Addition

Addition is the operation that combines two natural numbers into their sum. It is defined recursively using the successor function: n + 0 = n (adding zero does nothing) and n + S(m) = S(n + m) (adding a successor is the successor of adding). These two clauses, together with the induction principle, determine addition for all natural numbers.

Addition is commutative (a + b = b + a), associative ((a + b) + c = a + (b + c)), and has zero as its identity element. The natural numbers under addition form a commutative monoid. Extending to the integers (by including additive inverses) gives a commutative group.

Addition extends to larger number systems — integers, rationals, reals, complex numbers — always preserving commutativity and associativity. In a lattice, join plays a role analogous to addition: the join of two elements is their least upper bound, combining their information. In the Heyting algebra H, the join a ∨ b is the semantic analogue of combining two values.