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# 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:

- Any group.
- The set of all continuous selfmaps of a topological space with the operation of functional composition.
- The integers or natural numbers or complex numbers with multiplication as operation.
- The natural numbers with addition as operation.
- The set of all finite strings (including the empty string) over some fixed alphabet Σ, with string concatenation as operation. This is denoted by Σ* by theoretical computer scientists and called the "free monoid over Σ" by mathematicians.
- The elements of every ring with multiplication as operation. An example of this is the set of all n by n matrices with matrix multiplication.
- The set of all endomorphisms of a fixed group, with composition as operation.
- Pick an object of a category and consider the set of all morphisms from this object to itself, with composition as operation. (See category theory.)

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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