arXiv subjects
Publications and source records attributed to Bernd S. W. Schröder.
For every ordered set, we reconstruct the deck obtained by removal of the elements of rank r that are neither minimal nor maximal. Consequently, we also reconstruct the deck obtained by removal of the extremal, that is, minimal or maximal, elements. Finally, we reconstruct the ordered sets with a minmax pair of pseudo-similar points.
This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the structures which currently prevent the proof of such an exponential bound, or which indeed inflate the number of automorphisms beyond such a bound. This is a first step towards a possible resolution of the Automorphism Conjecture for ordered sets.
We introduce a recursive method to deconstruct the automorphism group of an ordered set. By connecting this method with deep results for permutation groups, we prove the Automorphism Conjecture for ordered sets of width less than or equal to $11$. Subsequent investigations show that the method presented here could lead to a resolution of the Automorphism Conjecture.
We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.