Idempotent semigroups and tropical algebraic sets

The tropical semifield, i.e., the real numbers enhanced by the operations ofaddition and maximum, serves as a base of tropical mathematics. Addition is anabelian group operation, whereas the maximum defines an idempotent semigroupstructure. We address the question of the geometry of idempotent semigroups, inparticular, tropical algebraic sets carrying the structure of a commutativeidempotent semigroup. We show that commutative idempotent semigroups arecontractible, that systems of tropical polynomials, formed from univariatemonomials, define subsemigroups with respect to coordinate-wise tropicaladdition (maximum); and, finally, we prove that the subsemigroups in theEuclidean space, which are tropical hypersurfaces or tropical curves in theplane or in the three-space, have the above polynomial description.