From list to free group

Given natural numbers \mathbb{N}, we could construct integers \mathbb{Z} by considering the equivalence relation on \mathbb{N} \times \mathbb{N} as follows,

  • (a, b)\sim(c, d) if a + d + r= b + c + r for some r

Obviously, it’s a equivalence relation from properties of commutativity. Thus we have equivalence class {[} ( a , b ){]} where a, b \in \mathbb{N}. Then, {[}(0, 0){]} = {[}(1, 1){]} = {[}(n , n){]}, and -{[}(a, b){]}={[}(b, a){]}. This construction from a commutative monoid to abelian group is called Grothendieck group. Continue reading