partialrange

Partialrange is a concept used to describe a contiguous subset of a totally ordered set where one or both endpoints may be unspecified. It generalizes the idea of an interval by allowing incomplete information about its boundaries while preserving the notion of a consecutive block of elements or values. A partialrange has a lower bound, an upper bound, and flags indicating whether the endpoints are inclusive or exclusive. If a bound is not specified, that side of the range is unbounded.

Formally, let S be a totally ordered set with order <=. A partialrange is defined by a triple (L, U, incL, incU), where L and U may be elements of S or undefined, and incL, incU denote inclusivity for the lower and upper bounds. A value x belongs to the partialrange if it satisfies the bound conditions: it is not less than L when L is defined (taking into account incL), and it is not greater than U when U is defined (taking incU into account). When both bounds are defined and L > U, the partialrange is empty; if L = U, the range may be a single point or empty depending on inclusivity.

Examples include [5, ∞) (lower bound 5, upper bound undefined), (−∞, 10] (upper bound 10, lower bound undefined), and [3, 7] (both bounds defined, inclusive). Empty partialranges arise when the bounds clash with inclusivity.

Applications of partialrange include modeling range filters in databases, time windows in streaming systems, and index-based slicing where endpoints may be unknown at query planning time. It relates to standard interval notions but accommodates incomplete information about bounds, making it useful in planning, propagation of partial information, and flexible querying. See also interval, range, open interval, closed interval, and boundary conditions.