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Transactions of the American Mathematical Society

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On realization of Björner's ``continuous partition lattice'' by measurable partitions


Author: Mark D. Haiman
Journal: Trans. Amer. Math. Soc. 343 (1994), 695-711
MSC: Primary 06C10; Secondary 28D99
DOI: https://doi.org/10.1090/S0002-9947-1994-1211408-0
MathSciNet review: 1211408
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Abstract: Björner [1] showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Björner's construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as a lattice of measurable partitions. This new lattice contains the Björner lattice and shares its key properties. Furthermore its automorphism group is the full automorphism group $ \pmod 0$ of the unit interval with Lebesgue measure, whereas, as we show, the Björner lattice possesses only a proper subgroup of these automorphisms.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1211408-0
Article copyright: © Copyright 1994 American Mathematical Society