unstructured":"A\u012dzen\u0161tat, A.Y.: The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sibirsk. Mat. Zh. 3, 161\u2013169 (1962) (in Russian)"},{"key":"10466_CR2","unstructured":"A\u012dzen\u0161tat, A.Y.: Homomorphisms of semigroups of endomorphisms of ordered sets. Uch. Zap. Leningr. Gos. Pedagog. Inst. 238, 38\u201348 (1962) (in Russian)"},{"key":"10466_CR3","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1007\/BF02676604","volume":"54","author":"VH Fernandes","year":"1997","unstructured":"Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: a new class of divisors. Semigroup Forum 54, 230\u2013236 (1997)","journal-title":"Semigroup Forum"},{"key":"10466_CR4","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s00233-010-9220-7","volume":"81","author":"VH Fernandes","year":"2010","unstructured":"Fernandes, V.H., Jesus, M.M., Maltcev, V., Mitchell, J.D.: Endomorphisms of the semigroup of order-preserving mappings. Semigroup Forum 81, 277\u2013285 (2010)","journal-title":"Semigroup Forum"},{"key":"10466_CR5","doi-asserted-by":"crossref","unstructured":"Fernandes, V.H., Paulista, T.: On the rank of monoids of endomorphisms of a finite directed path. Asian-Eur J. Math. 16, 2350069 (2023)","DOI":"10.1142\/S1793557123500699"},{"key":"10466_CR6","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1007\/s00233-010-9257-7","volume":"81","author":"VH Fernandes","year":"2010","unstructured":"Fernandes, V.H., Volkov, M.V.: On divisors of semigroups of order-preserving mappings of a finite chain. Semigroup Forum 81, 551\u2013554 (2010)","journal-title":"Semigroup Forum"},{"key":"10466_CR7","unstructured":"GAP \u2013 Groups, Algorithms, and Programming, Version 4.13.0; https:\/\/www.gap-system.org (2024)"},{"key":"10466_CR8","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1007\/BF03025769","volume":"45","author":"GMS Gomes","year":"1992","unstructured":"Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45, 272\u2013282 (1992)","journal-title":"Semigroup Forum"},{"key":"10466_CR9","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1142\/S0218196795000306","volume":"5","author":"PM Higgins","year":"1995","unstructured":"Higgins, P.M.: Divisors of semigroups of order-preserving mappings on a finite chain. Int. J. Algebra Comput. 5, 725\u2013742 (1995)","journal-title":"Int. J. Algebra Comput."},{"key":"10466_CR10","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1017\/S0013091500026936","volume":"17","author":"JM Howie","year":"1971","unstructured":"Howie, J.M.: Product of idempotents in certain semigroups of transformations. Proc. Edinb. Math. Soc. 17, 223\u2013236 (1971)","journal-title":"Proc. Edinb. Math. Soc."},{"key":"10466_CR11","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198511946.001.0001","volume-title":"Fundamentals of Semigroup Theory","author":"JM Howie","year":"1995","unstructured":"Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)"},{"key":"10466_CR12","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s00233-005-0553-6","volume":"72","author":"A Laradji","year":"2006","unstructured":"Laradji, A., Umar, A.: Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72, 51\u201362 (2006)","journal-title":"Semigroup Forum"},{"key":"10466_CR13","unstructured":"Ru\u0161kuc, N.: Semigroup Presentations. Ph.D. Thesis, University of St Andrews (1995)"},{"key":"10466_CR14","unstructured":"Vernitski\u012d, A.S., Volkov, M.V.: A proof and a generalisation of Higgins\u2019 division theorem for semigroups of order-preserving mappings. Izv. Vyssh. Uchebn. Zaved. Mat. 1, 38\u201344 (1995) (in Russian)"}],"container-title":["Semigroup Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00233-024-10466-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00233-024-10466-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00233-024-10466-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T21:02:22Z","timestamp":1727125342000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00233-024-10466-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,19]]},"references-count":14,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["10466"],"URL":"https:\/\/doi.org\/10.1007\/s00233-024-10466-2","relation":{},"ISSN":["0037-1912","1432-2137"],"issn-type":[{"value":"0037-1912","type":"print"},{"value":"1432-2137","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,19]]},"assertion":[{"value":"6 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}