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On k-CS-Transitive Cycle-Free Partial Orders with Finite Alternating Chains

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Abstract

The notion of cycle-free partial order (CFPO) was defined by R. Warren, and the major cases of the classification of the countable sufficiently transitive CFPOs were given, the finite and infinite chain cases, by Creed, Truss, and Warren. It is the purpose of this paper to complete the classification. The cases which remained untreated were CFPOs not embedding an infinite ‘alternating chain’ ALT (which can only happen in the finite chain case). It is shown that if a k-CS-transitive CFPO does not embed ALT, where k ≥ 3, then it does not embed any alternating chain of size k + 3, and this leads to the desired classification (which is only given explicitly for k = 3 and 4). The general result says that the class of k-CS-transitive CFPOs for k ≥ 3 not embedding ALT admits a recursive classification.

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References

  1. Creed, P., Truss, J. K. and Warren, R. (1999) The structure of k-CS-transitive cycle-free partial orders with infinite chains, Math. Proc. Cambridge Philos. Soc. 126, 175-194.

    Google Scholar 

  2. Droste, M. (1985) Structure of partially ordered sets with transitive automorphism groups, Mem. Amer. Math. Soc. 57(334).

  3. Droste, M., Truss, J. K. and Warren, R., Simple automorphism groups of cycle-free partial orders, to appear in Forum Mathematicum.

  4. MacNeille, H. M. (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416-460.

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  5. Rubin, M. (1993) The Reconstruction of Trees from their Automorphism Groups, Contemp. Math. 151, Amer. Math. Soc.

  6. Truss, J. K. (1996) Betweenness relations and cycle-free partial orders, Math. Proc. Cambridge Philos. Soc. 119, 631-643.

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  7. Warren, R. (1992) The structure of k-CS-transitive cycle-free partial orders, PhD Thesis, University of Leeds.

  8. Warren, R. (1997) The structure of k-CS-transitive cycle-free partial orders, Mem. Amer. Math. Soc. 614.

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  1. Department of Pure Mathematics, University of Leeds, LS2 9JT Leeds, U.K.

    J. K. Truss

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  1. J. K. Truss
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Truss, J.K. On k-CS-Transitive Cycle-Free Partial Orders with Finite Alternating Chains. Order 15, 151–165 (1998). https://doi.org/10.1023/A:1006172607771

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  • DOI: https://doi.org/10.1023/A:1006172607771

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