Averaging operators on the unit interval

Averaging Operators on the Unit Interval Mai Gehrke, Carol Walker, Elbert Walker New Mexico State University, Las Cruces, New Mexico, USA In this paper we look at algebras consisting of the unit interval [0; 1] with the standard partial order together with an averaging operator. In working with negations and t-norms, it is not uncommon to call upon the arithmetic of the real numbers even though that is not part of the structure of the unit interval as a bounded lattice. In particular, the operation of taking the average of two automorphisms of the unit interval is often called upon. In order to develop a self-contained system, we make the following de nition. DEFINITION An averaging operator is a binary operation _ + on the unit interval that (1) is commutative; (2) is strictly increasing in each variable; (3) is convex (continuous); (4) takes values between the two elements it is applied to; (5) satis es the pairwise exchange property (x _ +y) _ +(u _ +v) = (x _ +u) _ +(y _ +v). It follows from (4) that all averaging operators are idempotent. These are not \weighted" averages in the usual sense, although they share some of the basic properties. The averaging operators can be thought of as \skewed" averages. They provide a (continuous) scaling of the unit interval that is not provided by the lattice structure. We show that all averaging operators are isomorphic to the arithmetic mean by constructing an automorphism f of the unit interval (a generator for the averaging operator) that takes the given average of two elements x and y to the arithmetic mean of f(x) and f(y). The system consisting of the ordered unit interval together with an averaging operator has no nontrivial automorphisms. We show that each averaging operator on the unit interval naturally de nes a negation by the property: x _ + (x) = 0 _ +1. We relate properties of the averaging operator to properties of the nilpotent t-norms that determine the same negation. The Lukasiewicz t-norm and the arithmetic mean both lead to the standard negation 1 x. What happens in the general case? A given averaging operator on the unit interval induces an averaging operator on the group of automorphisms (and also on the set of antiautomorphisms) of the unit interval. We use these induced averaging operators to de ne special maps from the set of negations to the automorphism group of the unit interval, and from the automorphism group of the unit interval onto the centralizer of the negation . In this new setting, we generalize a number of the theorems from in an earlier paper (de Morgan Systems On the Unit Interval) where we proved those theorems for the arithmetic mean and its corresponding negation 1 x.