### Abstract

Original language | English |
---|---|

Pages (from-to) | 489-520 |

Number of pages | 32 |

Journal | Journal of the European Mathematical Society |

Volume | 14 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

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### Keywords

- tropical geometry
- polyhedral complexes
- tropical polynomials
- idempotent semigroups
- simple polynomials

### Cite this

*Journal of the European Mathematical Society*,

*14*(2), 489-520. https://doi.org/10.4171/JEMS/309

**Idempotent semigroups and tropical algebraic sets.** / Izhakian, Zur; Shustin , Eugenii .

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 14, no. 2, pp. 489-520. https://doi.org/10.4171/JEMS/309

}

TY - JOUR

T1 - Idempotent semigroups and tropical algebraic sets

AU - Izhakian, Zur

AU - Shustin , Eugenii

PY - 2012

Y1 - 2012

N2 - The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical polynomials, formed from univariate monomials, define subsemigroups with respect to coordinate-wise tropical addition (maximum); and, finally, we prove that the subsemigroups in the Euclidean space which are either tropical hypersurfaces, or tropical curves in the plane or in the three-space have the above polynomial description.

AB - The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical polynomials, formed from univariate monomials, define subsemigroups with respect to coordinate-wise tropical addition (maximum); and, finally, we prove that the subsemigroups in the Euclidean space which are either tropical hypersurfaces, or tropical curves in the plane or in the three-space have the above polynomial description.

KW - tropical geometry

KW - polyhedral complexes

KW - tropical polynomials

KW - idempotent semigroups

KW - simple polynomials

U2 - 10.4171/JEMS/309

DO - 10.4171/JEMS/309

M3 - Article

VL - 14

SP - 489

EP - 520

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 2

ER -