A note on supermodular sublattices in finite relatively complemented lattices (Q998786)

Let \(L\) be a lattice and let \(\mathbb{R}\) denote the set of real numbers. A function \(f: L\rightarrow \mathbb{R}\) is called supermodular if it satisfies \(f(x)+f(y)\leq f(x\wedge y)+f(x\vee y)\) for all \(x, y\in L.\) Note that supermodular functions play an important role in some applications, such as in mathematical economics, operations research or computer science. In the last domain, supermodular functions occur in the study of the complexity of certain optimisation problems called maximum constraint satisfaction problems. We need one more concept: Let \(K\) be a sublattice of \(L.\) \(K\) is said to be supermodular (in \(L\)) if, for any \(x\in K\) and \(y\in L\), at least one of the elements \(x\wedge y, x\vee y\) belongs to \(K.\) It is easy to verify that an ideal or a filter or a union of an ideal and a filter is a supermodular sublattice of \(L.\) The following result is also easy to derive: Let \(f\) be a 0-1 valued function on \(L\). Then \(f\) is supermodular on \(L\) iff \(\{x\in L: f(x)=1\}\) is a supermodular sublattice of \(L\). The main result describes the supermodular sublattices of a finite lattice \(L,\) where \(L\) is a direct product of non-trivial relatively complemented lattices.