Abstract algebra

Groups

Let us have a set of elements \(G\) with a defined binary operation \(\circ\) that satisfies the following properties.

  1. Closure: \(\forall a, b \in G\): \(a \circ b \in G\)

  2. Associativity: \(\forall a, b, c \in G\): \(a \circ \left( b \circ c \right) = \left( a \circ b \right) \circ c\)

  3. Identity element: \(\exists e \in G\) such that \(\forall a \in G\): \(e \circ a = a \circ e = a\)

  4. Inverse element: \(\forall a \in G\) \(\exists a^{-1} \in G\) such that \(a \circ a^{-1} = e\)

In this case \(\left(G, \circ\right)\) is called a group.

Therefore the group is a Monoid with inverse element property.

Consider a set of 2 elements: \(G = \left\{0, 1\right\}\) with the operation \(\circ\) defined by the table Group Z/2Z.

Cayley table for \(\mathbb{Z}/2\mathbb{Z}\)
\(\circ\) 0 1
0 0 1
1 1 0

The identity element is \(0\) i.e. \(e = 0\). Inverse element is the element itself because \(\forall a \in G\): \(a \circ a = 0 = e\).

Let us have a Group \(\left(G, \circ\right)\). The group is called an Abelian or commutative if \(\forall a, b \in G\) it holds \(a \circ b = b \circ a\).

Rings and Fields

Rings

Consider a set \(R\) with 2 binary operations defined. The first one \(\oplus\) (addition) and the elements of \(R\) form an Abelian group under this operation. The second one is \(\odot\) (multiplication) and the elements of \(R\) form a Monoid under the operation. The two binary operations are connected to each other via the following distributive law:

  • Left distributivity: \(\forall a,b,c \in R\): \(a \odot \left(b \oplus c\right) = a \odot b \oplus a \odot c\)

  • Right distributivity: \(\forall a,b,c \in R\): \(\left( a \oplus b \right) \odot c = a \odot c \oplus b \odot c\)

    The identity element for \(\left(R, \oplus\right)\) is denoted as \(0\) (additive identity). The identity element for \(\left(R, \odot\right)\) is denoted as \(1\) (multiplicative identity).

    The inverse element to \(a\) in \(\left(R, \oplus\right)\) is denoted as \(-a\)

In this case \(\left(R, \oplus, \odot\right)\) is called a ring.

The Ring is a generalization of the concept of integer numbers.

The set of integer numbers \(\mathbb{Z}\) forms a Ring under \(+\) and \(\cdot\) operations i.e. addition \(\oplus\) is \(+\) and multiplication \(\odot\) is \(\cdot\). Thus for integer numbers we have the following Ring: \(\left(\mathbb{Z}, +, \cdot\right)\)

Fields

The ring \(\left(R, \oplus, \odot\right)\) is called a field if \(\left(R \setminus \{0\}, \odot\right)\) is an Abelian group.

The inverse element to \(a\) in \(\left(R \setminus\{0\}, \odot\right)\) is denoted as \(a^{-1}\)

Note that \(\mathbb{Z}\) is not a field because an inverse does not exist for every integer number. But if we consider a set of fractions \(\mathbb{Q} = \left\{a/b \mid a \in \mathbb{Z}, b \in \mathbb{Z}\setminus\{0\}\right\}\) then it will be a field.

The inverse element to \(a/b\) in \(\left(\mathbb{Q}\setminus\{0\}, \cdot\right)\) will be \(b/a\).

Linear algebra

Let \(F\) is a Field. The set \(V\) is called as vector space under \(F\) if the following conditions are satisfied

  1. We have a binary operation \(V \times V \rightarrow V\) (addition): \((x,y) \rightarrow x + y\) with the following properties:

    1. \(x + y = y + x\)

    2. \((x + y) + z = x + ( y + z )\)

    3. \(\exists 0 \in V\) such that \(\forall x \in V: x + 0 = x\)

    4. \(\forall x \in V \exists -x \in V\) such that \(x + (-x) = x - x = 0\)

  2. We have a binary operation \(F \times V \rightarrow V\) (scalar multiplication) with the following properties

    1. \(1_F \cdot x = x\)

    2. \(\forall a,b \in F, x \in V\): \(a\cdot\left(b \cdot x\right) = \left(a b\right) \cdot x\).

    3. \(\forall a,b \in F, x \in V\): \((a+b)\cdot x = a \cdot x + b \cdot x\)

    4. \(\forall a \in F, x, y \in V\): \(a\cdot(x+y) = a\cdot x + a \cdot y\)