  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 tropical hypersurfaces or tropical curves in the
plane or in the three-space, have the above polynomial description.
