# User:Ascii/ProofWiki Sampling Notes for Theorems/Algebraic Structures

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## Contents

### Operations

- General Commutativity Theorem
- If an operation is commutative on $2$ entities, then it is commutative on any number of them.

- General Associativity Theorem
- If an operation is associative on $3$ entities, then it is associative on any number of them.

- Element Commutes with Product of Commuting Elements
- Associative Idempotent Anticommutative
- Let $\circ$ be a binary operation on a set $S$.
- Let $\circ$ be associative.
- Then $\circ$ is anticommutative if and only if:
- $(1): \quad \circ$ is idempotent
- and:
- $(2): \quad \forall a, b \in S: a \circ b \circ a = a$.

- Associative and Anticommutative
- Let $\circ$ be a binary operation on a set $S$.
- Let $\circ$ be both associative and anticommutative.
- Then:
- $\forall x, y, z \in S: x \circ y \circ z = x \circ z$

- Constant Operation is Commutative
- Let $S$ be a set.
- Let $x \left[{c}\right] y = c$ be a constant operation on $S$.
- Then $\left[{c}\right]$ is a commutative operation:
- $\forall x, y \in S: x \left[{c}\right] y = y \left[{c}\right] x$

- Constant Operation is Associative
- Let $S$ be a set.
- Let $x \left[{c}\right] y = c$ be a constant operation on $S$.
- Then $\left[{c}\right]$ is an associative operation:
- $\forall x, y, z \in S: \left({x \left[{c}\right] y}\right) \left[{c}\right] z = x \left[{c}\right] \left({y \left[{c}\right] z}\right)$

- Left Operation is Idempotent
- Right Operation is Idempotent
- Left Operation is Anticommutative
- Right Operation is Anticommutative
- Left Operation is Associative
- Right Operation is Associative
- Max and Min are Commutative
- Max and Min are Associative
- Max and Min are Idempotent
- Max and Min Operations are Distributive over Each Other

### Magmas

- Magma is Submagma of Itself
- Empty Set is Submagma of Magma
- Subset not necessarily Submagma
- Idempotent Magma Element forms Singleton Submagma

### Semigroups

- Restriction of Associative Operation is Associative
- Restriction of Commutative Operation is Commutative
- Subsemigroup Closure Test

### Subset Product

- Magma Subset Product with Self
- Subset Product within Semigroup is Associative
- Subset Product within Commutative Structure is Commutative
- Subset of Subset Product
- Left Cancellable Elements of Semigroup form Subsemigroup
- Right Cancellable Elements of Semigroup form Subsemigroup
- Cancellable Elements of Semigroup form Subsemigroup
- Left Cancellable Element is Left Cancellable in Subset
- Right Cancellable Element is Right Cancellable in Subset
- Cancellable Element is Cancellable in Subset
- Intersection of Subsemigroups
- Left Cancellable iff Left Regular Representation Injective
- Right Cancellable iff Right Regular Representation Injective
- Cancellable iff Regular Representations Injective
- Left Identity Element is Idempotent
- Right Identity Element is Idempotent
- Identity Element is Idempotent
- More than one Left Identity then no Right Identity
- More than one Right Identity then no Left Identity
- Left Operation is Associative
- Right Operation is Associative
- Left Operation is Idempotent
- Right Operation is Idempotent
- Left Operation is Anticommutative
- Right Operation is Anticommutative
- Left Operation All Elements Left Zeroes
- Right Operation All Elements Right Zeroes
- Element under Left Operation is Right Identity
- Element under Right Operation is Left Identity
- Left Operation is Right Distributive over All Operations
- Right Operation is Left Distributive over All Operations
- Left Operation is Distributive over Idempotent Operation
- Right Operation is Distributive over Idempotent Operation
- Left and Right Identity are the Same
- Identity is Unique
- Identity Property in Semigroup
- Identity of Monoid is Cancellable
- Identity of Cancellable Monoid is Identity of Submonoid
- Identity of Submonoid is not necessarily Identity of Monoid
- Set of all Self-Maps is Monoid
- Cancellable Elements of Monoid form Submonoid
- Idempotent Elements form Submonoid of Commutative Monoid

#### Inverses

- Left Inverse and Right Inverse is Inverse
- Inverse in Monoid is Unique
- Inverse of Inverse/General Algebraic Structure
- Inverse of Product/Monoid
- Equivalence of Definitions of Self-Inverse

#### Zeroes

- More than One Right Zero then No Left Zero
- Set System Closed under Union is Commutative Semigroup
- Identity of Power Set with Union
- Power Set with Union is Commutative Monoid
- Set System Closed under Intersection is Commutative Semigroup
- Identity of Power Set with Intersection
- Power Set with Intersection is Commutative Monoid
- Invertible Element of Associative Structure is Cancellable
- Right Cancellable Commutative Operation is Left Cancellable
- Left Cancellable Commutative Operation is Right Cancellable
- Commutation with Inverse in Monoid
- Commutation of Inverses in Monoid
- Inverse of Commuting Pair
- Product of Commuting Elements with Inverses
- Cancellation Laws
- Invertible Elements of Monoid form Subgroup of Cancellable Elements
- Structure Induced by Associative Operation is Associative
- Structure Induced by Commutative Operation is Commutative
- Induced Structure Identity
- Induced Structure Inverse

#### Quotient Structures

- Trivial Relation is Universally Congruent
- Equivalence Relation is Congruence for Constant Operation
- Equivalence Relation is Congruence for Left Operation
- Equivalence Relation is Congruence for Right Operation
- Quotient Structure is Well-Defined

#### Cartesian Product

- External Direct Product Closure
- External Direct Product Associativity
- External Direct Product of Semigroups
- External Direct Product Commutativity
- External Direct Product Identity
- External Direct Product Inverses
- External Direct Product Distributivity

#### Homomorphisms

- Morphism Property Preserves Closure
- Composite of Homomorphisms is Homomorphism/Algebraic Structure
- Morphism Property Preserves Cancellability
- Quotient Mapping on Structure is Canonical Epimorphism
- Restriction of Homomorphism to Image is Epimorphism
- Epimorphism Preserves Associativity
- Epimorphism Preserves Semigroups
- Homomorphism Preserves Subsemigroups
- Epimorphism Preserves Commutativity
- Epimorphism Preserves Identity
- Epimorphism Preserves Inverses
- Epimorphism Preserves Distributivity
- Homomorphism with Cancellable Codomain Preserves Identity
- Group Homomorphism Preserves Identity
- Group Homomorphism of Product with Inverse
- Homomorphism to Group Preserves Identity
- Homomorphism with Identity Preserves Inverses
- Homomorphism of External Direct Products
- Monomorphism Image is Isomorphic to Domain
- Inverse of Algebraic Structure Isomorphism is Isomorphism
- Isomorphism is Equivalence Relation

### Groups

- Group is not Empty
- Identity is only Idempotent Element in Group
- Group has Latin Square Property
- Self-Inverse Elements Commute iff Product is Self-Inverse
- Power Set of Group under Induced Operation is Semigroup
- Inverse of Product of Subsets of Group
- Set Equivalence of Regular Representations
- All Elements Self-Inverse then Abelian
- Commutation Property in Group
- Group Element Commutes with Inverse
- Identity Mapping is Automorphism
- Identity Mapping is Automorphism/Groups
- Identity Mapping is Group Endomorphism
- Inversion Mapping is Automorphism iff Group is Abelian
- Mapping to Square is Endomorphism iff Abelian
- Induced Group Product is Homomorphism iff Commutative
- Induced Group Product is Homomorphism iff Commutative/Corollary
- Automorphism Group is Subgroup of Symmetric Group
- Isomorphism Preserves Commutativity
- Inner Automorphisms form Subgroup of Symmetric Group
- Opposite Group is Group
- Symmetric Difference on Power Set forms Abelian Group
- One-Step Subgroup Test
- Two-Step Subgroup Test
- Identity of Subgroup
- Inverse of Subgroup
- Intersection of Subgroups is Subgroup
- Elements of Group with Equal Images under Homomorphisms form Subgroup
- Product of Subgroup with Itself
- Inverse of Subgroup
- One-Step Subgroup Test using Subset Product
- Two-Step Subgroup Test using Subset Product
- Group Homomorphism Preserves Subgroups

#### Conjugacy in Groups

#### Kernel

- Kernel of Group Homomorphism is Subgroup
- Kernel is Trivial iff Monomorphism/Group
- Commutative Semigroup is Entropic Structure
- Abelian Group Induces Entropic Structure

### Rings

- Ring is not Empty
- Cancellable Semiring with Unity is Additive Semiring
- Ring Product with Zero
- Product with Ring Negative
- Product of Ring Negatives
- Trivial Ring is Commutative Ring
- Null Ring is Trivial Ring
- Null Ring iff Zero and Unity Coincide
- Unity is Unit
- Group of Units is Group
- Unity and Negative form Subgroup of Units
- Negative of Product Inverse
- Zero Product with Proper Zero Divisor is with Zero Divisor
- Unit Not Zero Divisor
- Zero Divisor Product is Zero Divisor
- Product is Zero Divisor means Zero Divisor
- Ring Element is Zero Divisor iff not Cancellable
- Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors
- Idempotent Elements of Ring with No Proper Zero Divisors
- Division Ring has No Proper Zero Divisors
- Non-Zero Elements of Division Ring form Group
- Equivalence of Definitions of Integral Domain
- Null Ring and Ring Itself Subrings
- Subring Test
- Subdomain Test