Join-independent and meet-independent sets in complete lattices (Q1611296)

Let \(L\) be a complete lattice. A subset \(A\) of \(L\) is called join-independent if \(a\not \leq\sup A\setminus \{a\}\) for all \(a\in A\) (a meet-independent set is defined dually). For a set \(X\subseteq L\), let \(\psi (X)\) denote the supremum of \(X\) in \(L\). It is proved that \(A\subseteq L\) is join-independent if and only if for every \(x\in A\) and for every \(X\subseteq L\) we have \(x\in X\) whenever \(x\leq\sup X\), and it is equivalent to the fact that \(\psi\) is an order isomorphism and a complete join-embedding from the poset of all subsets of \(A\) into \(L\). Several properties of join-independent sets are given. Connections to closure spaces and incidence structures are discussed.