Congruences and coordinate semirings of tropical varieties
  In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, and the second based on the layered structure. We tie them to tropical geometry, especially in connection with the dimension of an affine variety. 
Zur Izhakian and Louis Rowen
math.AG math.AC math.RA