Journal of Commutative Algebra

Lattice-ordered abelian groups finitely generated as semirings

Vítězslav Kala

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}.

Article information

Source
J. Commut. Algebra, Volume 9, Number 3 (2017), 387-412.

Dates
First available in Project Euclid: 1 August 2017

https://projecteuclid.org/euclid.jca/1501574428

Digital Object Identifier
doi:10.1216/JCA-2017-9-3-387

Mathematical Reviews number (MathSciNet)
MR3685049

Zentralblatt MATH identifier
1373.06022

Citation

Kala, Vítězslav. Lattice-ordered abelian groups finitely generated as semirings. J. Commut. Algebra 9 (2017), no. 3, 387--412. doi:10.1216/JCA-2017-9-3-387. https://projecteuclid.org/euclid.jca/1501574428

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