The set of bisections forms a group, with the multiplication b 1 ⋅ ⋅ b 2 defined as ( b 1 ⋅ ⋅ b 2 ) ( x ) := b 1 ( b 2 ( x ) ) b 2 ( x ) . and inversion defined as b 1 − − 1 ( x ) := i ∘ ∘ b 1 ( ( t ∘ ∘ b 2 ) − − 1 ( x ) ) Note that the definition is given in such a way that, if t ∘ ∘ b 1 = ϕ ϕ 1 and t ∘ ∘ b 2 = ϕ ϕ 2 , then t ∘ ∘ ( b 1 ⋅ ⋅ b 2 ) = ϕ ϕ 1 ∘ ∘ ϕ ϕ 2 and t ∘ ∘ b 1 − − 1 = ϕ ϕ 1 − − 1 .