One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical Bezout's Theorem implies the Algebraic Bezout. 


What properties are there known that are true (or might be) in tropical geometry that don't imply that their algebraic version is true?



Properties from Tropical Geometry that do not imply their algebraic counterpart.
There is a simple nice fact which holds in the tropical plane that has no counterpart in algebraic geometry (nor in any kind of standard geometry I might think of): two tropical lines always "intersect" in a single point... even if they coincide!


Of course this property relies on the fact that "intersection" is not defined in the usual way. We define the "intersection" of two tropical curves $C\_1$ and $C\_2$ as follows: the union $C\_1\cup C\_2$ has a natural cellularization into vertices and edges, and "$C\_1\cap C\_2$" is the union of the vertices contained in the set-theoretic intersection $C\_1\cap C\_2$. One may also define a multiplicity on each intersection point. With this definition, the intersection of a tropical curve with itself is the union of its vertices.


Therefore two (possibly coinciding) tropical lines always intersect in a point. By defining anagously an appropriate (dual) notion of "span", the dual sentence is also true: two (possibly coinciding) points always span a single line.


