Semiring.js

A semiring is a mathematical construct that abstracts some notion of "plus" and "times" on mathematical objects. These objects can be numbers, but may also be other objects such as strings.

More formally, a semiring is a set $R$ equipped with two binary operations $\oplus$ and $\otimes$ , called addition and multiplication, such that:

• $(R, \oplus)$ is a commutative monoid with identity element $\bar 0$ :
  • $(a \oplus b) \oplus c = a \oplus (b \oplus c)$
  • $\bar0 \oplus a = a \oplus 0 = a$
  • $a+b=b+a$
• $(R, \otimes)$ is a monoid with identity element $\bar1$:
  • $(a\otimes b)\otimes c=a\otimes (b\otimes c)$
  • $\bar1 \otimes a=a\otimes \bar1 = a$
• Multiplication left and right distributes over addition:
  • $a \otimes (b+c)=(a\otimes b)+(a\otimes c)$
  • $(a+b)\otimes c=(a\otimes c)+(b\otimes c)$
• Multiplication by $\bar 0$ annihilates R:
  • $\bar0 \otimes a=a\otimes \bar0 = \bar0$

Different semirings are useful for different purposes. Select a semiring below to get more information.