Definitions for

**"Semiring"**One of the incomplete rings of the upper part of the bronchial tubes of most birds. The semirings form an essential part of the syrinx, or musical organ, of singing birds.

an algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian monoid operation, rather than an abelian group

an algebra with two operations and Theta, satisfying the usual ring axioms, but not necessarily having additive inverses