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Notes on the Length, the Structure and the Cardinality of a Chain

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Abstract

For a linearly ordered set (Z,≤) the length l(Z) of Z is the supremum of all cardinals that can be order-embedded or reverse order-embedded into Z. In this paper we give new proofs of two theorems relating the length and the cardinality of Z. The first one sets the following general inequality: |Z|≤2l(Z). The second one says that in the case that Z is a scattered chain (i.e. it does not contain rationals) we have |Z|=2l(Z).

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References

  1. Herden, G. and Pallack, A.: Interrelations between the length, the structure and the cardinality of a chain, Order 18 (2001), 191–200.

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  2. Kunen, K.: Set Theory. An Introduction to Independence Proofs, 1980.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Maciej Malicki

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  1. Maciej Malicki
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Correspondence to Maciej Malicki.

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Mathematics Subject Classifications (2000)

06A05, 03E04.

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Malicki, M. Notes on the Length, the Structure and the Cardinality of a Chain. Order 21, 201–205 (2004). https://doi.org/10.1007/s11083-004-6448-4

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  • DOI: https://doi.org/10.1007/s11083-004-6448-4

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