Ickemaximal

Ickemaximal is a coined term used in theoretical discussions of optimization and decision making to denote an element that is maximal with respect to a primary order while also optimizing a secondary quality measure referred to as the 'ick' score.

Definition: Let X be a set equipped with a partial order ≤ and let s: X → R be a real-valued score function called the ick score. An element x ∈ X is ickemaximal if x is maximal under ≤, and among all elements that are maximal under ≤, s(x) is as large as possible.

Existence and properties: In a finite set, a ickemaximal element exists whenever there is at least one maximal element, and the maximum of s over the set of maximal elements exists if s is real-valued. In infinite or disconnected spaces, existence may require compactness or boundedness assumptions.

Example: Consider X = {a, b, c} with the order defined by a ≤ b and a ≤ c, and no relation between b and c (they are incomparable). The maximal elements are b and c. If s(b) = 7 and s(c) = 5, then b is ickemaximal.

Relation to other concepts: Ickemaximal combines features of maximal elements in order theory with a secondary criterion akin to a scoring function used in multi-criteria decision making. It is related to Pareto efficiency when the secondary score is treated as a tiebreaker among maximal elements.

Applications and usage: The term is used primarily in theoretical expositions and thought experiments, rather than widely adopted in formal literature. It can serve as a conceptual tool for analyzing selection rules that require both a maximal position and high quality by an additional metric.

See also: maximal element, Pareto efficiency, multi-criteria optimization.