I'm a beginner in tropical geometry, and I'm running into the following question:

In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether an element of the ring in in an ideal given by some generators.

Suppose now I'm looking at an polynomial algebra over the max-plus semi-ring, and I have some relations, say $f_i = g_i$. Are there algorithms known (or better yet, has someone already written code) to determine whether a given relation, say $h=k$, lies in the the equivalence relation generated by the $f_i=g_i$?

I put "ideal" in quotes because as I understand it, kernels of semi-ring maps can not be thought of as subsets of the domain as in the case of rings, but rather are just equivalence relations.

So far Google searching hasn't turned up anything promising.