Basic Algebraic Structures

Algebra

$(\mathcal{S},☿,...)$ An Algebra is a set with any number of operations with any arities

Magma

A Magma is a set with a binary operation. an example of a magma: $(\varnothing ,\cdot )$

Semigroup

A Semigroup is a magma whose operation is associative.

Monoid

A Monoid is a semigroup with a nullary operation which takes a value $e$. This element acts as the identity. $e\cdot x=x\cdot e=x$

Group

A Group is a Monoid with invertibility. Meaning there is a unary operation called inversion that maps any element $x$ its inverse $\bar{x}$ such that $x\cdot \bar{x}=\bar{x}\cdot x=e$ To put it a different way, if $x$ exists in a group, it has an inverse $\bar{x}$ which satisfies the above equation.

Ring

A Ring is a more complex group. It has 4 additional properties

1. It is a group with an additional binary operation (Double Magma) $(\mathcal{S},+,\cdot )$.
2. $(\mathcal{S},+)$ must be commutative
3. The distributive law must hold for $(\mathcal{S},+,\cdot )$
4. $(\mathcal{S},\cdot )$ must be a semigroup It follows from the property of rings that $0$, the additive identity, functions as a destructor in $\cdot$.

Field

A Field is a more complex Ring.

1. $(\mathcal{S},\cdot )$ is commutative
2. All elements in $(\mathcal{S},\cdot )$ except $0$ must have an inverse
3. $(\mathcal{S},\cdot )$ must have an identity noted as $1$.