Chains, Antichains, and Fences

Bernd Schröder
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Bernd Schröder: University of Southern Mississippi, Department of Mathematics

Chapter Chapter 2 in Ordered Sets, 2016, pp 23-51 from Springer

Abstract: Abstract Chains and antichains are arguably the most common kinds of ordered sets in mathematics. The elementary number systems ℕ $$\mathbb{N}$$ , ℤ $$\mathbb{Z}$$ , ℚ $$\mathbb{Q}$$ , and ℝ $$\mathbb{R}$$ (but not ℂ $$\mathbb{C}$$ ) are chains. Chains are also at the heart of set theory. The Axiom of Choice 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.

Keywords: Well-ordered Set; Chain Decomposition Theorem; Dedekind Number; Maximal Antichain; Fixed Point Property (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-29788-0_2

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