Tour:Some variations of group

From Groupprops
Jump to: navigation, search
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 element
Expected time for this page: 10 minutes
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
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 S with a binary operation *: S \times S \to S. 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 a,b,c \in S, we have a * (b * c) = (a * b) * c
  • Neutral element (also termed identity element): An element e \in S is termed left neutral if e * a = a for all a, right neutral if a * e = a for all a. e 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 a \in S is termed left cancellative if a * b = a * c \implies b = c. Similarly a \in S is termed right cancellative if b * a = c * a \implies b = c. An element is termed cancellative if it is both left and right cancellative.
  • Invertible element: In a magma with neutral element e, an element a is said to be left invertible if there exists b such that b * a = e, and right invertible if there exists c such that a * c = e. If there exists a b such that a * b = b * a = e, 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