In 2016, Diane Maclagan and Felipe Rincón arrived at a seemingly natural definition of what an ideal should be in the Tropical Semiring - i.e. they satisfy a certain Monomial Elimination Axiom. As with most combinatorial/computational based algebra, we want to make explicit computations with these ideals. My question is,
- Do we yet have a good definition for the Colon Ideal $I:J$ of two tropical ideals $I$ and $J$?
- What about saturations and intersections?
Tropical Geometry has been a surprisingly fruitful field, but I get the impression that a large part of the foundations are yet to be decided. Has there been any meaningful progress on concepts like this in the last 7 years?