2 5 Se p 20 06 Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks ∗

We prove that for any n there is a pair (P n 1 , P n 2 ) of nonisomorphic ordered sets such that P n 1 and P n 2 have equal maximal and minimal decks, equal neighborhood decks, and there are n + 1 ranks k0, . . . , kn such that for each i the decks obtained by removing the points of rank ki are equal. The ranks k1, . . . , kn do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks 1 and 2, both without extremal elements, the decks obtained by removing the points of rank ri are not equal. AMS subject classification (2000): 06A07