Metric lattice

<h2 id="relation-to-other-notions">Relation to other notions</h2>

A <a href="/facts/Boolean_algebra/VkCToAmg">Boolean algebra</a> is a metric lattice; any <a href="/facts/Content_(measure_theory)/YsJgpdPt">finitely-additive measure</a> on its <a href="/facts/Stone%2527s_representation_theorem_for_Boolean_algebras/tN870P19">Stone dual</a> gives a valuation.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: 252–254 
Every metric lattice is a <a href="/facts/Modular_lattice/CLe2aDq7">modular lattice</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> c.f. lower picture. It is also a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>, with distance function given by<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 
 
 
 
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 {\displaystyle d(x,y)=v(x\vee y)-v(x\wedge y){\text{.}}}
 
 With that metric, the <a href="/facts/Join_and_meet/cYcLhRIR">join and meet</a> are <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> <a href="/facts/Contraction_mapping/VSu4qNqa">contractions</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 77  and so extend to the metric <a href="/facts/Completion_(metric_space)/lsckmU8G">completion (metric space)</a>. That lattice is usually not the <a href="/facts/Dedekind-MacNeille_completion/yCa3eCAL">Dedekind-MacNeille completion</a>, but it is <a href="/facts/Conditionally_complete_lattice/5ak6ufky">conditionally complete</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>: 80 

<h2 id="applications">Applications</h2>
In the study of <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a> and <a href="/facts/Interval_arithmetic/1ylnKraf">interval arithmetic</a>, the space of <a href="/facts/Continuous_uniform_distribution/XbnlVljT">uniform distributions</a> is a metric lattice.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Metric lattices are also key to von Neumann's construction of the <a href="/facts/Continuous_geometry/oKFUWHJ3">continuous projective geometry</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 126  A function satisfies the one-dimensional <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> if and only if it is a valuation for the lattice of spacetime coordinates with the natural partial order. A similar result should apply to any <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> solvable by the <a href="/facts/Method_of_characteristics/akP29vbW">method of characteristics</a>, but key features of the theory are lacking.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>: 150–151

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. pp. 20–22. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. pp. 20–22. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Kaburlasos, V. G. (2004). "FINs: Lattice Theoretic Tools for Improving Prediction of Sugar Production From Populations of Measurements." IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 34(2), 1017–1030. doi:10.1109/tsmcb.2003.818558 <a href="//doi.org/10.1109/tsmcb.2003.818558" target="_blank">//doi.org/10.1109/tsmcb.2003.818558</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Kaburlasos, V. G. (2004). "FINs: Lattice Theoretic Tools for Improving Prediction of Sugar Production From Populations of Measurements." IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 34(2), 1017–1030. doi:10.1109/tsmcb.2003.818558 <a href="//doi.org/10.1109/tsmcb.2003.818558" target="_blank">//doi.org/10.1109/tsmcb.2003.818558</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Metric lattice open-in-new

Metric lattice