View source for Disjunctive complement

''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 $A$.

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