Semirings (Mathematics)

In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction ∨ {\displaystyle \lor } as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers N {\displaystyle \mathbb {N} } (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.