Semiring

A semiring is a set equipped with two operations, addition (+) and multiplication (×), such that: (S, +, 0) is a commutative monoid, (S, ×, 1) is a monoid, multiplication distributes over addition from both sides, and 0 × a = a × 0 = 0. A semiring is like a ring but without requiring additive inverses — subtraction need not be possible.

The natural numbers are the prototypical semiring: they have addition and multiplication with the expected properties, but no negatives. Other examples include the nonnegative reals, the Boolean semiring ({0, 1} with OR and AND), and the tropical semiring (ℝ ∪ {∞} with min and +). Each captures a setting where “combining” and “scaling” make sense but “undoing” does not.

A ring is a semiring where (S, +, 0) is a group (additive inverses exist). A field is a ring where (S \ {0}, ×, 1) is also a group (multiplicative inverses exist for nonzero elements). The algebraic hierarchy — semiring → ring → field — adds increasingly strong inverse operations at each step.