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Partially Ordered Sets and Lattices

In: Lattice-ordered Rings and Modules

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  • Stuart A. Steinberg

    (University of Toledo, Department of Mathematics)

Abstract

In this chapter we present some basic facts about partially ordered sets and lattices which are fundamental for our study of lattice-ordered groups, rings, and modules. The material presented includes Zorn’s Lemma and some of its equivalences in Section 1.1, standard characterizations of distributive lattices and Boolean algebras in Section 1.2, and the construction of the MacNeille and Dedekind completions of a partially ordered set in Section 1.3. We also introduce some of the basic language of category theory and present enough of the subject of universal algebra so that we can establish the existence of free algebras in varieties. The symbols “⊆” and “⊇” will be used for set inclusion and “⊂” and “⊂” will be used for proper inclusion. The letters ℕ, ℤ, ℚ, ℝ, and ℂ denote the sets of natural numbers (excluding 0), integers, rational numbers, real numbers, and complex numbers, respectively. The symbols “|X|” or “card (X)” denote the cardinal number of the set X.

Suggested Citation

  • Stuart A. Steinberg, 2010. "Partially Ordered Sets and Lattices," Springer Books, in: Lattice-ordered Rings and Modules, edition 0, chapter 0, pages 1-31, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-1721-8_1
    DOI: 10.1007/978-1-4419-1721-8_1
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