Abstract

It is proved that the free distributive lattice on three generators is the smallest lattice among all finite distributive lattices L which have a subspace (in a normed space) realization L′ and an operator T in Alg L′ which cannot be written as a sum of rank T rank one operators from Alg L′ . Also some information concerning the form of such a lattice is given. If, furthermore, T has rank two, then L contains a sublattice with two Boolean lattices with three atoms each, and one of them is below the other.

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