On a class of lattices associated with n-cubes (Q792356)

Let \(L_ n\) be the lattice of faces of an n-cube and \(X\subset L_ n\) such that (i) \(x\in X\), \(x\leq y\Rightarrow y\in X,\) (ii) \(x,y\in X; x\leq y\Rightarrow\) the opposite face \(\Delta_ x(y)\) to y in the cube \([0,x]\subset L_ n\) belongs to X. The partial order on \(L_ n\) induces a lattice structure \(L(X)\) on \(X\cup \{0\}.\) It is shown that these lattices are, up to isomorphism, exactly the lattices of signed simplices of simplicial complexes such that the diagonal mappings \(\Delta_ x\) correspond to similar mappings in these lattices. Examples of lattices isomorphic to such an L(X) are finite lattices L with 0 and 1 in which any element other than 1 is an inf of co-atoms and in which for any \(x\in L\) there is an involutorial automorphism of [0,x].