Abstract Algebra

This is still WIP

General list of algebraic structures (Wikipedia)

Examples of algebraic structures with a single binary operation are:

• Magma
• Quasigroup
• Monoid
• Semigroup
• Group

Examples involving several operations include:

• Ring

• Field

• Module

• Vector space

• Algebra over a field

• Associative algebra

• Lie algebra

• Lattice

• Boolean algebra

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  More here: wikipedia.org/wiki/Abstract_algebra

Basic Algebraic Structures

Binary Operations

A binary operation on a set $S$ is a function $\cdot :S\times S\to S$. We typically write $ab$ instead of $a\cdot b$.

Magma

A magma is an ordered pair $(S,\cdot )$ consisting of a non-empty set $S$ and a binary operation $\cdot$ on $S$.

Semigroup

A semigroup is a magma $(S,\cdot )$ such that the binary operation $\cdot$ is associative: for all $a,b,c\in S$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.

Monoid

A monoid is a semigroup $(M,\cdot )$ with an identity element $e\in M$ such that for all $a\in M$: $a\cdot e=e\cdot a=a$. The identity element is unique.

Group

A group is a monoid $(G,\cdot ,e)$ where every element $a\in G$ has an inverse element $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$. The inverse of each element is unique.

Abelian Group

An abelian group (or commutative group) is a group $(G,\cdot ,e)$ where the binary operation $\cdot$ is commutative: for all $a,b\in G$, $a\cdot b=b\cdot a$.

Rings and Fields

These structures combine two binary operations: addition and multiplication.

Ring

A ring is an ordered tuple $(R,+,-,0,\cdot ,1)$ consisting of a set $R$ and two binary operations, $+$ and $\cdot$, such that:

1. $(R,+,-,0)$ is an abelian group (the additive group).
2. $(R,\cdot ,1)$ is a monoid (the multiplicative monoid).
3. The distributive laws hold: for all $a,b,c\in R$,
   • $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ (left distributive law)
   • $(b+c)\cdot a=(b\cdot a)+(c\cdot a)$ (right distributive law)

Commutative Ring

A commutative ring is a ring $(R,+,-,0,\cdot ,1)$ where the multiplicative monoid is also commutative: for all $a,b\in R$, $a\cdot b=b\cdot a$.

Integral Domain

An integral domain is a commutative ring $(D,+,-,0,\cdot ,1)$ that satisfies the following additional property: for all $a,b\in D$, if $a\cdot b=0$, then either $a=0$ or $b=0$ (or both). This property is also expressed by stating that there are no zero divisors.

Field

A field is a commutative ring $(F,+,-,0,\cdot ,1)$ where every nonzero element is a unit (invertible under multiplication). In other words, $(F\setminus \{0\},\cdot )$ is an abelian group. This implies that a field is also an integral domain.

Important Theorems

The following are significant theorems often used in proofs within abstract algebra:

• Lagrange’s Theorem: The order of a subgroup divides the order of the group.
• Isomorphism Theorems: Series of theorems that describe relationships between groups and their homomorphisms.