Abstract Algebra

Abstract algebra is the axiomatic study of algebraic structures.

Definition

Abstract algebra is the field of mathematics dedicated to the study of algebraic theories, and the algebraic structures that inhabit them. Abstract algebra focuses on the axiomatic rules that define how elements in a set interact through operations.

Examples

Idea

A group is an algebraic structure used to abstract the concept of symmetry. It consists of a set equipped with a single binary operation that is associative, has an identity element, and provides an inverse for every element.

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Idea

A ring is a algebraic generalization of the integers.

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Definition

A field is a ring $F$ in which the multiplication operation $\cdot$ is commutative and every non-zero element has a multiplicative inverse. Formally, a field is a structure $(F,+,\cdot ,0,1)$ such that:

• $(F,+,0)$ is an Group.
• $(F\setminus \{0\},\cdot ,1)$ is an Group.
• The distributive law holds: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$.

In the language of Category Theory and Type Theory, a field can be viewed as a commutative ring where the non-zero elements form a group under multiplication.

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Vector Space

A vector space is a mathematical object $(R:\text{Ring},V:\ast ,+:V\times V\to V,\cdot :R\times V\to V,\vec{0}:V)$ where $R$ is some Ring.

It must satisfy the following axioms:

$$\begin{array}{r}\forall \vec{x}:V.\vec{x}+\vec{0}=\vec{x}\\ \forall \vec{x},\vec{y}:V.\vec{x}+\vec{y}=\vec{y}+\vec{x}\\ \forall \vec{x},\vec{y},\vec{z}:V.(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})\\ \forall \vec{x}:V.0_R\cdot x=\vec{0}\\ \forall a,b:R,\vec{x}:V.a\cdot \vec{x}+b\cdot \vec{x}=(a+b)\cdot x\end{array}\text{\hspace{1em}(1)(2)(3)(4)(5)}$$Link to original

The Abstract Approach

The “abstract” in abstract algebra refers to the fact that these structures are studied independently of their specific elements. For instance, the Group Axioms apply equally to the rotations of a polygon, permutations of a set, and the addition of integers. This universality allows mathematicians to prove a theorem once for a general structure and apply it to every specific instance across mathematics and science.

Applications

Number Theory: Understanding the properties of integers using ring theory. Physics and Chemistry: Using group theory to describe the symmetry of crystals or the behavior of subatomic particles. Computer Science: Utilizing finite fields for error-correcting codes and cryptography (e.g., the AES encryption standard).

Would you like to see a formal comparison table of the axioms for groups, rings, and fields, or perhaps explore how these relate to the Universal Algebra concepts we discussed previously?

Agebraic Structure

An algebraic structure is a collection of four kinds of objects:

• Sorts
• Operations on sorts
• Distinguished elements
• Laws or equations that the operations must follow.

Instances of an algebra must provide:

• A set for each sort.
• A function for each operation
• A value in a sort for each distinguished element
• And a proof that each law is satisfied.

Example: Groups

For instance a group is made up of:

• A single sort $G$.
• A binary operation $\cdot :G\text{texttimes}G\to G$
• A unary operation ${\_}^{-1}:G\to G$
• A constant $e:G$
• An associativity law $\forall x,y,z:G.x\cdot (y\cdot z)=(x\cdot y)\cdot z$
• Left and right unit laws $\forall x:G.x\cdot e=e\cdot x=x$.
• Left and right inverse laws: $\forall x:G.x\cdot x^{-1}=x^{-1}\cdot x=e$.