This paper deals with a domain of Artificial Intelligence known under the name of "qualitative simulation" or "qualitative physics", to which special volumes of "Artificial Intelligence" (1984) and or "IEEE Transactions on Systems, Man and Cybernetics" (1987) have been devoted.
It defines the concept of "qualitative frame" of a set, which allows to introduce strict, large and dual confluence frames of a finite dimensional vector-space.
After providing a rigorous definition of standard, lower and upper qualitative solutions in terms of confluences introduced by De Kleer, it provides a duality criterion for the existence of a strict standard solution to both linear and non linear equations.
It also furnishes a dual characterization of the existence or upper and lower qualitative solutions to a linear equation.
These theorems are extended to the case or "inclusions", where single-valued maps are replaced by set-valued maps. This may be useful for dealing with qualitative properties of maps which are not precisely known, or which are defined by a set of properties, a requirement which is at the heart of qualitative simulation
