The boxes (Q805238)

Take a set \(E\) and a family \(L\) of subsets of \(E\). A subsection of \(L\) is a subset of \(E\) whose intersections with each element of \(L\) consists of at most one element. A pair \((L,C)\) of families of subsets of \(E\) is non-ambiguious if \(L\) is the set of subsections of \(C\) and \(C\) is the set of subsections of \(L\). Given such a pair \((L,C)\), denote by \((L_0,C_0)\) the pair of families of maximal elements of \((L,C)\), respectively (maximal with respect to inclusion). This pair \((L_0,C_0)\) is called a box. The paper defines these new combinatorial objects, the boxes (and the related objects, ``the trunks'') and investigates their fundamental (combinatorial and algebraic) properties, presenting also applications (in code theory).