Lattice-ordered abelian groups finitely generated as semirings

Abstract

A lattice-ordered group (an $\ell $-group) $G(\oplus , \vee , \wedge )$ can naturally be 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~\cite {BCM}. Then, we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell $-groups associated to rooted trees (Theorem \ref {classify}).

This classification result has a number of interesting 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~\ref {main-conj} discussed in \cite {BHJK, JKK}.