Semiring

A semiring or rig is a "Ring without negatives". In other words, it is a Monoid Object in the Category of Commutative Monoids with respect to the Tensor Product of Commutative Monoids.

Explicitly, a semiring is a H-Set $A$ equipped with a pair of Binary Operations $+:A\times A\to A$, $\cdot:A\times A\to A$ and a pair of constants $0,1:A$ such that

• $(A,+,0)$ is a Commutative Monoid
• $(A,\cdot,1)$ is a Monoid
• $0\cdot x=0$
• $x\cdot 0=0$
• $x\cdot(y+z)=x\cdot y+x\cdot z$
• $(x+y)\cdot z=x\cdot z+y\cdot z$

Arguably, semirings are a more fundamental notion than Ring.

Examples

• The Natural Numbers are the Initial Object in the Category of Semirings.
• Bounded Lattices are semirings where both multiplication and addition are Idempotent.