Difference between revisions of "Disjunctive complement"
From Encyclopedia of Mathematics
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| − | ''of a set | + | ''of a set $A$ in a [[vector lattice]]'' |
| − | The set | + | 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
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