tieredordered

Tieredordered is a term used to describe a structured ordering in which elements are partitioned into discrete tiers or levels, and an explicit order is maintained within each tier. In a tieredordered system, the primary criterion for comparing two elements is their tier; elements in lower-numbered tiers come before those in higher-numbered tiers. If two elements share the same tier, a secondary, intra-tier order determines their relative position. The concept is a natural generalization of lexicographic order, combining a tier function with an internal comparator.

Formalization: Let X be a set, T: X -> {1,...,k} assign a tier to each element, and ≤i be an order within each tier. Then a ≤tiered X b if T(a) < T(b) or (T(a) = T(b) and a ≤i b).

Applications: tieredordered structures appear in data sorting, priority-based task queues, and user interfaces that present items grouped by priority with a defined sequence inside each group. They are also used in reporting and analytics where categories (tiers) must be shown in a fixed order but items within each category should be sortable by date, value, or other metrics.

Examples: A to-do list may label tasks with tiers 1–3 by urgency; within each tier tasks are ordered by due date. A product catalog may group items by stock status with an internal price-based order.

See also: lexicographic order, hierarchical ordering, multi-criteria sorting, posets.