(a) if a1,a2,…,an,… is a countable sequence of elements of S then a1 + a2 + … + an + …,
exists and is unique; the order in which the various elements are added is irrelevant;
(b) the operation · (see semiring) distributes over countably infinite sums as well as finite sums.
A special unary operation called closure can be defined on closed semirings. Given an element a in S, powers can be defined in the expected manner: a0 = 1 an = a·an–1 for all n > 0
Then the closure a* can be defined as follows:
a* = 1 + a + a2 + … + an + …
The properties of a semiring imply that 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.
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