Metapartition

Metapartition is a term used in some theories to denote a partition of partitions or a partition at a higher level than a standard set partition. In its simplest form, given a set X and a partition P = {B_i} of X, a metapartition M may refer to a partition of the index set I labeling the blocks, or equivalently a partition of the blocks into meta-blocks. This yields a two-level decomposition of X: X is partitioned into blocks B_i, and the blocks are grouped into meta-blocks J_k by M.

Formal variants: 1) Metapartition of a partition: M = {J_k} is a partition of I; the meta-blocks J_k correspond to groups of blocks. 2) Metapartition of a family of partitions: given a collection {P^t} of partitions of X, a metapartition is a partition of the collection into classes of partitions that share a specified property, such as similar block structure or similarity measure.

Applications: In ensemble clustering, metapartitions describe a higher-order grouping of clusterings to assess consensus or stability. In knowledge organization and data modeling, metapartitions capture multi-level taxonomies or hierarchies.

Relation to related concepts: It relates to the lattice of partitions, where partitions are ordered by refinement, and to meta-analysis when combining results across partitions.

See also: partition, equivalence relation, lattice of partitions, ensemble clustering, meta-analysis.