I've come across this symbol $\bot$ when reading a description of euler's totient function:
$\varphi(n) = |\{m \in \mathbb{N} | m < n \land m \bot n\}|$
I am not familiar with the $\bot$ symbol and I've googled for a while and I see it can mean perpendicular or orthogonal to (I can't see how either of those apply in this context, but if so, please explain) or logically false. However, I'm seeking just a fundamental explanation of what it means when there are two variables $a$ and $b$ and it is said that $a \bot b$.
The only meaning that makes sense is that $m\bot n\iff \gcd(m.n) = 1$. Though I must admit that I never saw this notation before, and without the context I wouldn't have guessed what it's supposed to mean here.
We can make sense of it by noting that we can represent a natural number as an infinite-dimensional vector of it's prime powers, i.e. let
$$n=\prod_{k\in\Bbb N} p_k^{n_k}$$ where $p_k$ is the $k$-th prime number. We then could represent $n$ uniquely as
$$v(n) := (n_1, n_2, n_3, \dots) \in\Bbb N_0^\infty$$ where only finitely many $n_k$ are non-zero. With this notation we have
$$\gcd(m, n) = 1 \quad\iff\quad v(m)\cdot v(n) = 0$$ where "$\cdot$" is the usual vector product with the notation
$$v(n)\cdot v(m) = 0 \quad\iff\quad v(n)\bot\ v(m)$$
But this all is just made up by me. My recommendation is to take the effort of typing "gcd" and stick with well-established notation. And for the lazy folks there's already $(a,b)$ as shorthand for $\gcd(a,b)$.
What also speaks against the notation is that it doesn't scale well: Many theorems, formulae or proofs refer to the gcd like "Let $d=\gcd(a,b)$" where $\bot$ is useless and one has to resort to established notation, anyway.