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Retractions

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Ordered Sets

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Abstract

Retractions are an important tool to explore the structure of ordered sets. For example, for the fixed point property, certain retractions can reduce the problem to a problem on smaller, easier to handle structures. Viewed “in the opposite direction,” retractions can also be seen as a tool to build larger examples of ordered sets with certain properties. Similar statements hold for lexicographic sum decompositions, see Chapter 7, and for products and sets P Q, see Chapter 12 Sometimes investigations via retractions can have surprising consequences, as can be seen in Theorem 4.48. For an excellent survey on retractions, see [245]. For a recent survey on the use of retractions in fixed point theory, see [289].

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Notes

  1. 1.

    This choice of language comes from lattice theory and will be motivated in Proposition 8.29.

  2. 2.

    Although the letter \(\mathcal{C}\) is used in Definition 1.33, there will be no confusion, as comparative retractions will not be used in the context of reconstruction.

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  1. Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, USA

    Bernd Schröder

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Schröder, B. (2016). Retractions. In: Ordered Sets. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29788-0_4

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