Minimum matrix representation of Sperner systems Original Research Article

LetX be ann-element set and2X be the family of subsets ofx. IfK⊂ 2x such that for anyK1,K2 εK,k1 ≠K2 impliesK1 ≠K2 thenKis called a Sperner system. LetM be anm×n matrix and letX denote the set of columns ofM. IfA ⊂X,b εX andM contains no two rows equal inA but different inb, then we say thatA impliesb and the closure ofA, denoted byLM(A), is defined byLM(A) = {b: b ε X, A implies b} . We say that anm×nmatrixM represents a given Sperner systemKifK= {K ε 2X:LM(K) = X, andK is minimal for this property}. Let s(K= min {m: ε an m × n matrix M representingK}. It is well known that this parameter plays an important role in database theory. In this paper, we give a necessary and sufficient condition for the existence of anm ×n matrix M to represent a given Sperner systemK. We are then able to obtain some useful results relating to the determination of s(K).