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Lattice disjoint

In mathematics, specifically in order theory and functional analysis, two elements x and y of a vector lattice X are lattice disjoint or simply disjoint if inf { | x | , | y | } = 0 {\displaystyle \inf \left\{|x|,|y|\right\}=0} , in which case we write x ⊥ y {\displaystyle x\perp y} , where the absolute value of x is defined to be | x | := sup { x , − x } {\displaystyle |x|:=\sup \left\{x,-x\right\}} . We say that two sets A and B are lattice disjoint or disjoint if a and b are disjoint for all a in A and all b in B, in which case we write A ⊥ B {\displaystyle A\perp B} . If A is the singleton set { a } {\displaystyle \{a\}} then we will write a ⊥ B {\displaystyle a\perp B} in place of { a } ⊥ B {\displaystyle \{a\}\perp B} . For any set A, we define the disjoint complement to be the set A ⊥ := { x ∈ X : x ⊥ A } {\displaystyle A^{\perp }:=\left\{x\in X:x\perp A\right\}} .

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Characterizations

Two elements x and y are disjoint if and only if sup { | x | , | y | } = | x | + | y | {\displaystyle \sup\{|x|,|y|\}=|x|+|y|} . If x and y 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 z, z + := sup { z , 0 } {\displaystyle z^{+}:=\sup \left\{z,0\right\}} and z − := sup { − z , 0 } {\displaystyle z^{-}:=\sup \left\{-z,0\right\}} .

Properties

Disjoint complements are always bands, but the converse is not true in general. If A is a subset of X such that x = sup A {\displaystyle x=\sup A} exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from { x } {\displaystyle \{x\}} .4

Representation as a disjoint sum of positive elements

For any x in X, 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 x as the difference of disjoint elements that are ≥ 0 {\displaystyle \geq 0} .5 For all x and y in X, | 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\}} .6 If y ≥ 0 and x ≤ y then x+ ≤ y. 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}} .7

See also

Sources

References

  1. 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. https://search.worldcat.org/oclc/840278135 ↩

  2. 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. https://search.worldcat.org/oclc/840278135 ↩

  3. 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. https://search.worldcat.org/oclc/840278135 ↩

  4. 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. https://search.worldcat.org/oclc/840278135 ↩

  5. 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. https://search.worldcat.org/oclc/840278135 ↩

  6. 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. https://search.worldcat.org/oclc/840278135 ↩

  7. 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. https://search.worldcat.org/oclc/840278135 ↩