Lattice-ordered abelian groups finitely generated as semirings

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].