For $m,n,k,d\in \Bbb{Z}, m\leq n$, define for interval of integers $(m,n)$:
$$ \mathcal{F}_d(m,n,k) := \Big\{ x \in (m,n): x^2 = k^2 \pmod d\Big\} $$
Then $\mathcal{F}_d(m,n, i)\cdot\mathcal{F}_d(m', n',j) \subset \mathcal{F}_d(mm',nn', ij)$ where $\cdot$ is elementwise multiplication in $(\Bbb{Z}, \cdot)$ the monoid.
However if $c\mid d$ then there is inclusion $\mathcal{F}_d(m,n,k) \subset \mathcal{F}_c(m,n,k)$ since if the given modular congruence $x^2 = k^2 \pmod d$ is true, then it's also certainly true $\pmod c$.
This reminds me of adjoints in category theory, but I think some other structure is formed, something that hasn't been heavily studied & generalized to yet.
Question 1. Can you figure out what structure is formed? By that I mean do the two poset structures shown interact in any way?
Application. The sums $$\theta(o,m,n,k) := \sum_{d \ \mid\ o\#} \mu(d)\cdot |\mathcal{F}_d(m,n,k)|$$ seem to carry information about counting $2k$-gap prime pair averages sitting in the interval $(m,n)$. In other words, depending on your careful choice of $m,n$, the count is either exact or at least very close to this count, i.e. only sometimes off by (maximally) some relatively small constant such as $1$.
By the above two properties we have that $\mathcal{F}_d(m,n,i) \cdot \mathcal{F}_{d'}(m',n',j) \subset \mathcal{F}_{\gcd(d, d')}(mm',nn', ij)$.
Question 2. Do the sets $\mathcal{F}_d(m,n,k)$ form a base for a topology on $\Bbb{Z}$?
I can't figure out: if $x^2 = k^2 \pmod d,$ and $x^2 = k'^2 \pmod {d'}$, then does there exist $k''$ such that $x^2 = k''^2 \pmod {\text{lcm}(d,d')}$ or something?