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''independent elements''

Two elements  $  x \in X $
and  $  y \in X $
of a [[Vector lattice|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|disjunction]] and the [[Conjunction|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,   B.Z. Vulikh,   A.G. Pinsker,   "Functional analysis in semi-ordered spaces" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.Z. Vulikh,   "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR></table>

====Comments====
The phrase  "disjunctive sets"  also has a different meaning, cf. [[Disjunctive family of sets|Disjunctive family of sets]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>