Balance in Systems of Finite Sets with Applications

An extension of balance notion from the theory of signed graphs to the case of finite sets systems is presented. For a finite set T, a subset S ⊆ T and a family F of subsets of T we denote by δm (S|F) respectively δM (S|F) the minimum/maximum number of changes (addition or deletion of elements), without repetition, which transforms S into a set from F. We are especially interested in the particular case in which F is the group generated by a family of subsets X1,..., Xn ⊆ T with symmetric difference operation. The obtained results are applied to the theory of signed graphs.