Abstract
Let (Z,≤) be a chain and let (Z′,≤′) be its dual chain. Then the length l(Z) of (Z,≤) is the least upper bound of all cardinal numbers which can be order-embedded into (Z,≤) or (Z′,≤′). In particular, a chain is said to be short if its length is not greater than the smallest infinite cardinal. In this paper we shall prove that the cardinality |Z| of a chain (Z,≤) cannot be smaller than l(Z) and not greater than 2l(Z). The inequality |Z|≤2l(Z) is an immediate consequence of a general theorem which combines the structure of a chain with its length. In case of a short chain it follows that its structure may be rather complicated but that its cardinality cannot be greater than the cardinality of the real line.
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Herden, G., Pallack, A. Interrelations between the Length, the Structure and the Cardinality of a Chain. Order 18, 191–200 (2001). https://doi.org/10.1023/A:1011967103457
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DOI: https://doi.org/10.1023/A:1011967103457