Finitely maxitive T-conditional possibility theory: Coherence and extension

• Representation of a full T-conditional possibility through a (unique) T-nested class.
• Characterization of coherence for a T-conditional possibility assessment.
• Coherent extension of a coherent T-conditional possibility assessment.
• Topological properties of the set of coherent extensions.
• Problems related to conditioning for necessity measures.

Starting from the axiomatic definition of finitely maxitive T-conditional possibility (where T is a continuous triangular norm), the paper aims at a comprehensive and self-contained treatment of coherence and extension of a possibilistic assessment defined on an arbitrary set of conditional events. Coherence (or consistence with a T-conditional possibility) is characterized either in terms of existence of a linearly ordered class of finitely maxitive possibility measures (T-nested class) agreeing with the assessment, or in terms of solvability of a finite sequence of nonlinear systems for every finite subfamily of conditional events. Coherence reveals to be a necessary and sufficient condition for the extendibility of an assessment to any superset of conditional events and, in the case of T equal to the minimum or a strict t-norm, the set of coherent values for the possibility of a new conditional event can be computed solving two optimization problems over a finite sequence of nonlinear systems for every finite subfamily of conditional events.