Monoid - Properties

Properties

In a monoid, one can define positive integer powers of an element x : x1 = x, and xn=x*...*x (n times) for n > 1 . The rule of powers xn+p=xn * xp is obvious.

Directly from the definition, one can show that the identity element e is unique. Then, for any x, one can set x0=e and the rule of powers is still true with nonnegative exponents.

It is possible to define invertible elements: an element x is called invertible if there exists an element y such that x*y = e and y*x = e. The element y is called the inverse of x . If y and z are inverses of x, then by associativity y = (zx)y = z(xy) = z. Thus inverses, if they exist, are unique.

If y is the inverse of x, one can define negative powers of x by setting x−1 = y and x−n = y*...*y (n times) for n > 1 . And the rule of exponents is still verified for all n,p rational integers. This is why the inverse of x is usually written x−1. The set of all invertible elements in a monoid M, together with the operation *, forms a group. In that sense, every monoid contains a group (if only the trivial one consisting of the identity alone).

However, not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements a and b exist 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 (or is cancellative) 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 commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck construction. That's how the additive group of the integers (a group with operation +) is constructed from the additive monoid of natural numbers (a commutative monoid with operation + and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.

If a monoid has the cancellation property and is finite, then it is in fact a group. Proof: Fix an element x in the monoid. Since the monoid is finite, xn = xm for some m > n > 0. But then, by cancellation we have that xm−n = e where e is the identity. Therefore x * xm−n−1 = e, so x has an inverse.

The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. obviously include the identity and not so obviously are closed under the operation). This means that the cancellative elements of any commutative monoid can be extended to a group.

An inverse monoid is a monoid where for every a in M, there exists a unique a−1 in M such that a=a*a−1*a and a−1=a−1*a*a−1. If an inverse monoid is cancellative, then it is a group.