Date: Tue, Nov 14, 2017
Time: 15:00 - 16:00
Venue: Middle Meeting Room
Title: Relations among partitions (I): Partition of a finite set
First I consider a single partition of a finite set. Usually it is uniform, which means that all of its parts have the same size. If it has $n$ parts and the set has size $N$ then we have an $N \times n$ incidence matrix and an $N \times N$ relation matrix. The partition defines an $n$-dimensional vector subspace of the $N$-dimensional space defined by the whole set, as well as the matrix of orthogonal projection onto that subspace.
There is a partial order on partitions called refinement, which is related to properties of the vector subspaces and their projection matrices. This leads to the definitions of the supremum and infimum of two partitions.
Orthogonality is a nice relation between two partitions. I will give some equivalent definitions. These lead to families of mutually orthogonal partitions, such as orthogonal arrays and orthogonal block structures.
Slides: View slides