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 [math]\displaystyle{ \inf \left\{ |x|, |y| \right\} = 0 }[/math], in which case we write [math]\displaystyle{ x \perp y }[/math], where the absolute value of x is defined to be [math]\displaystyle{ |x| := \sup \left\{ x, - x \right\} }[/math]. 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 [math]\displaystyle{ A \perp B }[/math]. If A is the singleton set [math]\displaystyle{ \{ a \} }[/math] then we will write [math]\displaystyle{ a \perp B }[/math] in place of [math]\displaystyle{ \{ a \} \perp B }[/math]. For any set A, we define the disjoint complement to be the set [math]\displaystyle{ A^{\perp} := \left\{ x \in X : x \perp A \right\} }[/math].

Characterizations

Two elements x and y are disjoint if and only if [math]\displaystyle{ \sup\{ | x |, | y | \} = | x | + | y | }[/math]. If x and y are disjoint then [math]\displaystyle{ | x + y | = | x | + | y | }[/math] and [math]\displaystyle{ \left(x + y \right)^{+} = x^{+} + y^{+} }[/math], where for any element z, [math]\displaystyle{ z^{+} := \sup \left\{ z, 0 \right\} }[/math] and [math]\displaystyle{ z^{-} := \sup \left\{ -z, 0 \right\} }[/math].

Properties

Disjoint complements are always bands, but the converse is not true in general. If A is a subset of X such that [math]\displaystyle{ x = \sup A }[/math] exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from [math]\displaystyle{ \{ x \} }[/math].

Representation as a disjoint sum of positive elements

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

See also

• Solid set
• Locally convex vector lattice
• Vector lattice

Sources

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.