Half-Open Rectangles form Semiring of Sets

Theorem
The half-open $n$-rectangles $\mathcal{J}_{ho}^n$ form a semiring of sets.

Proof
By definition, $\varnothing$ is considered to be a half-open $n$-rectangle.

That $\mathcal{J}_{ho}^n$ is $\cap$-stable follows from Half-Open Rectangles Closed under Intersection.

Thus, it remains to show condition $(3')$ for a semiring of sets:


 * $(3'):\quad$ If $A, B \in \mathcal{J}_{ho}^n$, then there exists a finite sequence of pairwise disjoint half-open $n$-rectangles $A_1, A_2, \ldots, A_m \in \mathcal{J}_{ho}^n$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^m A_k$.

To prove this, proceed by induction on $n$:

Basis for the Induction
Suppose that $n = 1$, and let $I := \left[{a \,.\,.\, b}\right)$ and $J := \left[{c \,.\,.\, d}\right)$ be half-open intervals.

It is to be demonstrated that $I \setminus J$ is a finite union of pairwise disjoint half-open intervals.

Equivalently, by the above verification of the other axioms, that $\mathcal{J}_{ho}^1$ is a semiring of sets.

By swapping the roles of $I$ and $J$ if necessary, it may be arranged that $a \le c$.

Now if $b \le c$ as well, then $I \cap J = \varnothing$.

Subsequently, this implies $I \setminus J = I$, and the statement trivially holds.

Next, consider the other case, i.e., $c < b$.

If also $d < b$, then we may write $I$ as the following disjoint union:


 * $I = \left[{a \,.\,.\, b}\right) = \left[{a \,.\,.\, c}\right) \cup \left[{c \,.\,.\, d}\right) \cup \left[{d \,.\,.\, b}\right)$

The middle term equals $J$, and we immediately obtain:


 * $I \setminus J = \left[{a \,.\,.\, c}\right) \cup \left[{d \,.\,.\, b}\right)$

verifying the statement in this case.

Lastly, suppose that $b \le d$.

Then $I \cap J = \left[{a \,.\,.\, b}\right) \cap \left[{c \,.\,.\, d}\right) = \left[{c \,.\,.\, b}\right)$.

Therefore, by Set Difference with Intersection is Difference:


 * $I \setminus J = I \setminus \left({I \cap J}\right) = \left[{a \,.\,.\, b}\right) \setminus \left[{c \,.\,.\, b}\right) = \left[{a \,.\,.\, c}\right)$

and, having verified this last case, the result follows from Proof by Cases.

This constitutes the induction basis.

Induction Hypothesis
Now assume the induction hypothesis.

That is, for some fixed $n \ge 1$, assume that $\mathcal{J}_{ho}^n$ is a semiring of sets.

Next, it is to be shown that $\mathcal{J}_{ho}^{n+1}$ is also a semiring of sets.

Induction Step
The induction step goes as follows.

By definition of half-open rectangle, it holds that:


 * $\mathcal{J}_{ho}^{n+1} = \mathcal{J}_{ho}^n \times \mathcal{J}_{ho}^1$

Further, we have that $\mathcal{J}_{ho}^n$ and $\mathcal{J}_{ho}^1$ are semirings of sets.

Hence by Cartesian Product of Semirings of Sets, $\mathcal{J}_{ho}^{n+1}$ is a semiring of sets, too.