On the poset of computation rules for nonassociative calculus

TL;DR

This paper studies the structure of computation rules for nonassociative extensions of the maximum operation, revealing an uncountably infinite poset with complex ordering properties.

Contribution

It introduces a quasi-order on computation rules for nonassociative max extensions and characterizes the poset's structure, including its infinite size and embedding properties.

Findings

The poset of equivalence classes of computation rules is uncountably infinite.

There are infinitely many maximal elements and atoms in the poset.

The poset embeds the powerset of natural numbers ordered by inclusion.

Abstract

The symmetric maximum, denoted by v, is an extension of the usual max operation so that 0 is the neutral element, and -x is the symmetric (or inverse) of x, i.e., x v(-x)=0. However, such an extension does not preserve the associativity of max. 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 v(-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.