Content-Type: text/html Wikipedia: Monoid

[Home]Monoid

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A monoid is a pair (M,*), where M is a set and * is a binary operation on M, obeying the following rules:
  1. closure: for all a, b in M, a*b is in M.
  2. identity: there exists an element e in M, such that for all a in M, a*e = e*a = a.
  3. associativity: * is an associative operation; that is, for all a, b, c in M, (a*b)*c = a*(b*c)

In other words, a monoid is a semigroup with an identity element.

Some examples of monoids:

Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an element x is called invertible if there exists an element y such x*y = e and y*x = e. It turns out that the set of all invertible elements, together with the operation *, forms a group. In that sense, every monoid contains a group.

However, not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which exist two elements a and b and such that a*b = a holds even though b is not the identity element. Such a monoid cannot be embedded in a group, because in the group we could multiply both sides with the inverse of a and would get that b = e, which isn't true. A monoid (M,*) has the cancellation property if for all a, b and c in M, a*b = a*c always implies b = c and b*a = c*a always implies b = c. A monoid with the cancellation property can always be embedded in a group. That's how the integers (a group with operation +) are constructed from the natural numbers (a monoid with operation + and cancellation property).

If a monoid has the cancellation property and is finite, then it is in fact a group.


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Last edited August 16, 2001 5:18 am (diff)
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