Multiplication

Multiplication is the operation that combines two natural numbers into their product. It is defined recursively using addition and the successor function: n × 0 = 0 and n × S(m) = n + (n × m). Multiplication reduces to repeated addition, just as addition reduces to repeated successor.

Multiplication is commutative (a × b = b × a), associative ((a × b) × c = a × (b × c)), and distributes over addition (a × (b + c) = a × b + a × c). The number 1 is the identity element: n × 1 = n. Together with addition, multiplication makes the natural numbers a commutative semiring.

In a lattice, meet plays a role analogous to multiplication: it is the “conjunctive” operation that takes two elements and produces their greatest lower bound. The distributive law in a distributive lattice (a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)) mirrors how multiplication distributes over addition. In a Heyting algebra, this distributivity is guaranteed by the lattice structure.