# semiring

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 (http://planetmath.org/Distributivity) 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 (http://planetmath.org/Idempotency) operation, then $\preceq$ is a partial order. Addition and (left and right) multiplication are order-preserving operators (http://planetmath.org/Poset).