## Quick Reference

A semiring *S* with two additional properties:*a*_{1},*a*_{2},…,*a** _{n}*,… is a countable sequence of elements of

*S*then

*a*

_{1}+

*a*

_{2}+ … +

*a*

*+ …,exists and is unique; the order in which the various elements are added is irrelevant;*

_{n}*a*in

*S*, powers can be defined in the expected manner:

*a*

^{0}= 1

*a*

*=*

^{n}*a*·

*a*

^{n}^{-1}for all

*n*> 0 Then the closure

*a** can be defined as follows:

*a** = 1 +

*a*+

*a*

^{2}+ … +

*a*

*+ … The properties of a semiring imply that*

^{n}*a** = 1 +

*a*·

*a**

*a*_{1},*a*_{2},…,*a** _{n}*,… is a countable sequence of elements of

*S*then

*a*

_{1}+

*a*

_{2}+ … +

*a*

*+ …,exists and is unique; the order in which the various elements are added is irrelevant;*

_{n}*a*_{1} + *a*_{2} + … + *a** _{n}* + …,

*a*^{0} = 1

*a** ^{n}* =

*a*·

*a*

^{n}^{-1}for all

*n*> 0

*a** = 1 + *a* + *a*^{2} + … + *a** ^{n}* + …

*a** = 1 + *a*·*a**

Closed semirings have applications in various branches of computing such as automata theory, the theory of grammars, the theory of recursion and fixed points, sequential machines, aspects of matrix manipulation, and various problems involving graphs, e.g. finding shortest-path algorithms within graphs.

*Subjects:*
Computing.