\begin{definition}[Definition:Interval/Ordered Set/Right Half-Open]
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $a, b \in S$.
The '''right half-open interval between $a$ and $b$''' is the set:
:$\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$
where:
:$a^\succcurlyeq$ denotes the upper closure of $a$
:$b^\prec$ denotes the strict lower closure of $b$.
\end{definition}