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Lattice disjoint
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<h2 id="characterizations">Characterizations</h2> <p>Two elements <i>x</i> and <i>y</i> are disjoint if and only if sup { | x | , | y | } = | x | + | y | {\displaystyle \sup\{|x|,|y|\}=|x|+|y|} . If <i>x</i> and <i>y</i> are disjoint then | x + y | = | x | + | y | {\displaystyle |x+y|=|x|+|y|} and ( x + y ) + = x + + y + {\displaystyle \left(x+y\right)^{+}=x^{+}+y^{+}} , where for any element <i>z</i>, z + := sup { z , 0 } {\displaystyle z^{+}:=\sup \left\{z,0\right\}} and z − := sup { − z , 0 } {\displaystyle z^{-}:=\sup \left\{-z,0\right\}} . </p> <h2 id="properties">Properties</h2> <p>Disjoint complements are always <a href="/facts/Band_(order_theory)/gfUmQjed">bands</a>, but the converse is not true in general. If <i>A</i> is a subset of <i>X</i> such that x = sup A {\displaystyle x=\sup A} exists, and if <i>B</i> is a subset lattice in <i>X</i> that is disjoint from <i>A</i>, then <i>B</i> is a lattice disjoint from { x } {\displaystyle \{x\}} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h3>Representation as a disjoint sum of positive elements</h3> <p>For any <i>x</i> in <i>X</i>, let x + := sup { x , 0 } {\displaystyle x^{+}:=\sup \left\{x,0\right\}} and x − := sup { − x , 0 } {\displaystyle x^{-}:=\sup \left\{-x,0\right\}} , where note that both of these elements are ≥ 0 {\displaystyle \geq 0} and x = x + − x − {\displaystyle x=x^{+}-x^{-}} with | x | = x + + x − {\displaystyle |x|=x^{+}+x^{-}} . Then x + {\displaystyle x^{+}} and x − {\displaystyle x^{-}} are disjoint, and x = x + − x − {\displaystyle x=x^{+}-x^{-}} is the unique representation of <i>x</i> as the difference of disjoint elements that are ≥ 0 {\displaystyle \geq 0} .<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> For all <i>x</i> and <i>y</i> in <i>X</i>, | x + − y + | ≤ | x − y | {\displaystyle \left|x^{+}-y^{+}\right|\leq |x-y|} and x + y = sup { x , y } + inf { x , y } {\displaystyle x+y=\sup\{x,y\}+\inf\{x,y\}} .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> If <i>y ≥ 0</i> and <i>x</i> ≤ <i>y</i> then <i>x</i>+ ≤ <i>y</i>. Moreover, x ≤ y {\displaystyle x\leq y} if and only if x + ≤ y + {\displaystyle x^{+}\leq y^{+}} and x − ≤ x − 1 {\displaystyle x^{-}\leq x^{-1}} .<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Solid_set/0TIYIsVI">Solid set</a></li> <li><a href="/facts/Locally_convex_vector_lattice/T6c39Lac">Locally convex vector lattice</a></li> <li><a href="/facts/Vector_lattice/FX7Mwfxe">Vector lattice</a></li></ul> <h2 id="sources">Sources</h2> <ul><li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 3. New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Schaefer & Wolff 1999, pp. 204–214. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>