Maximal superiority is generated through constrained maximals.

The notions of secure and perfect maximality are introduced.

Conditions for the existence of secure and perfects maximals are shown.

The structure of secure and perfect maximals is examined.

The paper introduces a refinement of maximality, called secure maximality, and a refinement of secure maximality, called perfect maximality. The effectivity of these refinements and the connection with other relevant optimality notions are investigated. Furthermore, necessary and sufficient conditions are provided for the secure maximality of all maximals and for the perfect maximality of all maximals as well as for the perfect maximality of all secure maximals. Several sufficient conditions for (as well as two characterizations of) the existence of secure and perfect maximals are established. The precise structure of the entire sets of secure and perfect maximals is examined for some specific classes of relations like interval orders that admit a certain type of representability by means of two real-valued functions, relations induced by cones and relations that admit linear multi-utility representations.

Maximal

Secure maximal

Perfect maximal

Right trace

Existence

Representation

© 2025 The Author. Published by Elsevier Inc.