The max-plus algebra of the natural numbers has no finite equational basis

Abstract

This paper shows that the collection of identities which hold in the algebra $\text{N}$ of the natural numbers with constant zero, and binary operations of sum and maximum is not finitely based. Moreover, it is proven that, for every n, the equations in at most n variables that hold in $\text{N}$ do not form an equational basis. As a stepping stone in the proof of these facts, several results of independent interest are obtained. In particular, explicit descriptions of the free algebras in the variety generated by $\text{N}$ are offered. Such descriptions are based upon a geometric characterization of the equations that hold in $\text{N}$, which also yields that the equational theory of $\text{N}$ is decidable in exponential time.