Commutative monoid: Difference between revisions
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{{monoid property}} | |||
==Definition== | ==Definition== | ||
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==Related notions== | ==Related notions== | ||
For a monoid with all elements invertible, i.e. a [[group]], the related notion is an [[abelian group]]. | ===Weaker than=== | ||
For a monoid with all elements invertible, i.e. a [[group]], the related notion is an [[Weaker than::abelian group]]. | |||
==Examples== | |||
* Any [[abelian group]]. | |||
* The [[additive monoid of natural numbers]]. | |||
* The [[multiplicative monoid of non-zero integers]]. | |||
Latest revision as of 00:59, 12 January 2024
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a monoid property, viz a property that can be evaluated for any monoid. Recall that a monoid is a set with an associative binary operation, having a neutral element (viz multiplicative identity)
Definition
A monoid in which all elements commute is called a commutative monoid. That is, a commutative monoid satisfies for all in the monoid.
Related notions
Weaker than
For a monoid with all elements invertible, i.e. a group, the related notion is an abelian group.
Examples
- Any abelian group.