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, in which case we write, where the absolute value of x is defined to be.
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.
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.

Characterizations

Two elements x and y are disjoint if and only if.
If x and y are disjoint then and, where for any element z, and.

Properties

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.

Representation as a disjoint sum of positive elements

For any x in X, let and, where note that both of these elements are and with.
Then and are disjoint, and is the unique representation of x as the difference of disjoint elements that are.
For all x and y in X, and.
If y ≥ 0 and x ≤ y then x+ ≤ y.
Moreover, if and only if and.