semiring

right annihilator

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 $(R,+)$ isrequired 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 $⪯$given by $a⪯b$ 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 thanany other element. Thus if $A$ has a nonzero element $a$ with an additiveinverse, then the elements $-a$, $0$, $a$ form a cycle with respect to $⪯$.If $+$ is an idempotent (http://planetmath.org/Idempotency) operation,then $⪯$ is a partial order.Addition and (left and right) multiplication areorder-preserving operators (http://planetmath.org/Poset).