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This yields a characterization of generalized fences:\n                    <jats:italic>F<\/jats:italic>\n                    is a generalized fence if and only if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\dim (P\\times F) \\leqslant \\dim (P) +1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>\u2a7d<\/mml:mo>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for every poset\n                    <jats:italic>P<\/jats:italic>\n                    . Whether\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\dim (P \\times Q) \\geqslant \\dim (P) + \\dim (Q) -2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>\u2a7e<\/mml:mo>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for all posets\n                    <jats:italic>P<\/jats:italic>\n                    and\n                    <jats:italic>Q<\/jats:italic>\n                    is a long-standing question in dimension theory. Having understood products where one factor is a fence we further investigate the dimension of products where the factors are crowns and other 3-dimensional posets. We reconsider the\n                    <jats:italic>covering property<\/jats:italic>\n                    defined by Reuter for Ferrers relations and show that in many cases the gap between\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\dim (P \\times Q)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\dim (P) + \\dim (Q)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>dim<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    can be explained in terms of the covering properties of one of the factors.\n                  <\/jats:p>","DOI":"10.1007\/s11083-025-09710-3","type":"journal-article","created":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T07:32:12Z","timestamp":1760772732000},"page":"811-827","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Order Dimension, Grids, and Products"],"prefix":"10.1007","volume":"42","author":[{"given":"Stefan","family":"Felsner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Torsten","family":"M\u00fctze","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maximilian","family":"Wittmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,18]]},"reference":[{"key":"9710_CR1","unstructured":"Baker, KA.: Dimension, join-independence, and breadth in partially ordered sets. 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