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||Mean according to a composition law
||Compatibility, Idempotence law, Lattice, Magma, n-ary application, Order isomorphism
||Algebra, Lattices, Orders and preorders, Statistics
||This article presents an algebraic model of the concept of mean, in the discontinuous and finite case, but not necessarily restricted to numerical calculation. The approach which has been used is strictly formal and axiomatic even though it is, of course, issued from empirical models i.e. Pythagorean means (type r classical means).
In view of possible applications in the human sciences, there has been an attempt to break away from strict numerical data, without excluding them entirely. We have determined a process which is able to embody the concept of average, by minimizing the mathematical tools needed for its application. As a starting point for its construction, we have used only two notions, those of latice, and of composition law. The first notion enables us to formalize the idea of intermediate value and the second the idea of aggregation according to the criterion of constant ; we think that these two principles make up the very foundation of the theory of averages.
We begin by studying the means which operate on two objects only. We set out a few elementary characteristics before proceeding with an examination of the obtention conditions. We then generalize to more than two objects by pointing out the barycentric characteristics and weighted means. |
||151, Fall 2000 |
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