arXiv subjects
Publications and source records attributed to Bernd Schröder.
We present a representation for permutation groups as the automorphism group of an ordered set $U$ such that the automorphism group's action on a subset $T\subseteq U$ is the permutation group itself. For many imprimitive permutation groups, the representation uses fewer points than the smallest representations to date.
We prove the automorphism conjecture for ordered sets of width less than or equal to 11. The proof supports the meta conjecture that a large number of automorphisms is achievable only as some type of product of independent automorphisms on highly symmetric subsets.
We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n $ and $P_2 ^n $ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n $ such that for each $i$ the decks obtained by removing the points of rank $k_i $ are equal. The ranks $k_1 , \ldots , k_n $ 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 $r_i $ are not equal.