Algebra
Algebraic Structures
One can arrange algebraic structures by increasing gemerality. The most
commonly known structures are
-
Monoids
-
Groups
-
Rings
-
Fields
This describes a hierarchy, such that for instance groups are more general
than monoids, but have a monoid structure as well as a group structure.
Sloppily once could write
$$
\large{\text{Field}\supset\text{Ring}\supset
\text{Group}\supset\text{Monoid}}
$$
Sets
Sets are at the heart of modern mathematics. Sets are collections of
objects, such that an object's membership in a set is defined by some
predicate(s). They were formalized by Dedekind and Cantpr in the late
19th century and have become an indispensible construct of modern
mathematics.
For instance, the collection ..., -1, 0, 1, ... is the set
of all integers, written $\mathbb{Z}$. This set is a subset of rational
numbers $\mathbb{Q}$ and the rationals are called a superset of the
integers, expressed by
$$
\large{\mathbb{Z}\subset\mathbb{Q}\quad\text{and}\quad
\mathbb{Q}\supset\mathbb{Z}}
$$
Operations like $\text{Union}\ \cup$ or $\text{Intersection}\ \cap$
can be performed on pairs of sets, thus creating new sets. The intersection
of two sets S and T, for instance, consiste of all objects that belong to
S as well as T.
Mappings
Mappings, together with the sets they operate on, are the second
of two buildong blocks used to define algebraic structures. A mapping
between two sets S and T can be written as
$$
\large{
\begin{gather}
m: S\mapsto T\\
x\in S\Rightarrow m(x)\in T
\end{gather}}
$$
One distinguishes between three types of mapping, surjective, injective
and bijective:
$$
\text{m is}
\begin{cases}
\text{surjective} & \forall y\in T\ \exists x\in S\\ & m(x) = y\\
\text{injective} & \forall x,\,y\in S\\ & m(x) = m(y)\Rightarrow x = y\\
\text{bijective} & \text{surjective and injective}
\end{cases}
$$
Basic Algebraic Structures
With sets and mappings between sets explained, one can proceed to define
algebraic structures. Mappings between such structures are usually called
homomorphisms when they preserve the algebraic structure of the sets they
operate on.
The basics of the structures mentioned above can be defined by a few simple
axioms. When defined they are abstract, but have far-reaching uses in
modern physics. Groups in particular are an indispensible tool in pjysics.
Together with Noether's theorem they lead to all the conservation
laws of physics, conservation of energy, linear and angular momentum and
so on.
Monoids
Monoids in general are not as familiar as the other three structures
mentioned above. Simply put, they are groups
without imverse elements. Therefore grpups are also monoids.
They consist of a set M and a mapping $\circ : M\times M\mapsto M$ with
the following properties
$$
\begin{gather}
\circ:\ M\times M\mapsto M\quad (a,\,b)\mapsto\ a\circ b\\
\forall a,\,b\in M\quad a\circ b\in M\\
a\circ(b\circ c) = (a\circ b)\circ c\\
\exists e\in M\ \forall a\in M\quad e\circ a = a\circ e = a
\end{gather}
$$
It is easy to show that the unit element e is unique along the lines
one can often find in mathematical texts:
"The proof is trivial and left as an exercise for the reader."
Monoids are not as rare as one might think. For instance, the set of
integer numbers $\mathbb{Z}$ with multiplication as the mapping is a
monoid, but not a group. On the other hand $\mathbb{Z}$ with addition
is a group.
Groups
Next in the hierrachy of algebraic structures are groups. Groups are an
extremely important structure in modern physics and can occur in
many manifestations. There are finite groups with a defined number of
elements $|G| < \infty$ and groups with an infinite number of elements.
They extend monoids by introducing an inverse.
$${\forall g\in G\ \exists h\in G\ \quad g\circ h = e}$$
Again it is eas to show that the inverse of every element is unique and
that $g\circ h = h\circ g$.
When $\forall a,\,b\in G\quad a\circ b = b\circ a$,
the group is called commutative or abelian. Not every group is abelian.
Examples
$\mathbb{Z}$ with adition is an infinite abelian group. The simplest
finite group may be $\{-1,\,1\}$ with multiplication. 1 is the unit element
and each element is its own inverse. Such objects are called idempotent
$a\circ a = e$. This subset of integers is the only set of integers that
forms a group under multiplication. It is not a group under addition,
because $1 + 1 = 2\notin \{-1,\,1\}$.
$2\mathbb{Z}$, the subset of even integers, is a group under addition,
but the subset of odd integers $2\mathbb{Z}+1$ is not, because the sum
of two odd integers is even. $2\mathbb{Z}$ is called a subgroup of
$\mathbb{Z}$. It is a normal subgroup defined by
$$
\large{
\begin{gather}
2\mathbb{Z}\subset\mathbb{Z}\\
\forall n\in\mathbb{Z}\quad n+2\mathbb{Z} = 2\mathbb{Z}+n
\end{gather}}
$$
Rings
Monoids and groups are sets endowed with one kind of mapping. Rings and
Fields extend the concept by introducing a second kind of mapping. In the
structures that most people are familiar with, rational, real and complex
numbers, these mappings are called addition and multiplication.
The simplest way to characterize the ring structure of a set R with
mappings + and $\times$ written as $(R, +, \times)$, is:
- (R, +) is an abelian group
- $\forall a,\,b,\,c\in R\,: a\times (b\times c) = (a\times b)\times c$
- $\forall a,\,b,\,c\in R\,: a\times (b + c) = a\times b + a\times c$
- $\forall a,\,b,\,c\in R\,: (a + b)\times c = a\times c + b\times c$
Rule 4 is necessary because (R, $\times$) need not be commutative.
An example is a ring formed by matrices that may or may not have
multiplicative inverses for some or all elements (e.g. det M = 0).
The existence of a multiplicative unit element is not required,
but also not excluded. The latter care clled rings with unity.
Hence with respect to multiplication R may be less than a monoid.
Other examples of rings are $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$,
the second one being the ubtegere modulo n. a finite ring, as the
underlying set consists of the integers from 0 to n - 1 inclusive.
In bozj cases, as already mentioned in the Group section, there are no
multiplicative inverses.
Elements $a\in R$ of a ring such that $\exists b\ne 0 \in R$ with
$a\times b = 0$ are called zero divisors. An example would be
$\mathbb{Z}/4\mathbb{Z}$, since $2\times 2 = 0$ modulo 4. More generally
any n that is the square of another number, $n = k^2\quad k\times k = 0$
modulo n.
Fields
If, additionally, a ring (R, + $\times$) under multiplication, is not just
associative, but a group, it is called a field. The additive unit element
must however be excluded from the multiplicative set, but not from the
field itself. Multiplication of any element of R with 0 is 0.
The definition of (F, +, $\times$) thus is:
- F is a ring
- (F \ {0}, $\times$) is an abelian group
In the hierarchy
$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$
all of them are rings, but fields start with $\mathbb{Q}$.
A less familiar example of a field is $\mathbb{Z}/p\mathbb{Z}$, the set of
integers modulo a prime number p. 1 is always its own inverse and for p = 5
we have 2 x 3 = 3 x 2 = 4 x 4 = 1. 0 and p are not part of the multiplicative
set, 0 by definition and p because p = 0 modulo p. The number of elements in
the set is p - 1.