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Semimodular lattice
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<h2 id="birkhoffs-condition">Birkhoff's condition</h2> <p>A lattice is sometimes called weakly semimodular if it satisfies the following condition due to <a href="/facts/Garrett_Birkhoff/5jeI7Wmr">Garrett Birkhoff</a>: </p> Birkhoff's condition If <i>a</i> ∧ <i>b</i> <: <i>a</i> and <i>a</i> ∧ <i>b</i> <: <i>b</i>, then <i>a</i> <: <i>a</i> ∨ <i>b</i> and <i>b</i> <: <i>a</i> ∨ <i>b</i>. <p>Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains) <a href="/facts/Atom_(order_theory)/krCynTXI">relatively atomic</a> lattices. </p> <h2 id="mac-lanes-condition">Mac Lane's condition</h2> <p>The following two conditions are equivalent to each other for all lattices. They were found by <a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Saunders Mac Lane</a>, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation. </p> Mac Lane's condition 1 For any <i>a, b, c</i> such that <i>b</i> ∧ <i>c</i> < <i>a</i> < <i>c</i> < <i>b</i> ∨ <i>a</i>, there is an element <i>d</i> such that <i>b</i> ∧ <i>c</i> < <i>d</i> ≤ <i>b</i> and <i>a</i> = (<i>a</i> ∨ <i>d</i>) ∧ <i>c</i>. Mac Lane's condition 2 For any <i>a, b, c</i> such that <i>b</i> ∧ <i>c</i> < <i>a</i> < <i>c</i> < <i>b</i> ∨ <i>c</i>, there is an element <i>d</i> such that <i>b</i> ∧ <i>c</i> < <i>d</i> ≤ <i>b</i> and <i>a</i> = (<i>a</i> ∨ <i>d</i>) ∧ <i>c</i>. <p>Every lattice satisfying Mac Lane's condition is semimodular. The converse is true for lattices of finite length, and more generally for <a href="/facts/Atom_(order_theory)/krCynTXI">relatively atomic</a> lattices. Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric. </p> <h2 id="notes">Notes</h2> <ul><li>Fofanova, T. S. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Semi-modular_lattice">"Semi-modular lattice"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>. (The article is about M-symmetric lattices.)</li> <li>Stern, Manfred (1999), <a href="https://archive.org/details/semimodularlatti0000ster"><i>Semimodular Lattices</i></a>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-46105-4.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/SemimodularLattice">"Semimodular lattice"</a>. <i><a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></i>.</li> <li>OEIS <a href="https://oeis.org/A229202">sequence A229202 (Number of unlabeled semimodular lattices with <i>n</i> elements)</a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Antimatroid/hYBJb89z">Antimatroid</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices. Most authors only deal with the finite case, in which both definitions are equivalent to semimodular and atomistic. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>For instance, Fofanova (2001). <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>