Algebraic Structures

Once we have defined our set we want to define the operations, normally binary operations, on the elements of such set. The operations we define must also satisfy a finite set of identities called axioms.

The combination of a nonempty set, the operations on the elements and the axioms for these operation is what we call algebric structure.

Binary operations

A binary operation on a set $S$ is a function

$$ \cdot : S \times S \to S $$

that takes two elements of $S$ and returns another element of $S$.

Semigroups

A semigroup is a pair $(S, \cdot)$ where $S$ is a set and $\cdot$ is a binary associative operation:

$$ \forall a, b, c \in S \implies (a \cdot b) \cdot c = a \cdot (b \cdot c) $$

Associativity is the key structural property here.

For brevity, we will often omit the symbol $\cdot$ and write $ab$ instead of $a \cdot b$, whenever the meaning is clear from the context.

Two elements $a, b \in S$ are said to commute if $ab = ba$. A semigroup is called abelian (or commutative semigroup) if every $a, b \in S$ commute, such that:

$$ \forall a, b \in S, \quad ab = ba $$

The term abelian is more commonly used in the context of groups, but the notion of commutativity applies to any algebraic structure with a binary operation.

We can also use the addition notation $+$, with it the associative property can be defined as the following:

$$ \forall a, b, c \in S \implies (a + b) + c = a + (b + c) $$

Cancellation Laws

Given a semigroup $(S, \cdot)$ and an element $x \in S$, we say that:

$x$ is left cancellable if, $\forall a, b \in S$, $$ xa = xb \implies a = b $$
$x$ is right cancellable if, $\forall a, b \in S$, $$ ax = bx \implies a = b $$

A semigroup in which every element is left and right cancellable is called a cancellative semigroup.

Not all semigroups satisfy the cancellation law. This is a strong additional property.

Powers in a Semigroup

Let $a \in S$, we define the positive powers of an element $a$ in the following way:

$$ a^{n} = \begin{cases} a^1 = a \ a^{n+1} = a^na \end{cases} $$

Because of associativity, expressions like $a \cdot a \cdot a$ do not depend on how we place parentheses, so $a^n$ is well-defined.

This is one of the first concrete consequences of associativity.

An element $x \in S$ is called idempotent if $x^2 = x$.

Example: In $(\mathbb{N}, +)$, the element $0$ is idempotent since $0 + 0 = 0$.

With the addition notation $+$ the powers are called multiplies, defined as

$$ a^{n} = \begin{cases} a^1 = a \ a^{n+1} = a^n + a \end{cases} $$

Monoids

A semigroup $(M, \cdot)$ is called a monoid if there exists $e \in M$ (called the identity element) such that

$$ \forall a \in M \quad ea = ae = a $$

This element is unique, indeed let $e, e' \in M$ two identity elements, then

$$ e = ee' = e' $$

Because of this, we can denote a monoid by the triple $(M, \cdot, e)$.

Powers in a Monoid

The existence of the identity allows us to extend the definition of powers. For every $a \in M$, we define: $$ a^0 = e $$

Together with the recursive definition $$ a^{n+1} = a^na $$ this gives a consistent definition of $a^n$ for all $n \in \mathbb{N}$.

The identity element is idempotent, indeed

$$ e^2 = ee = e $$

Invertible elements

Let $(M, \cdot, e)$ be a monoid, an element $a \in M$ is called invertible if there exists an element $b \in M$ such that

$$ ab = ba = e $$

The element $b$ is called the inverse of a, and it is unique, indeed let $b, c \in M$ be two inverse of $a$, then

$$ b = be = b(ac) = (ba)c = ec = c $$

The uniqueness of the inverse of an element $a$ allow us to denote it by $a^{-1}$.

The name inverse is used withing the multiplication notation, if we use the addition notation, then the inverse element is called the opposite

Groups

A group is a monoid $(G, \cdot, e)$ where every element has an inverse.

Formally, a group $(G, \cdot, e)$ satisfies:

Associativity of the $\cdot$ operation
There exists the identity element $e$
For every $a \in G$, there exists $a^{-1} \in G$ such that $$ aa^{-1} = a^{-1}a = e $$

If the operation is also commutative, the group is called abelian, of course a group is abelian if the underlying semigroup is commutative.

We can now demonstrate that every group satisfies the cancellation law, indeed if $ab = ac$ then

$$ b = eb = (a^{-1}a)b = a^{-1}(ab) = a^{-1}(ac) = (a^{-1}a)c = ec = c $$

similarly, if $ba = ca$ then

$$ b = be = b(aa^{-1}) = (ba)a^{-1} = (ca)a^{-1} = c(aa^{-1}) = ce = c $$

Rings

A ring is a triple $(R, +, \cdot)$ where $R$ is a set, $+$ is a binary operation and $\cdot$ is an associative binary operation such that the following properties hold:

$(R, +, 0)$ is an abelian group
$(R, \cdot)$ is a semigroup
Distributive laws hold: $$ a(b + c) = ab + ac $$ $$ (a + b)c = ac + bc $$

Given two elements $a, b \in R$ we say that they commute if $ab = ba$. A ring is called commutative ring if $\forall a, b \in R$, $ab = ba$.

Unitary rings

The $\cdot$ operation in our ring $(R, + \cdot)$ may not have an identity element 1 such that $\forall a \in R$, $a \cdot e = e \cdot a = a$, if this element does exist, $e$ is called the unity of the ring.

Like for groups, this element is unique and with it, the subgroup $(R, \cdot)$ becomes a monoid $(R, \cdot, 1)$, we can then call the ring $(R, +, \cdot)$ a unitary ring.

Commutative rings

Given a ring $(R, +, \cdot)$ we already know that the group under the $+$ operation is abelian, such that $\forall a, b \in R$, $a + b = b + a$.

We did not put such constraint for the $\cdot$ operation, if such property holds, thus $\forall a, b \in R$, $a \cdot b = b \cdot a$ we call the ring $(R, +, \cdot)$ a commutative ring.

Integral domains

To define an integral domain we must first take a step back and define what a zero divisor is. A zero divisor is an element $a \ne 0 \in R$ such that there exists $x \ne 0 \in R$ with $ax = 0$.

The basic set of numbers we can think of is the integers $\mathbb{Z}$ which does not include zero divisor, indeed $\forall a, b \in \mathbb{Z}, ab = 0 \iff a = 0 \text{ or } b = 0$.

We can see an example in $\mathbb{Z}/4\mathbb{Z}$ through a Cayley table:

·	1	2	3
0	0	0	0
1	1	2	3
2	2	0	2
3	3	2	1

We note that $2 \cdot 2 = 0$ so $2$ is a zero divisor, with it we can now finally define an integral domain.

An integral domain is a commutative ring $(R, +, \cdot)$ without zero divisors, like $\mathbb{Z}$.

Division rings

A division ring is a unitary ring $(R, +, \cdot)$ such that every non-zero element in $R$ have an inverse under the $\cdot$ operation, meaning that $(R - {0}, \cdot, 1)$ is a group.

The commutative property is not required, if it is present, then the structure becomes a field.

Fields

A field is a commutative unitary ring $(R, +, \cdot)$ such that every non-zero element in $R$ have an inverse under the $\cdot$ operation, meaning that $(R - {0}, \cdot, 1)$ is an abelian group.

Typo?