{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T12:42:08Z","timestamp":1772196128490,"version":"3.50.1"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","license":[{"start":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T00:00:00Z","timestamp":1772150400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T00:00:00Z","timestamp":1772150400000},"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":["Period Math Hung"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper, we consider the inverse submonoids\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {AOR}_n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>AOR<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of oriented transformations and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {AOP}_n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>AOP<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of orientation-preserving transformations of the alternating inverse monoid\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal{A}\\mathcal{I}_n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    on a chain with\n                    <jats:italic>n<\/jats:italic>\n                    elements. We compute the cardinalities, describe the Green\u2019s structures and the congruences, and calculate the ranks of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {AOR}_n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>AOR<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {AOP}_n.$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>AOP<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                  <\/jats:p>","DOI":"10.1007\/s10998-026-00702-3","type":"journal-article","created":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T11:32:44Z","timestamp":1772191964000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On oriented alternating inverse monoids"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1057-4975","authenticated-orcid":false,"given":"V\u00edtor Hugo","family":"Fernandes","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,27]]},"reference":[{"issue":"2","key":"702_CR1","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/s00233-023-10396-5","volume":"107","author":"I Dimitrova","year":"2023","unstructured":"I. Dimitrova, V.H. Fernandes, J. Koppitz, T.M. Quinteiro, Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set. Semigroup Forum 107(2), 315\u2013338 (2023). https:\/\/doi.org\/10.1007\/s00233-023-10396-5","journal-title":"Semigroup Forum"},{"key":"702_CR2","volume-title":"Abstract Algebra","author":"D Dummit","year":"1999","unstructured":"D. Dummit, R. Foote, Abstract Algebra (John Wiley & Sons Inc, Hoboken, NJ, 1999)"},{"issue":"7","key":"702_CR3","doi-asserted-by":"publisher","first-page":"3401","DOI":"10.1080\/00927870008827033","volume":"28","author":"VH Fernandes","year":"2000","unstructured":"V.H. Fernandes, The monoid of all injective orientation preserving partial transformations on a finite chain. Commun. Algebra 28(7), 3401\u20133426 (2000). https:\/\/doi.org\/10.1080\/00927870008827033","journal-title":"Commun. 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