Lattice-ordered abelian groups finitely generated as semirings

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].

16 pages; revised and slightly extended version

References in corpus (1)

• Representation of Perfect and Local MV-algebras

Cited by in corpus (1)

• Idempotence of finitely generated commutative semifields