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 coordinatewise tropical addition (maximum); and, finally, we prove that the subsemigroups in Rn 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 coordinatewise tropical addition (maximum); and, finally, we prove that the subsemigroups in Rn which are either tropical hypersurfaces, or tropical curves in the plane or in the three-space have the above polynomial description.

KW - Idempotent semigroups

KW - Polyhedral complexes

KW - Simple polynomials

KW - Tropical geometry

KW - Tropical polynomials

UR - http://www.scopus.com/inward/record.url?scp=84857338485&partnerID=8YFLogxK

U2 - 10.4171/JEMS/309

DO - 10.4171/JEMS/309

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AN - SCOPUS:84857338485

SN - 1435-9855

VL - 14

SP - 489

EP - 520

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 2

ER -