  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].
