individe

Individe is a theoretical operation used to partition a structure into its most basic, non-decomposable components under a specified decomposition relation. The term blends indivisible and divide, and it appears in speculative mathematics and discussions about data organization and modular design.

Formally, let X be a structure and D a relation that describes allowable subdecompositions (such as a substructure relation, a similarity predicate, or another criterion for breaking X apart). An individe of X with respect to D is a decomposition of X into D-atomic blocks—blocks that admit no nontrivial D-subdecomposition. Existence and uniqueness of an individe depend on the nature of X and the chosen relation D. In well-behaved settings, the D-atomic decomposition is unique up to isomorphism and permutation of blocks; in others, it may fail to exist or be non-unique.

Examples and analogies help intuition. In graph theory, if D corresponds to the relation of connectivity, the individe of a graph is its partition into connected components. In data analysis, using a homogeneity or similarity criterion as D can yield a set of clusters that cannot be further split without violating the criterion. In algebra, choosing D as a subobject relation may lead to a decomposition into simple or irreducible subobjects, analogous to prime or atomic factorization in other contexts.

See also: partition, decomposition, irreducible, atoms, connected components, factorization. Individe is a theoretical construct and is not a standardized term in all mathematical communities.