## Journal of Commutative Algebra

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

### 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}.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 1 August 2017

**Permanent link to this document**

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

**Subjects**

Primary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] 12K10: Semifields [See also 16Y60]

Secondary: 06D35: MV-algebras 16Y60: Semirings [See also 12K10] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

**Keywords**

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

#### 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