Lattice disjoint explained

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 \left\{ |x|, |y| \right\} = 0 , in which case we write x \perp y , where the absolute value of x is defined to be |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 \perp B . If A is the singleton set \{ a \} then we will write a \perp B in place of \{ a \} \perp B . For any set A, we define the disjoint complement to be the set 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 | . If x and y are disjoint then | x + y | = | x | + | y | and \left(x + y \right) ^{+} = x ^{+} + y ^{+} , where for any element z, z ^{+} : = \sup \left\{ z, 0 \right\} and 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 exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from \{ x \} .

Representation as a disjoint sum of positive elements

For any x in X, let

x ^{+} : = \sup \left\{ x, 0 \right\} and x ^{-} : = \sup \left\{ - x, 0 \right\} , where note that both of these elements are \geq 0 and x = x ^{+} - x ^{-} with | x | = x ^{+} + x ^{-} . Then x ^{+} and x ^{-} are disjoint, and x = x ^{+} - x ^{-} is the unique representation of x as the difference of disjoint elements that are \geq 0 . For all x and y in X, \left| x ^{+} - y ^{+} \right| \leq | x - y | and x + y = \sup\{ x, y \} + inf \{ x, y \} . If y ≥ 0 and x ≤ y then x⁺ ≤ y. Moreover, x \leq y if and only if x ^{+} \leq y ^{+} and x ^{-} \leq x ^{-1} .

See also

• Solid set
• Locally convex vector lattice
• Vector lattice

Sources

• Book: Helmut H. Schaefer

. Schaefer . Helmut H. . Helmut H. Schaefer . Wolff . Manfred P. . Topological Vector Spaces . Springer New York Imprint Springer . . 3 . New York, NY . 1999 . 978-1-4612-7155-0 . 840278135 .