The Max-Plus Algebra of the Natural Numbers has no Finite Equational Basis

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