Measure and integral with purely ordinal scales

We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 0 on the real interval [−1, 1]. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and their inner product. 1 Motivation and survey Measuring and aggregation or integration techniques have a very long tradition. Here numbers play an important role. But how do humans perceive numbers? The numbers, say the set R of reals, support two basic structures, the algebraic structure defined by + and ×, and the ordinal structure given by ≤. There are many situations where only order is relevant, cardinals being used merely by tradition and convenience. During the last years the interest in ordinal aggregation has increased, see e.g. [4, 5, 6, 14, 19, 20]. We are interested in the question if aggregation or integration can be done in purely ordinal terms and what results can be obtained. Of course many partial results are already available. Since often they are formulated in terms of numbers, we ask what can be sustained if one ignores the algebraic structure or what weaker additional structure has to be imposed on the linear ordinal scale in order to formulate some well known important issues. It turns out that enough structure is given by an order reversing bijection of the scale leaving one point fixed. Thus the scale decomposes into two symmetric parts. This can be interpreted as the first step to numbers. Since, repeating the procedure with each resulting part of the scale infinitely often, one ends up with the