The tropical Nullstellensatz for congruences

We develop the foundations of a theory of algebraic geometry for semirings, concentrating mainly on the semiring of tropical polynomials. Replacing ideals with the more general notion of congruences, we establish a relationship between congruences on the semiring of tropical polynomials and subsets of tropical space. We ultimately establish analogues of the weak and strong forms of Hilbert's Nullstellensatz.