Definition:Interval/Ordered Set/Right Half-Open

Definition
Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $a, b \in S$.

The right half-open interval between $a$ and $b$ is the set:


 * $\left[{a \,.\,.\, b}\right) := a^\succcurlyeq \cap b^\prec = \left\{{s \in S: \left({a \preccurlyeq s}\right) \land \left({s \prec b}\right)}\right\}$

where:
 * $a^\succcurlyeq$ denotes the upper closure of $a$
 * $b^\prec$ denotes the strict lower closure of $b$.

Also defined as
Some sources require that $a \preccurlyeq b$.