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Holland’s Embedding Theorem

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Lattice-Ordered Groups

Part of the book series: Reidel Texts in the Mathematical Sciences ((RTMS,volume 4))

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Abstract

Recall example 1.1.15: Let T be a totally ordered set and A(T) the ℓ-group of orderpreserving permutations of T, where α ∈ A(T) is positive if tα ≥ t for all tT. Birkhoff [B] asked what ℓ-groups can be constructed in this manner. Holland [63] gave a partial answer to this question and in the process provided a new perspective from which ℓ-groups can be studied. (Bigard, Keimel and Wolfenstein [BKW] refer to this as “l’école américaine”.) Holland’s main result is that every ℓ-group is ℓ-isomorphic to an ℓ-subgroup of the ℓ-group of order-preserving permutations of some totally ordered set. In this chapter we will derive Holland’s theorem, along with some immediate consequences. For a more complete study of groups viewed in this manner see [G]. The reader should note that after Holland’s theorem is proved we will employ multiplicative notation extensively.

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© 1988 D. Reidel Publishing Company, Dordrecht, Holland

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Anderson, M., Feil, T. (1988). Holland’s Embedding Theorem. In: Lattice-Ordered Groups. Reidel Texts in the Mathematical Sciences, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2871-8_5

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  • DOI: https://doi.org/10.1007/978-94-009-2871-8_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7792-7

  • Online ISBN: 978-94-009-2871-8

  • eBook Packages: Springer Book Archive

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