Lattice-ordered abelian groups finitely generated as semirings

TL;DR

This paper classifies abelian lattice-ordered groups that are finitely generated as semirings, revealing their structure in terms of rooted trees and applying this to ideal-simple semirings.

Contribution

It provides a complete classification of finitely generated abelian ll-groups as semirings using rooted tree structures, extending previous results.

Findings

Classification in terms of rooted trees (Theorem 4.1)

Implications for finitely generated ideal-simple semirings

Progress towards Conjecture 1.1 in semiring theory

Abstract

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing that each such $\ell$-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici [8]. Then we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell$-groups associated to rooted trees (Theorem 4.1). This classification result has a number of important applications: for example it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot)$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture 1.1 discussed in [2], [15].