Question

Consider the multiset $X=left{n_{1} cdot a_{1}, n_{2} cdot a_{2}, ldots, n_{k} cdot a_{k} ight}$ of $k$ distinct elements with positive repetition numbers $n_{1}, n_{2}, ldots, n_{k}$. We introduce a partial order on the combinations of $X$ by stating the following relationship: If $A=left{p_{1} cdot a_{1}, p_{2} ight.$. $left.a_{2}, ldots, p_{k} cdot a_{k} ight}$ and $B=left{q_{1} cdot a_{1}, q_{2} cdot a_{2}, ldots, q_{k} cdot a_{k} ight}$ are combinations of $X$, then $A leq B$ provided that $p_{i} leq q_{i}$ for $i=1,2, ldots, k .$ Prove that this statement defines a partial order on $X$ and then compute its Möbius function.

          Consider the multiset $X=left{n_{1} cdot a_{1}, n_{2} cdot a_{2}, ldots, n_{k} cdot a_{k}
ight}$ of $k$ distinct elements with positive repetition numbers $n_{1}, n_{2}, ldots, n_{k}$. We introduce a partial order on the combinations of $X$ by stating the following relationship: If $A=left{p_{1} cdot a_{1}, p_{2}
ight.$. $left.a_{2}, ldots, p_{k} cdot a_{k}
ight}$ and $B=left{q_{1} cdot a_{1}, q_{2} cdot a_{2}, ldots, q_{k} cdot a_{k}
ight}$ are combinations of $X$, then
$A leq B$ provided that $p_{i} leq q_{i}$ for $i=1,2, ldots, k .$ Prove that this statement defines a partial order on $X$ and then compute its Möbius function.

Added by Alba H.

Question

Please give Ace some feedback