On Sublattices of Partition Lattices

Author: Chase, Phillip John

Year: 1965

Degree: Dissertation (Ph.D.)

Advisor: Dilworth, Robert P.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/WY75-ZV20

Abstract

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We characterize strongly independent sets in an arbitrary geometric lattice in terms of properties of minimal dependent sets of points. The minimal dependent sets of points in partition lattices are identified. It turns out that strongly independent sets in the partition lattice on S correspond in a one - one fashion with systems of subsets of S characterized by certain properties. Lattice properties of partitions are obtained through application of a careful study of these subset systems. We show, for example, that the complete sublattice of a partition lattice generated by the ideals corresponding to a strongly independent set is isomorphic to the direct union of those ideals. Necessary and sufficient conditions are given for two ideals [alpha]/0 and [beta]/0 in a partition lattice to generate the entire ideal [alpha] [union] [beta]/0. The problem dual to this one is also solved. We characterize a large class of complete sublattices of a partition lattice, namely, those in which the union of all of the points of the partition lattice contained in the sublattice is the unit partition. The characterization takes the form of a system of subsets of S, of the type mentioned above, together with a suitable equivalence relation between the subsets comprising that system.

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