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. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring^† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.