Definition:Interval/Ordered Set/Right Half-Open

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

Let $a, b \in S$.

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

It is written $[a \,.\,.\, b)$.

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