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.
Read more about this topic: Monoid
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)