On the poset of computation rules for nonassociative calculus

Miguel Couceiro () and Michel Grabisch
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Miguel Couceiro: LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique

Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) from HAL

Abstract: 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.

Date: 2013
Note: View the original document on HAL open archive server: https://hal.science/hal-00787750v1
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Published in Order, 2013, pp.269-288

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Working Paper: On the poset of computation rules for nonassociative calculus (2013)
Working Paper: On the poset of computation rules for nonassociative calculus (2013)
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