Finite basis classification between the TR₆ variety and the finite truncation variety

Determine which subvarieties of the max-plus variety that contain the variety generated by the six-element semiring TR₆, lie below the variety generated by the finite truncation A₂, and are not contained in the greatest hereditarily finitely based subvariety are finitely based; in particular, determine all finitely based subvarieties in the interval [V(TR₆), V(A₂)].

Background

The paper establishes that V(TR₆) is the unique limit subvariety of the max-plus variety V(N), while the greatest hereditarily finitely based subvariety is characterized separately. It also constructs a descending sequence of finitely based varieties whose intersection is V(TR₆), showing that finite basedness is not preserved under such intersections.

The unresolved classification concerns the intermediate subvarieties containing V(TR₆) and lying below V(A₂), particularly those outside the already classified hereditarily finitely based portion. The paper notes that all varieties in this range are proper subvarieties of V(N) and therefore locally finite, but local finiteness does not settle finite basedness.