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2017 Lattice-ordered abelian groups finitely generated as semirings
Vítězslav Kala
J. Commut. Algebra 9(3): 387-412 (2017). DOI: 10.1216/JCA-2017-9-3-387

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}.

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Vítězslav Kala. "Lattice-ordered abelian groups finitely generated as semirings." J. Commut. Algebra 9 (3) 387 - 412, 2017. https://doi.org/10.1216/JCA-2017-9-3-387

Information

Published: 2017
First available in Project Euclid: 1 August 2017

zbMATH: 1373.06022
MathSciNet: MR3685049
Digital Object Identifier: 10.1216/JCA-2017-9-3-387

Subjects:
Primary: 06F20 , 12K10
Secondary: 06D35 , 16Y60 , 52B20

Keywords: finitely generated , Lattice-ordered abelian group , MV-algebra , order-unit , parasemifield , semiring

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.9 • No. 3 • 2017
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