(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.
"closed semiring." A Dictionary of Computing. . Encyclopedia.com. (March 21, 2019). https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/closed-semiring
"closed semiring." A Dictionary of Computing. . Retrieved March 21, 2019 from Encyclopedia.com: https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/closed-semiring
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:
Modern Language Association
The Chicago Manual of Style
American Psychological Association
- Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
- In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.