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 $A$.