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).
Similar content being viewed by others
References
Herden, G. and Pallack, A.: Interrelations between the length, the structure and the cardinality of a chain, Order 18 (2001), 191–200.
Kunen, K.: Set Theory. An Introduction to Independence Proofs, 1980.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000)
06A05, 03E04.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-004-6448-4