Dichotomy of the addition of natural numbers

TL;DR

This paper explores the arithmetic of trees and its connection to the Tamari poset, including methods to realize this poset as a polytope, contributing to the understanding of combinatorial structures in algebra.

Contribution

It introduces an elementary approach to tree arithmetic linked to the Tamari poset and investigates polytope realizations of this poset, including a method inspired by Tamari's thesis.

Findings

Establishes a relationship between tree arithmetic and the Tamari poset

Proposes various realizations of the Tamari poset as a polytope

Includes a novel approach inspired by Tamari's thesis

Abstract

This is an elementary presentation of the arithmetic of trees. We show how it is related to the Tamari poset. In the last part we investigate various ways of realizing this poset as a polytope (associahedron), including one inferred from Tamari's thesis.