Remarks on the Möbius function of a subposet (Q1115458)

In this brief note the author shows via an interesting observation leading to a simple computation that \[ \mu_ P(x,y)=\mu_ R(x,y)+\sum_{A}\mu_ R(x,s)\mu_ P(t,u)\mu_ R(v,y) \] where P is a poset, D is a subposet and \(R=P-D\) with induced ordering, \(\mu_ X\) is the Möbius function of the poset X and \[ A=\{(s,t,u,v)\in R\times D\times D\times R| \quad s\leq t,\quad u\leq v\}. \] From this, several useful corollaries are then easily obtained.