Monotonic computation rules for nonassociative calculus
Résumé
In this paper we revisit the so-called computation rules for calculus using a single nonassociative binary operation over possibly infinite sequences of integers. In this paper we focus on the symmetric maximum that is an extension of the usual maximum ∨ so that 0 is the neutral element, and −x is the symmetric (or inverse) of x, i.e., x (−x) = 0. However, such an extension does not preserve the associativity of ∨. This fact asks for systematic ways of bracketing terms of a sequence using , and which we refer to as computation rules. These computation rules essentially reduce to deleting terms of sequences based on the condition x (−x) = 0, and they can be quasi-ordered as follows: say that rule 1 is below rule 2 if for all sequences of numbers, rule 1 deletes more terms in the sequence than rule 2. As it turns out, this quasi-ordered set is extremely complex, e.g., it has infinitely many maximal elements and atoms, and it embeds the powerset of natural numbers by inclusion. Local properties of computation rules have also been presented by the authors, in particular, concerning their canonical representations. In this paper we address the problem of determining those computation rules that preserve the monotonicity of ∨, and present an explicit description of mono-tonic computation rules in terms of their factorized irredundant form. 1 Motivation This short contribution is the continuation of the work initiated in [1, 2], and we refer the reader to these references for further motivation. Let L be a totally ordered set with bottom element 0, and let −L := {−a : a ∈ L} be its "symmetric" copy endowed with the reversed order. Consider the symmetric ordered structureL := L ∪ (−L) \ {−0}, a bipolar scale analogous to the real line where the zero acts as a neutral element and such that a + (−a) = 0 (symmetry). In particular, −(−a) = a. The symmetric maximum is intended to extend the maximum on L with 0 as neutral element, while fulfilling symmetry. However, this symmetry requirement immediately entails that any extension of the maximum operator ∨ cannot be associative. To illustrate this point, let L = N and observe that (2 3) (−3) = 3 (−3) = 0 whereas 2 (3 (−3)) = 2 0 = 2. Nonetheless, Grabisch [3] showed that the "best" definition of (see Theorem 1 below) is: a b = −(|a| ∨ |b|) if b = −a and |a| ∨ |b| = −a or = −b 0 if b = −a |a| ∨ |b| otherwise. (1) In other words, if b = −a, then a b returns the element that is the larger in absolute value among the two elements a and b. Moreover, it is not difficult to see that satisfies the following properties:
Origine : Fichiers produits par l'(les) auteur(s)
Loading...