M-TF equivalences in the real Grothendieck groups

For an abelian length category A with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence in the dual real Grothendeick group K₀(A)R^*=HomR(K₀(A)R,R), which are defined by semistable subcategories and semistable torsion pairs in A associated to elements θ ∈ K₀(A)R^*. In this paper, we introduce the M-TF equivalence for each object M ∈ A as a systematic way to coarsen the TF equivalence. We show that the set Σ(M) of the closures of M-TF equivalence classes is a finite complete fan in K₀(A)R^*, and that Σ(M) is the normal fan of the Newton polytope N(M) in K₀(A)R.