Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.