Algebraic Structures¶

Note

Any set \(X\) can be endowed with rich albegraic structures.
In the following, we assume that all the elements are in \(X\).
In each of the following operations, we assume that it creates a closure, i.e. it maps to some other element in \(X\) itself.

Group¶

Note

We have an addition \(+\) defined for \(X\) which follows associativity, i.e.
- \(x+(y+z)=(x+y)+z\).
There is a unique identity element \(0\) w.r.t \(+\) such that
- \(x+0=0+x=x\)
For every \(x\), there is a unique inverse element w.r.t \(+\), \(-x\) such that
- \(x+(-x)=(-x)+x=0\)

Tip

If we enforce group structure to the set of numbers, we extend naturals (\(\mathbb{N}\)) to the set of integers (\(\mathbb{Z}\))

Note

Note

It is an Abelian group w.r.t the addition operator \(+\).
It also has a multiplication \(\cdot\) defined so that
- It is associative, \(x\cdot (y\cdot z)=(x\cdot y)\cdot z\)
- There is a unique identity element \(1\) w.r.t \(\cdot\) such that
  - \(x\cdot 1=1\cdot x=x\)
- For every \(x\), there is a unique inverse element w.r.t \(\cdot\), \(x^{-1}\) such that
  - \(x\cdot x^{-1}=x^{-1}\cdot x=1\)
Addition and multiplication follow the distributive property
- \(x\cdot(y+z)=x\cdot y+x\cdot z\)

Note

It is a ring.
It is an Abelian group w.r.t the addition operator \(+\) as well as the multiplication opeartor \(\cdot\).

Tip

If we enforce field structure to the set of numbers, we extend integers (\(\mathbb{Z}\)) to the set of rationals (\(\mathbb{Q}\))
To extend rationals (\(\mathbb{Q}\)) to reals (\(\mathbb{R}\)), we need to introduce topological properties.