Disjunctive elements
independent elements
Two elements and
of a vector lattice
with the property
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Here
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which is equivalent to
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The symbols and
are, respectively, the disjunction and the conjunction. Two sets
and
are called disjunctive if any pair of elements
,
is disjunctive. An element
is said to be disjunctive with a set
if the sets
and
are disjunctive. A disjunctive pair of elements is denoted by
or
; a disjunctive pair of sets is denoted by
or
, respectively.
Example of disjunctive elements: The positive part and the negative part
of an element
.
If the elements ,
, are pairwise disjunctive, they are linearly independent; if
and
are disjunctive elements, the linear subspaces which they generate are also disjunctive; if
,
, and
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exists, then . For disjunctive elements, several structural relations are simplified; e.g., if
, then
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for , etc.
The concept of disjunctive elements may also be introduced in more general partially ordered sets, such as Boolean algebras.
References
[1] | L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in semi-ordered spaces" , Moscow-Leningrad (1950) (In Russian) |
[2] | B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian) |
[3] | N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) |
Comments
The phrase "disjunctive sets" also has a different meaning, cf. Disjunctive family of sets.
References
[a1] | W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971) |
Disjunctive elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_elements&oldid=46742