intervallejas

Intervallejas are a theoretical construct in order theory and data modeling that generalizes the classical notion of an interval. An intervalleja is a finite, linearly ordered set equipped with two distinguished endpoints and optional state labels on interior elements, used to represent a bounded segment with extra internal information.

Formally, an intervalleja I consists of a finite linearly ordered set (I, ≤) with endpoints L and R ∈ I such that L ≤ x ≤ R for all x ∈ I. Each x ∈ I carries a state s(x) from a secondary totally ordered set S. The interval determined by a ≤ b is {x ∈ I : a ≤ x ≤ b} with the induced state data. Endpoints may be included or excluded, yielding closed, open, or half-open intervallejas.

Operations on intervallejas mirror those on ordinary intervals but incorporate state data: union and intersection combine endpoint constraints and merge states with a defined join on S; refinement inserts new interior points while preserving order. Some accounts define a concatenation operation that glues adjacent intervallejas along matching endpoints.

Usage and scope: Intervallejas appear in theoretical discussions as a framework for analyzing interval-based data structures, for pedagogical illustration of endpoint versus interior information, and in speculative or fictional contexts as a generalized interval concept.

See also: Intervals; posets; lattices; interval algebra; order theory.