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The Automorphism Conjecture for Small Sets and Series Parallel Sets

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Abstract

The automorphism conjecture for ordered sets states that the automorphism to endomorphism ratio will tend to zero as the size of the ordered set goes to infinity. We show by computer enumeration that up to size 11 the ratio is largest for weakly ordered sets. Subsequently, we derive exact recursive formulas for the number of homomorphisms between two related types of weakly ordered sets and we prove a strong automorphism conjecture for series-parallel ordered sets. We conclude with an example that shows that the automorphism to endomorphism ratio can exceed \(\displaystyle{ \left( .5244 \right) ^{|P|}} \) for arbitrarily large \(|P|\).

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Authors and Affiliations

  1. Program of Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA

    Bernd S. W. Schröder

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  1. Bernd S. W. Schröder
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Correspondence to Bernd S. W. Schröder.

Additional information

This work was sponsored by Louisiana Board of Regents RCS grant LEQSF(1999-02)-RD-A-27.

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Schröder, B.S.W. The Automorphism Conjecture for Small Sets and Series Parallel Sets. Order 22, 371–387 (2005). https://doi.org/10.1007/s11083-005-9024-7

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  • DOI: https://doi.org/10.1007/s11083-005-9024-7

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