A semiring is a set $A$ with two operations, $+$ and $\cdot$ such that $0\in A$ makes $(A,+)$ into a commutative monoid, $1\in A$ makes $(A,\cdot)$ into a monoid, the operation $\cdot$ distributes over $+$ and for any $a\in A$ $0\cdot a=a\cdot 0=0$ Usually, $a\cdot b$ is instead written $ab$

A ring $(R,+,\cdot)$ can be described as a semiring for which $(R,+)$ is required to be a group. Thus every ring is a semiring. The natural numbers $\mathbb{N}$ form a semiring, but not a ring, with the usual multiplication and addition.

Every semiring $A$ has a quasiorder $\preceq$ given by $a\preceq b$ if and only if there exists some $c\in A$ such that $a+c=b$ Any element $a\in A$ with an additive inverse is smaller than any other element. Thus if $A$ has a nonzero element $a$ with an additive inverse, then the elements $-a$ $0$ $a$ form a cycle with respect to $\preceq$ If $+$ is an idempotent operation, then $\preceq$ is a partial order. Addition and (left and right) multiplication are order-preserving operators.