Idempotent semigroups and tropical algebraic sets

Izhakian, 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 -