Lattice-ordered abelian groups finitely generated as semirings
| Autor: | Vítězslav Kala |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
finitely generated
Lattice (group) Structure (category theory) MV-algebra semiring 01 natural sciences Semiring Combinatorics Lattice-ordered abelian group 06D35 0101 mathematics Abelian group Commutative property 16Y60 Mathematics parasemifield Group (mathematics) 010102 general mathematics 52B20 Mathematics - Commutative Algebra 010101 applied mathematics Primary 06F20 12K10 Secondary 06D35 16Y60 52B20 Idempotence 12K10 06F20 order-unit Mathematics - Group Theory |
| Zdroj: | J. Commut. Algebra 9, no. 3 (2017), 387-412 |
| ISSN: | 1939-2346 |
| DOI: | 10.1216/jca-2017-9-3-387 |
| Popis: | 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]. Comment: 16 pages; revised and slightly extended version |
| Databáze: | OpenAIRE |
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