On Levfs Duality between Permutations and Convergent Series

This paper concerns a duality between conditionally convergent real series and permutations of their indices. It is widely known that if one is given a series E ^ which is conditionally convergent but not absolutely convergent, then there is a permutation n such that £«< # £#*(*)• Various authors have studied extensions of this result of Riemann (for example, Smith [8], Steinitz [9], Threlfall [10], Wald [11,12]), but in our opinion a more challenging problem is to find those permutations which do not change the value of 2>*, and, dually, given a permutation to find those series whose sum is unaffected by the permutation. F. W. Levi [5] was apparently the first to consider such problems, and he introduced an interesting duality between subsets of C, denoting the set of all convergent real series, and subsets of P, denoting the set of permutations of the counting numbers N = {1, 2, ...}. Given a set A £ C, let A x = {7i G P: Ea, = Efl«(i) for all aeA}, and given a ? c p ) let P = { E ^ e C : ^ = Zan(i) for all neP}. Levi called A x + the closure of A, and P the closure of P, and noted that x and + are inverses of each other when considered as maps between closed sets of series and closed sets of permutations. In particular, this duality can be used to show that the closed sets of permutations and series each form a lattice, where