| Summary: | Let G be a finite transitive permutation group on
$\Omega $
. The G-invariant partitions form a sublattice of the lattice of all partitions of
$\Omega $
, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an orthogonal block structure, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a poset block structure, whose automorphism group is a generalised wreath product. We examine permutation groups with these properties, which we call the OB property and PB property respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties.
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