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Continuous lattices of partitions and lattices of continuous partitions. (English) Zbl 0997.06004

Uraltseva, N. N. (ed.), Proceedings of the St. Petersburg Mathematical Society. Vol. VII. Transl. from the Russian by Tamara Rozhkovskaya. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 203, 1-15 (2001).
The paper describes connections between the theory of continuous (measurable) partitions and the asymptotic theory of finite partition lattices. In particular: (i) the Björner continuous partition lattice is realized as a subsemilattice of measurable partitions in the sense of V. A. Rokhlin; (ii) for any homogeneous equivalence relation a continuous lattice is constructed and a rank function on this lattice is defined, (iii) the representation of these lattices in the von Neumann factors of type \(\text{II}_1\) is described. This representation generalizes linear realizations of continuous matroids.
For the entire collection see [Zbl 0968.00033].

MSC:

06B35 Continuous lattices and posets, applications
28B05 Vector-valued set functions, measures and integrals
05A18 Partitions of sets

Citations:

Zbl 0811.06009
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