Titlebook: Ordered Sets; An Introduction Bernd S. W. Schr?der Textbook 20031st edition Springer Science+Business Media New York 2003 Algebraic topolog

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Enumeration, such a question. A counting question can be motivated by pure curiosity or, as the Kelly Lemma (cf. Proposition 1.5.14) in reconstruction shows, it can be asked as a step in proving something else. The two most natural counting questions for ordered sets are still unanswered.

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The Dimension of Ordered Sets,Dimension theory is a prominent area in ordered sets. So far, we have looked at specific ordered structures and investigated their properties. In dimension theory one represents orders using the orders that occur most frequently, namely total orders.

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The Basics,y known as “mathematical maturity” should have been developed to the point that the reader can read and understand proofs and produce simple proofs. A text that develops these skills is for example [103]. A background in graph theory helps, but is not necessary.

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Upper and Lower Bounds,roofs 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

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Truncated Lattices,nly the top and the bottom are gone. However in terms of order-theoretical properties there is a significant change. Note that both the proof of reconstructibility of finite lattices as well as the characterization of the fixed point property for lattices heavily relied on the existence of the small

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