Introduction to Monoids
Abstract algebra is a branch of mathematics which studies the algebraic structures that occur throughout different parts of mathematics. In this post I will go over a simple algebraic structure known as a monoid.
In this post I will define a monoid and give some important examples of these objects. A monoid is a set M with a binary operation, which is usually written as multiplication, such that the binary operation is associative and the set M must contain an element, usually denoted by e, which acts as an identity element for the multiplication.
We can write the above properties as follows
- For all x, y, z in M we have (xy)z = x(yz)
- There exists an element e in M such that ex = xe = x for all x in M.
For those familiar with groups we can say that a group is a monoid in which each element has an inverse. Thus the concept of a monoid is a generalization of the idea of a group. A simple example of a monoid which is not a group is the set of natural numbers under addition.
A more interesting example of a monoid can be found in the study of compact surfaces. Here we take M to be the set of homeomorphism classes of compact surfaces. We define the binary operation on M to be the connected sum. It can be shown that the connected sum is well defined up to homeomorphism and thus this gives a well defined operation on M. The identity element in this monoid is the 2-sphere.
Another important example of a monoid is the set of square matrices over some field. The operation here is ordinary matrix multiplication. For those familiar with linear algebra it is clear that this multiplication is associative and the identity matrix serves as the identity element in this monoid. Furthermore, this monoid is not a group because matrices with zero determinant are not invertible.
In mathematics it is important to not only study the objects but the maps between them. A homomorphism between monoids M and N is a function f from M to N such that f(xy) = f(x)f(y) for all x, y in M and where the multiplications are done in the proper monoid and f(e) = e. Thus a homomorphism must respect multiplication and map the identity element of M to the identity element of N.
For those familiar with groups it is clear that a homomorphism of monoids is the same thing as a homomorphism of groups when M and N are groups.
In this post we have introduced the concept of a monoid and have given several important and interesting examples of these objects. The idea of a monoid can be generalized in category theory and we will go over this idea in a later post.
References:
https://en.wikipedia.org/wiki/Monoid
http://mathworld.wolfram.com/Monoid.html
Lang, Serg (2002) Algebra, New York, NY, Springer GTM
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