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Sets P Q = Hom(Q, P) and Products

In: Ordered Sets

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  • Bernd S. W. Schröder

    (Lousiana Tech University, Program of Mathematics and Statistics)

Abstract

Order-preserving mappings from one ordered set to another form a natural ordered set under the pointwise order. Since products in general are defined similar to homomorphism sets, we shall investigate homomorphism sets and products of ordered sets closely together. In this chapter we introduce some of the salient results on these sets such as the fixed point theorem for products of two finite ordered sets (cf. Theorem 10.2.11), Hashimoto’s refinement theorem (cf. Theorem 10.4.4) and the cancelation property for exponents (cf. Theorem 10.5.9). The automorphism conjecture (cf. Open Question 11.5.1) as well as the open problems at the end of this chapter show that there are interesting problems related to homomorphism sets and products that remain open.

Suggested Citation

  • Bernd S. W. Schröder, 2003. "Sets P Q = Hom(Q, P) and Products," Springer Books, in: Ordered Sets, chapter 10, pages 227-262, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0053-6_10
    DOI: 10.1007/978-1-4612-0053-6_10
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