A note on inclusion-exclusion on semilattices (Q2725016)

A \(\vee\)-semilattice is a partially ordered set \(P\) containing the least upper bound \(x\vee y\) of any \(x,y\in P\). Let \(P\) be a finite \(\vee\)-semilattice and let \(\widehat P\) be the lattice obtained by adjoining a least element \(\widehat 0\) to \(P\). The Möbius function of \(\widehat P\), \(\widehat\mu:\widehat P\to \mathbb{Z}\), is defined recursively by \(\widehat\mu(\widehat 0)= 1\) and \(\widehat\mu(p)= -\sum_{q< p} \widehat\mu(q)\), if \(p>\widehat 0\). The author derives the following consequence of an inclusion-exclusion variant of \textit{H. Narushima} [J. Comb. Theory, Ser. A 17, 196-203 (1974; Zbl 0289.05013)]. Theorem. Let \(P\) be a finite \(\vee\)-semilattice and \(\{A_p\}_{p\in P}\) a family of finite sets such that \(A_x\cap A_y= A_{x\vee y}\) for any \(x,y\in P\), and let \(\widehat\mu\) denote the Möbius function of \(\widehat P\). Then \(|\bigcup_{p\in P} A_p|=- \sum_{p\in P} \widehat\mu(p)|A_p|\). The author also deduces a well-known result about the Euler function of number theory.