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Research draft

homomorphism

vr.tr.homomorphism · XCT.QLT

Let an agent define homomorphisms for various structures, explain kernels, images and isomorphism theorems, verify whether a map is a homomorphism, and relate the concept to category theory.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent define homomorphisms for various structures, explain kernels, images and isomorphism theorems, verify whether a map is a homomorphism, and relate the concept to category theory.

A structure-preserving map between two algebraic structures of the same kind, such as groups, rings, vector spaces or modules, sending operations in one to the corresponding operations in the other, including special cases such as endomorphisms, isomorphisms, quotient maps, ring homomorphisms and chain maps; homomorphisms are the morphisms of algebraic categories and underpin much of abstract algebra.

What it is for: Structure-preserving maps in algebra.

It can be define homomorphisms for a given structure; check whether a map is a homomorphism; explain kernels, images and isomorphism theorems; relate to category theory.

Distinguishing features

Structure-preserving

Composable

Has kernel and image

Category-theoretic morphism

What it looks like

Not physical; a function between algebraic structures.

How it is recognised

Preserves operations

Between structures of the same kind

An arbitrary function need not preserve structure

Related models

is a kind of - category

function

is related to - a source structure

group

is related to - functions as sets of pairs

ordered pair

is related to - an example of a ring automorphism

complex conjugate

In practice

Families and kinds

group homomorphisms

ring and rng homomorphisms

linear and affine maps

module and algebra homomorphisms

endomorphisms, isomorphisms and quotient maps

Standards and regulation

Mathematical notation conventions

Failure modes and hazards

Forgetting to check preservation of all operations

Confusing homomorphism and homeomorphism

Ignoring identity preservation for rings

Also called

endomorphismquotient mappingring homomorphismrng homomorphismchain mapaffine mapbimodule homomorphismfield homomorphismaugmentationmodule homomorphismadditive mapgraph homomorphismalgebra homomorphismgroup homomorphismgroup representationnilpotent endomorphismring endomorphismFrobenius endomorphismformally smooth mapgenus of a multiplicative sequencesemilinear mapBoustrophedon transformadditive functionOne-parameter groupisogenymultiplicative characterJ-homomorphismLie group homomorphismprojective representationgroup isomorphismunitary representationreducible representationundecomposable representationspin representationrepresentation on coordinate ringsdual representationInduced representationrepresentation of the quaternion groupcuspidal representationsubrepresentation

+18

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Define What a homomorphism is.

Definitions.

Definition

Definition by structure.

Definition

Definition.

  1. What must a map preserve to be a homomorphism of groups, rings, vector spaces or modules? definition
  2. What are endomorphisms, automorphisms and isomorphisms? definition

Check

Checking a map.

Check

Checking.

  1. Is this given map a homomorphism, and what is its kernel and image? action
  2. Is it injective, surjective or bijective? action
Theory Key results.

Theorems.

Theorems

Isomorphism theorems.

Theorems

Theorems.

  1. What do the isomorphism theorems say, and how do quotient maps arise? definition
  2. How is a quotient structure constructed from a kernel? definition

Category

Category theory view.

Category

Category theory.

  1. How are homomorphisms the morphisms of algebraic categories, and what are functors between them? definition
  2. What are chain maps? definition
Apply Uses.

Applications.

Examples

Important examples.

Examples

Examples.

  1. Which homomorphisms are fundamental, such as determinant, exponential map or evaluation maps? definition
  2. How are affine maps related to linear maps? definition

Computing

Homomorphisms in computing.

Computing

Computing.

  1. How do homomorphisms appear in cryptography, such as homomorphic encryption, and in coding theory? definition
  2. Which entry fits those topics? action
Learn Teaching.

Education.

Teach

Teaching homomorphisms.

Teach

Teaching.

  1. How can homomorphisms be taught with concrete examples before abstraction? action
  2. Which misconceptions arise? provenance

History

History.

History

History.

  1. How did the concept of homomorphism develop in nineteenth and twentieth century algebra? provenance
  2. Who introduced the terminology? provenance

What the second pass must settle

  • Should each structure homomorphism be a separate entry?
  • How should textbooks be linked?
  • How should proof assistants be linked?
Draft as JSON