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

Authors:
Luca Aceto;Zoltán Ésik;Anna Ingólfsdóttir
Affiliations:
Basic Research in Computer Science (BRICS), Department of Computer Science, Centre of the Danish National Research Foundation, Aalborg University, Fr. Bajersvej 7E, 9220 Aalborg Ø, Denmark;Department of Computer Science, A. József University, Árpád tér 2, 6720 Szeged, Hungary;Basic Research in Computer Science (BRICS), Department of Computer Science, Centre of the Danish National Research Foundation, Aalborg University, Fr. Bajersvej 7E, 9220 Aalborg Ø, Denmark
Venue:
Theoretical Computer Science
Year:
2003

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Abstract

This paper shows that the collection of identities which hold in the algebra 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 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 N are offered. Such descriptions are based upon a geometric characterization of the equations that hold in N, which also yields that the equational theory of N is decidable in exponential time.