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
{
|
,
|
}
, in which case we write
x ⊥ y
, where the absolute value of x is defined to be
|
}
{\displaystyle |x|:=\sup \left\{x,-x\right\}}
[1] 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
⊥
{\displaystyle A\perp B}
[2] If A is the singleton set
then we will write
in place of
For any set A, we define the disjoint complement to be the set
[2] Two elements x and y are disjoint if and only if
If x and y are disjoint then
, where for any element z,
Disjoint complements are always bands, but the converse is not true in general.
If A is a subset of X such that
exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from
, where note that both of these elements are
are disjoint, and
is the unique representation of x as the difference of disjoint elements that are
x + y = sup { x , y } + inf { x , y }