Disjunctive elements

independent elements

Two elements $ x \in X $ and $ y \in X $ of a vector lattice $ X $ with the property

$$ | x | \wedge | y | = 0 . $$

Here

$$ | x | = x \lor ( - x ) , $$

which is equivalent to

$$ | x | = \sup ( x , - x ) . $$

The symbols $ \wedge $ and $ \lor $ are, respectively, the disjunction and the conjunction. Two sets $ A \subset X $ and $ B \subset X $ are called disjunctive if any pair of elements $ x \in A $, $ y \in B $ is disjunctive. An element $ x \in X $ is said to be disjunctive with a set $ A \subset X $ if the sets $ \{ x \} $ and $ A $ are disjunctive. A disjunctive pair of elements is denoted by $ x \perp y $ or $ xdy $; a disjunctive pair of sets is denoted by $ A \perp B $ or $ AdB $, respectively.

Example of disjunctive elements: The positive part $ x _ {+} = x \lor 0 $ and the negative part $ x _ {-} = ( - x ) \lor 0 $ of an element $ x $.

If the elements $ x _ {i} $, $ i= 1 \dots n $, are pairwise disjunctive, they are linearly independent; if $ A $ and $ B $ are disjunctive elements, the linear subspaces which they generate are also disjunctive; if $ x _ \alpha \perp y $, $ \alpha \in \mathfrak A $, and

$$ \sup _ \alpha x _ \alpha = x $$

exists, then $ x \perp y $. For disjunctive elements, several structural relations are simplified; e.g., if $ x \perp y $, then

$$ | x + y | = | x | + | y | , $$

$$ ( x + y ) \wedge z = x \wedge z + y \wedge z $$

for $ z > 0 $, 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)