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Lewis and Souza isibility order on the interval of integers <jats:inline-formula><jats:alternatives><jats:tex-math>$$[N\/\\kappa , N]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>[<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mi>\u03ba<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>]<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is bounded above by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa (\\log \\kappa )^{1+o(1)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03ba<\/mml:mi>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mi>o<\/mml:mi>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and below by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega ((\\log \\kappa \/\\log \\log \\kappa )^2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We improve the upper bound to <jats:inline-formula><jats:alternatives><jats:tex-math>$$O((\\log \\kappa )^3\/(\\log \\log \\kappa )^2).$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>.<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.<\/jats:p>","DOI":"10.1007\/s11083-023-09653-7","type":"journal-article","created":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T05:01:35Z","timestamp":1700629295000},"page":"693-707","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Dimension of Divisibility Orders and Multiset Posets"],"prefix":"10.1007","volume":"41","author":[{"given":"Milan","family":"Haiman","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,22]]},"reference":[{"key":"9653_CR1","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/BF01108597","volume":"11","author":"GR Brightwell","year":"1994","unstructured":"Brightwell, G.R., Kierstead, H.A., Kostochka, A.V., Trotter, W.T.: The dimension of suborders of the Boolean lattice. 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