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\}} .
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\}} .
… excerpt ends here. Continue reading the full article.
