Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. An algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
 n. One of the incomplete rings of the upper part of the bronchial tubes of most birds, forming an essential part of the syrinx of songbirds.
from the GNU version of the Collaborative International Dictionary of English
 n. 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.
from The Century Dictionary and Cyclopedia
 n. In zoology, a tracheal or bronchial halfring. See tracheal rings (under ring), and cut under pessulus.
Etymologies
Examples

Because of this asymmetry, the algebraic structure of tropical arithmetic is classified as a semiring (whereas ordinary arithmetic on \ (\mathcal {R} \) is a ring).

One of the things that Vicary shows is that, for a category with tensor unit $I$ and all finite dagger limits, the semiring of scalars $I \rightarrow I$ has a natural embedding into a characteristic zero field.

Strictly speaking, it's more like a semiring module monad as Int32 isn't a field.

Here's another bit of jargon for jargon's sake: the tropical semiring is idempotent.

As I understand it, Imre Simon and those French mathematicians who take a tropical view of Brazil were interested in the tropical semiring mainly in connection with automata theory.
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