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Chains, Antichains, and Fences

In: Ordered Sets

Author

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  • Bernd Schröder

    (University of Southern Mississippi, Department of Mathematics)

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.

Suggested Citation

  • Bernd Schröder, 2016. "Chains, Antichains, and Fences," Springer Books, in: Ordered Sets, edition 2, chapter 0, pages 23-51, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-29788-0_2
    DOI: 10.1007/978-3-319-29788-0_2
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