Sandpile groups and Laplacian matrices

Proof: (R ⊂ κ(G, s)). Let I ⊂ M(G, s) be a nonempty ideal. Fix ψ ∈ I. Given σ ∈ R, there exists a sandpile φ such that σ = (φ + ψ)◦ by defintion. Since I is an ideal, this implies that σ ∈ I. Hence R ⊂ I. We conclude that R ⊂ ⋂ I ideal of M(G,s) I = κ(G, s). (κ(G, s) ⊂ R). Recall that R 6= ∅. Since κ(G, s) is the minimal ideal of M(G, s), it is enough to show that R is an ideal. Consider σ ∈ R and τ ∈ M(G, s). We want to show that (σ+τ)◦ is recurrent. Let ψ ∈M(G, s) and choose a sandpile φ such that σ = (ψ+φ)◦. Set φ′ = φ+ τ . By the abelian property of sandpile stabilization, we have (ψ + φ′)◦ = (ψ + φ+ τ)◦ = ((ψ + φ)◦ + τ)◦ = (σ + τ)◦.