Upper and Lower Bounds

Abstract

Upper and lower bounds have already been defined in Definitions 2.1.4 and 2.3.1. From their use in Zorn's Lemma, as well as their occurrences in the proofs of Dilworth's Chain-Decomposition Theorem 2.5.7 and Proposition 2.6.7 (in both proofs, sets were defined in terms of their upper bounds), the reader can already infer that bounds of sets play an important role in ordered sets. In this chapter we consider various types of bounds and relate them to open problems as well as to each other.