Abstract
Chains and antichains are arguably the most common kinds of ordered sets in mathematics. The elementary number systems \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) (but not \(\mathbb{C}\)) are chains. Chains are also at the heart of set theory. The Axiom of Choice is equivalent to Zorn’s Lemma, which we will adopt as an axiom, and the Well-Ordering Theorem. The latter two results are both about chains.
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Notes
- 1.
Sometimes, sets that intersect every maximal antichain are called fibers. The overall upper bound on the size of such a fiber is | P | , as can be seen considering chains. Upper and lower bounds on the fiber size for individual ordered sets in given classes of ordered sets can be interesting.
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Schröder, B. (2016). Chains, Antichains, and Fences. In: Ordered Sets. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29788-0_2
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