The "Semicontinuity" aspect typically refers to the behavior of dimensions (like the rank of a semimodule) under deformations.

Unlike traditional modules over a ring, are defined over semirings (like the

A key feature is the adaptation of and Tor functors. Since you cannot always "subtract" to find boundaries, homological algebra here often uses:

This framework provides the "linear algebra" for tropical varieties. Just as homological algebra helps classify manifolds, semimodule homology helps classify and understand the intersections of tropical hypersurfaces.