Abstract
Let \(\lambda \in \left (0,\frac {1}{2} \right ) \). We prove that, for ordered sets P of order dimension 2 and for interval orders, the ratio of the number of automorphisms to the number of endomorphisms is asymptotically bounded by \(2^{-|P|^{\lambda } } \). The key to the proof is to establish this bound for certain types of lexicographic sums.
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Schröder, B.S.W. The Automorphism Conjecture for Ordered Sets of Dimension 2 and Interval Orders. Order 38, 271–281 (2021). https://doi.org/10.1007/s11083-020-09540-5
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DOI: https://doi.org/10.1007/s11083-020-09540-5