Let $\mathcal{J}$ be the system of all intervals of the form $(a,b] \subseteq (0,1]$ argue why this is not an algebra.
My argument is:
$$ \text{let } 0< a_1 < b_1 < a_2 < b_2 \leq 1, \text{ and } (a_1, b_1] \cap (a_2, b_2] = \emptyset\text{ and } (a_1, b_1], (a_2, b_2] \in \mathcal{J} $$
then: $$ (a_1, b_1] \cup (a_2, b_2] \notin \mathcal{J} $$
because $(a_1, b_1] \cup (a_2, b_2]$ is not a half open interval. Am i correct?
You are correct if you additionally assume that $b_1 \neq a_2$ and $b_2 \neq a_1$, as otherwise the union of these intervals will be an interval.
Anyway, technically you should find two specific elements of $\mathcal{J}$ such that their union is not in $\mathcal{J}$, e.g. $\left( 0, \frac{1}{3} \right]$ and $\left( \frac{2}{3}, 1 \right]$.