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