Signed posets

We define a new object, called a signed poset, that bears the same relation to the hyperoctahedral group B_n (i.e., signed permutations on n letters), as do posets to the symmetric group S_n. We then prove hyperoctahedral analogues of the following results: (1) the generating function results from the theory of P-partitions; (2) the fundamental theorem of finite distributive lattices (or Birkhoff's theorem) relating a poset to its distributive lattice of order ideals; (3) the edgewise-lexicographic shelling of upper-semimodular lattices; (4) MacMahon's calculation of the distribution of the major index for permutations.