Lattice disjoint

In mathematics, specifically in <a href="/facts/Order_theory/gn6fM3oJ">order theory</a> and <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, two elements x and y of a <a href="/facts/Vector_lattice/FX7Mwfxe">vector lattice</a> 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\}}
 
.

Lattice disjoint open-in-new

Lattice disjoint