# Idempotent semigroups and tropical algebraic sets

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 2, page 489-520
- ISSN: 1435-9855

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topIzhakian, Zur, and Shustin, Eugenii. "Idempotent semigroups and tropical algebraic sets." Journal of the European Mathematical Society 014.2 (2012): 489-520. <http://eudml.org/doc/277527>.

@article{Izhakian2012,

abstract = {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.},

author = {Izhakian, Zur, Shustin, Eugenii},

journal = {Journal of the European Mathematical Society},

keywords = {tropical geometry; polyhedral complexes; tropical polynomials; idempotent semigroups; simple polynomials; tropical geometry; polyhedral complexes; tropical polynomials; idempotent semigroups; simple polynomials},

language = {eng},

number = {2},

pages = {489-520},

publisher = {European Mathematical Society Publishing House},

title = {Idempotent semigroups and tropical algebraic sets},

url = {http://eudml.org/doc/277527},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Izhakian, Zur

AU - Shustin, Eugenii

TI - Idempotent semigroups and tropical algebraic sets

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 2

SP - 489

EP - 520

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.

LA - eng

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

UR - http://eudml.org/doc/277527

ER -

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