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Difference between revisions of "Disjunctive complement"

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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332701.png" />''
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''of a set $A$ in a [[vector lattice]]''
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332702.png" /> of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332703.png" /> of a [[Vector lattice|vector lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332704.png" /> which are disjunctive with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332705.png" /> (cf. [[Disjunctive elements|Disjunctive elements]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332707.png" />; moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332708.png" /> is a conditionally-complete vector lattice (cf. [[Conditionally-complete lattice|Conditionally-complete lattice]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d0332709.png" /> is the smallest component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d03327010.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033270/d03327011.png" />.
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The set $A^{\mathrm{d}} = \{x \in X : x \perp A \}$ of all elements $x$ of a vector lattice $X$ which are disjunctive with the set $A$ (cf. [[Disjunctive elements]]). For any $A$, $A \subseteq A^{\mathrm{d\,d}} = (A^{\mathrm{d}})^{\mathrm{d}}$; moreover, if $X$ is a conditionally-complete vector lattice (cf. [[Conditionally-complete lattice]]), then $A^{\mathrm{d\,d}}$ is the smallest component of $X$ containing $A$.
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Latest revision as of 18:34, 3 September 2017

of a set $A$ in a vector lattice

The set $A^{\mathrm{d}} = \{x \in X : x \perp A \}$ of all elements $x$ of a vector lattice $X$ which are disjunctive with the set $A$ (cf. Disjunctive elements). For any $A$, $A \subseteq A^{\mathrm{d\,d}} = (A^{\mathrm{d}})^{\mathrm{d}}$; moreover, if $X$ is a conditionally-complete vector lattice (cf. Conditionally-complete lattice), then $A^{\mathrm{d\,d}}$ is the smallest component of $X$ containing $A$.

How to Cite This Entry:
Disjunctive complement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_complement&oldid=18795
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article