Domain basis: Difference between revisions
Cmloegcmluin (talk | contribs) m fix link now that page exists and I can actually confirm them |
Cmloegcmluin (talk | contribs) →Terminology: interval basis vs. subgroup: "we" → "this article" in key places |
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== Simple vs. advanced math: linear algebra vs. group theory == | == Simple vs. advanced math: linear algebra vs. group theory == | ||
Regarding the choice between these two internally consistent versions of this term, then, | Regarding the choice between these two internally consistent versions of this term, then, this article prefers "subspace basis". This is because group theory is a relatively obscure and advanced field of mathematics, and this article prefers to leverage terminology from the more well-known and basic field of linear algebra whenever possible. | ||
"Subgroup" and "subspace" are indeed analogous, but due to differences between group theory and linear algebra, they are not completely synonymous. Essentially, group theory takes some of the convenient assumptions which we rely on when doing linear algebra and sets them aside. Doing so can be powerful, and some argue that RTT cannot be sufficiently described using only linear algebra. This article, however, prioritizes pedagogy of the basics over any potential considerations arising from such advanced RTT problems. | "Subgroup" and "subspace" are indeed analogous, but due to differences between group theory and linear algebra, they are not completely synonymous. Essentially, group theory takes some of the convenient assumptions which we rely on when doing linear algebra and sets them aside. Doing so can be powerful, and some argue that RTT cannot be sufficiently described using only linear algebra. This article, however, prioritizes pedagogy of the basics over any potential considerations arising from such advanced RTT problems. | ||
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== Generic math terms vs. specialized application terms == | == Generic math terms vs. specialized application terms == | ||
For the previous two reasons — consistency, and simplicity — choosing "subspace basis" over "subgroup basis" would be preferable. But | For the previous two reasons — consistency, and simplicity — choosing "subspace basis" over "subgroup basis" would be preferable. But this article thinks it can do even better. | ||
Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. | Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure. | ||
A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces, as we discussed in the earlier section [[User:Cmloegcmluin/Interval basis#Interval subspaces as subspaces of other interval subspaces]]. If the name for the specialized RTT object was simply "subspace" instead of "interval subspace", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses). | A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces, as we discussed in the earlier section [[User:Cmloegcmluin/Interval basis#Interval subspaces as subspaces of other interval subspaces]]. If the name for the specialized RTT object was simply "subspace" instead of "interval subspace", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses). | ||
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== A fresh start re: not excluding standard prime-limit interval subspaces == | == A fresh start re: not excluding standard prime-limit interval subspaces == | ||
"Subgroup" in its typical RTT usage is apparently intended to exclude the standard prime-limit subgroups. | "Subgroup" in its typical RTT usage is apparently intended to exclude the standard prime-limit subgroups. This article sees several issues with this: | ||
* It is confusing because prime-limit subgroups are still very much subgroups — of the entire space of primes, for one example. | * It is confusing because prime-limit subgroups are still very much subgroups — of the entire space of primes, for one example. | ||
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* And it hurts by making it more difficult than necessary to communicate about the standard prime-limit interval bases. | * And it hurts by making it more difficult than necessary to communicate about the standard prime-limit interval bases. | ||
For all these three reasons, the interval basis terminology makes no such exclusion. | For all these three reasons, the interval basis terminology makes no such exclusion. | ||
= Footnotes = | = Footnotes = | ||