# Tour:Some variations of group

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Introduction two (beginners)|UP: Introduction two (beginners)|NEXT: Equality of left and right neutral elementExpected time for this page: 10 minutes

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WHAT YOU NEED TO DO: Read, and understand, all the definitions presented below. These involve variations on the notion of group.

- Magma: A magma is a set with a binary operation . There is
*no*condition of associativity, there is no requirement that an identity element exist, and there is no condition for inverses of any kind to exist. - Semigroup: This is a magma where the associativity condition is satisfied. For any , we have
- Neutral element (also termed
**identity element**): An element is termed*left neutral*if for all , right neutral if for all . is termed neutral if it is both left and right neutral. A neutral element is also termed an*identity element*. - Monoid: A monoid is a semigroup with a neutral element.
- Cancellative element: An element is termed left cancellative if . Similarly is termed right cancellative if . An element is termed
*cancellative*if it is both left and right cancellative. - Invertible element: In a magma with neutral element , an element is said to be left invertible if there exists such that , and right invertible if there exists such that . If there exists a such that , the element is termed invertible.
- Group: A group is a monoid where every element is invertible.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Introduction two (beginners)|UP: Introduction two (beginners)|NEXT: Equality of left and right neutral element