Idempotent Semigroups and Tropical Algebraic Sets

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.