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 .[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 .[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]

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 .[2]

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 .[2] For all x and y in X, and .[3] If y ≥ 0 and xy then x+y. Moreover, if and only if and .[2]

See also

References

  1. Schaefer & Wolff 1999, pp. 204–214.
  2. Schaefer & Wolff 1999, pp. 74–78.
  3. Schaefer & Wolff 1999, pp. 74-78.

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.CS1 maint: ref=harv (link)
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