Generator: Difference between revisions
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Cmloegcmluin (talk | contribs) →Generators in math and JI subgroups: basis is not a synonym for minimal generating set of a subgroup; it is the analogous structure for a subspace of a vector space |
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A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]], a [[regular temperament]] based on a JI subgroup, or any [[MOS]]-based harmony) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''. | A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]], a [[regular temperament]] based on a JI subgroup, or any [[MOS]]-based harmony) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''. | ||
A ''' | A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2, 3, 5/3} are bases for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a basis: 15 = 3 · 5 so we can take out 15 from this generating set. | ||
If the abelian group is written additively, then if <math>\lbrace g_1, g_2, \ldots g_k \rbrace</math> is the generating set, every element <math>g</math> of the group can be written | If the abelian group is written additively, then if <math>\lbrace g_1, g_2, \ldots g_k \rbrace</math> is the generating set, every element <math>g</math> of the group can be written | ||