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}).\n¶ 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}.