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Math Help - monoid v group

  1. #1
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    monoid v group

    Is a monoid the same thing as a group but without the requirement for elements to have an inverse element?
    Thanks!
    Chad.
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  2. #2
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    Quote Originally Posted by Chad950 View Post
    Is a monoid the same thing as a group but without the requirement for elements to have an inverse element?
    Thanks!
    Chad.

    Yes...at least according to the standard definition I know.

    Tonio
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  3. #3
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    Essentially you're correct, but I wouldn't use the expression "same thing." Every group is a monoid, but not every monoid is a group. The integers under multiplication is an example of an abelian monoid which isn't a group (as are the rationals and reals).

    A monoid has less structure than a group, so more structures are monoids, but monoids (that aren't groups) don't behave as nicely as groups.
    Last edited by DrSteve; February 5th 2011 at 02:41 AM.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by DrSteve View Post
    Essentially you're correct, but I wouldn't use the expression "same thing." Every group is a monoid, but not every monoid is a group. The integers under multiplication is an example of an abelian monoid which isn't a group (as are the rationals and reals).

    A monoid has less structure than a group, so more structures are monoids, but monoids (that aren't groups) don't behave as nicely as groups.
    Yes, but only groups can be abelian. Monoids are just commutative...

    Another useful example of a monoid is the set of all (partial) functions from A to A. There is always an identity function, and some functions are bijections so have inverses. In general though, your functions will not have inverses. However, injections and surjections will have left or right inverses.
    Last edited by Swlabr; February 7th 2011 at 06:13 AM.
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