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Lexicographic Sums

In: Ordered Sets

Author

Listed:
  • Bernd Schröder

    (University of Southern Mississippi, Department of Mathematics)

Abstract

In this chapter, we investigate a construction, lexicographic sums, that uses existing ordered sets to build new ordered sets. The pictorial idea is very simple: Take an ordered set T and replace each of its points t with an ordered set P t . The resulting structure will be a new, larger ordered set. It is then natural to ask how various order-theoretical properties and parameters behave under lexicographic constructions. We will revisit lexicographic sums in later chapters to specifically answer this question in a variety of contexts. At the end of this chapter, in Section 7.6, we will explain the definition of “hard” problems that was mentioned at the start of Section 5.4 (see Definition 7.30). We will then use a construction that is close to the lexicographic sum construction (see Section 7.5), to show that determining if an ordered set has the fixed point property is “hard.”

Suggested Citation

  • Bernd Schröder, 2016. "Lexicographic Sums," Springer Books, in: Ordered Sets, edition 2, chapter 0, pages 173-197, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-29788-0_7
    DOI: 10.1007/978-3-319-29788-0_7
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