Free monoid on 1 generator: f̂(aʲ · aᵏ) − (f̂(aʲ) + f̂(aᵏ)) ≡ 0

Free monoid on 1 generator: F_{Mon}({a}) ≅ (N, +, 0) via aᵏ ↔ k. Universal property with target (Z, +): for any n ∈ Z, the map f: {a} → Z, a ↦ n, extends uniquely to the homomorphism f̂: N → Z, aᵏ ↦ n·k. Homomorphism property: f̂(aʲ ·…