On realization of Björner’s “continuous partition lattice” by measurable partitions
Mark D. Haiman · Transactions of the American Mathematical Society · 1994
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 ( mod 0 ) \pmod 0 of the unit interval with Lebesgue measure, whereas, as we show, the Björner lattice possesses only a proper subgroup of these automorphisms.