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

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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

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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

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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  .[2] If y ≥ 0 and xy then x+y. Moreover,   if and only if   and  .[2]

See also

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References

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  1. ^ Schaefer & Wolff 1999, pp. 204–214.
  2. ^ a b c d e f Schaefer & Wolff 1999, pp. 74–78.

Sources

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