On the poset of computation rules for nonassociative calculus - Université Paris 1 Panthéon-Sorbonne Accéder directement au contenu
Article Dans Une Revue Order Année : 2013

On the poset of computation rules for nonassociative calculus

Résumé

The symmetric maximum, denoted by $\svee$, is an extension of the usual maximum $\vee$ operation so that 0 is the neutral element, and $-x$ is the symmetric (or inverse) of $x$, i.e., $x\svee(-x)=0$. However, such an extension does not preserve the associativity of $\vee$. This fact asks for systematic ways of parenthesing (or bracketing) terms of a sequence (with more than two arguments) when using such an extended maximum. We refer to such systematic (predefined) ways of parenthesing as computation rules. As it turns out there are infinitely many computation rules each of which corresponding to a systematic way of bracketing arguments of sequences. Essentially, computation rules reduce to deleting terms of sequences based on the condition $x\svee(-x)=0$. This observation gives raise to a quasi-order on the set of such computation rules: 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. In this paper we present a study of this quasi-ordering of computation rules. In particular, we show that the induced poset of all equivalence classes of computation rules is uncountably infinite, has infinitely many maximal elements, has infinitely many atoms, and it embeds the powerset of natural numbers ordered by inclusion.
Fichier principal
Vignette du fichier
order11-2.pdf (218.01 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00787750 , version 1 (13-02-2013)

Identifiants

  • HAL Id : hal-00787750 , version 1

Citer

Miguel Couceiro, Michel Grabisch. On the poset of computation rules for nonassociative calculus. Order, 2013, pp.269-288. ⟨hal-00787750⟩
251 Consultations
223 Téléchargements

Partager

Gmail Facebook X LinkedIn More