Residueja

Residueja is a term used in mathematical and computational literature to denote a structured set of residual values that emerge when a process partitions a domain into equivalence classes and retains additional information within each class. The word combines the English residue with a plural suffix -ja, a stylistic choice found in some theoretical writings.

Definition and construction. Given a set X and a function f: X -> Y, the domain X is partitioned into fibers f^{-1}(y) for y in Y. A residueja is the paired construction (y, r_y) in which r_y is a data container that records residual information associated with the fiber f^{-1}(y). The collection of all r_y over y in Y forms the residueja corresponding to f. In the simplest form, r_y may be the cardinality of the fiber or a representative statistic; in richer models it may include distributions, moments, or a stored sample of elements from the fiber.

Properties. Residuejas support a projection-like operation that maps the residueja to a coarser partition while preserving the residual data, enabling analysis of how residual information evolves under changes to the partition. The framework is designed to be compatible with standard notions of equivalence relations, partitions, and fiber structure, while allowing extra structure to be attached to each fiber.

Applications and scope. The concept is used in theoretical computer science to model state aggregation and in data encoding to manage residual statistics. It also appears in algebraic combinatorics and in abstract modeling of hierarchical data where both the primary classification and secondary residue data are relevant.

See also. Residue, Residue class, Equivalence relation, Partition.