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''independent elements''
 
''independent elements''
  
Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332802.png" /> of a [[Vector lattice|vector lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332803.png" /> with the property
+
Two elements $  x \in X $
 +
and $  y \in X $
 +
of a [[Vector lattice|vector lattice]] $  X $
 +
with the property
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332804.png" /></td> </tr></table>
+
$$
 +
| x | \wedge | y |  = 0 .
 +
$$
  
 
Here
 
Here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332805.png" /></td> </tr></table>
+
$$
 +
| x |  = x \lor ( - x ) ,
 +
$$
  
 
which is equivalent to
 
which is equivalent to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332806.png" /></td> </tr></table>
+
$$
 +
| x |  = \sup ( x , - x ) .
 +
$$
  
The symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332808.png" /> are, respectively, the [[Disjunction|disjunction]] and the [[Conjunction|conjunction]]. Two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d0332809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328010.png" /> are called disjunctive if any pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328012.png" /> is disjunctive. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328013.png" /> is said to be disjunctive with a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328014.png" /> if the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328016.png" /> are disjunctive. A disjunctive pair of elements is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328018.png" />; a disjunctive pair of sets is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328019.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328020.png" />, respectively.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328021.png" /> and the negative part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328022.png" /> of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328023.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328025.png" />, are pairwise disjunctive, they are linearly independent; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328027.png" /> are disjunctive elements, the linear subspaces which they generate are also disjunctive; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328029.png" />, and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328030.png" /></td> </tr></table>
+
$$
 +
\sup _  \alpha  x _  \alpha  = x
 +
$$
  
exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328031.png" />. For disjunctive elements, several structural relations are simplified; e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328032.png" />, then
+
exists, then $  x \perp  y $.  
 +
For disjunctive elements, several structural relations are simplified; e.g., if $  x \perp  y $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328033.png" /></td> </tr></table>
+
$$
 +
| x + y |  = | x | + | y | ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328034.png" /></td> </tr></table>
+
$$
 +
( x + y ) \wedge z  = x \wedge z + y \wedge z
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033280/d03328035.png" />, etc.
+
for $  z > 0 $,  
 +
etc.
  
 
The concept of disjunctive elements may also be introduced in more general partially ordered sets, such as Boolean algebras.
 
The concept of disjunctive elements may also be introduced in more general partially ordered sets, such as Boolean algebras.
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 19:36, 5 June 2020


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)
How to Cite This Entry:
Disjunctive elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_elements&oldid=46742
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article