  Fairness in AI has garnered quite some attention in research, and
increasingly also in society. The so-called "Impossibility Theorem" has been
one of the more striking research results with both theoretical and practical
consequences, as it states that satisfying a certain combination of fairness
measures is impossible. To date, this negative result has not yet been
complemented with a positive one: a characterization of which combinations of
fairness notions are possible. This work aims to fill this gap by identifying
maximal sets of commonly used fairness measures that can be simultaneously
satisfied. The fairness measures used are demographic parity, equal
opportunity, false positive parity, predictive parity, predictive equality,
overall accuracy equality and treatment equality. We conclude that in total 12
maximal sets of these fairness measures are possible, among which seven
combinations of two measures, and five combinations of three measures. Our work
raises interest questions regarding the practical relevance of each of these 12
maximal fairness notions in various scenarios.
