I've seen this partially ordered set in our combinatorics script and it says that it is obvious how to calculate the möbius function $\mu(a_1,a_i)$ for all $i \in \{1, ..., 10\}$.
Here's the Hasse diagram of the Poset, but I don't know how to calculate the möbius function.
I know that the möbius function is multiplicative and that the sum of the Möbius function over all positive divisors of $n$ is zero except when $n=1$. But that still doesn't help me.
You are confusing two different (but related) meanings of the term Möbius function: the number-theoretic Möbius function and the poset-theoretic Möbius function.
The poset-theoretic Möbius function is defined recursively on interval $[x,y]$ of a poset as follows: $\mu([x,x]) = 1$ and when $x < y$, $$ \mu([x,y]) = - \sum_{x \leq z < y} \mu([x,z]). $$
One famous poset is the divisibility poset, in which the elements are the natural numbers (excluding zero), and $x \leq y$ if $x \mid y$ (i.e., if $x$ divides $y$). This poset is self-similar, in the sense that the subposet consisting of all elements above some $x$ is isomorphic to the entire poset, and for this reason $\mu([x,y])$ depends only on $y/x$, i.e., there is a function $\mu_{\mathbb{N}}$ such that $\mu([x,y]) = \mu_{\mathbb{N}}(y/x)$. This function is the number-theoretic Möbius function.