Definition:Interval/Ordered Set/Left Half-Open

Definition
Let $(S, \preceq)$ be an ordered set.

Let $a, b \in S$.

Then $\left\{{ s \in S: (a \prec s) \land (s \preceq b) }\right\} = {\dot\uparrow} a \cap {\bar\downarrow} b$ is called the left half-open interval between $a$ and $b$.

It is written $(a \,.\,.\, b]$.

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