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

Operations
Let $(S, \circ)$ be a semigroup.
 * 1) General Commutativity Theorem
 * 2) General Associativity Theorem
 * If an operation is associative on $3$ entities, then it is associative on any number of them.
 * 1) Element Commutes with Product of Commuting Elements

Let $x, y, z \in S$.

If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$. Let $\circ$ be a binary operation on a set $S$.
 * 1) Associative Idempotent Anticommutative

Let $\circ$ be associative.

Then $\circ$ is anticommutative :
 * $(1): \quad \circ$ is idempotent

and:
 * $(2): \quad \forall a, b \in S: a \circ b \circ a = a$.

Let $\circ$ be a binary operation on a set $S$.
 * 1) Associative and Anticommutative

Let $\circ$ be both associative and anticommutative.

Then:
 * $\forall x, y, z \in S: x \circ y \circ z = x \circ z$

Let $S$ be a set.
 * 1) Constant Operation is Commutative

Let $x \left[{c}\right] y = c$ be a constant operation on $S$.

Then $\left[{c}\right]$ is a commutative operation:


 * $\forall x, y \in S: x \left[{c}\right] y = y \left[{c}\right] x$

Let $S$ be a set.
 * 1) Constant Operation is Associative

Let $x \left[{c}\right] y = c$ be a constant operation on $S$.

Then $\left[{c}\right]$ is an associative operation:


 * $\forall x, y, z \in S: \left({x \left[{c}\right] y}\right) \left[{c}\right] z = x \left[{c}\right] \left({y \left[{c}\right] z}\right)$


 * 1) Left Operation is Idempotent
 * 2) Right Operation is Idempotent
 * 3) Left Operation is Anticommutative
 * 4) Right Operation is Anticommutative
 * 5) Left Operation is Associative
 * 6) Right Operation is Associative
 * 7) Max and Min are Commutative
 * 8) Max and Min are Associative
 * 9) Max and Min are Idempotent
 * 10) Max and Min Operations are Distributive over Each Other

Magmas

 * 1) Magma is Submagma of Itself
 * 2) Empty Set is Submagma of Magma
 * 3) Subset not necessarily Submagma
 * 4) Idempotent Magma Element forms Singleton Submagma

Semigroups

 * 1) Restriction of Associative Operation is Associative
 * 2) Restriction of Commutative Operation is Commutative
 * 3) Subsemigroup Closure Test

Subset Product

 * 1) Magma Subset Product with Self
 * 2) Subset Product within Semigroup is Associative
 * 3) Subset Product within Commutative Structure is Commutative
 * 4) Subset of Subset Product
 * 5) Left Cancellable Elements of Semigroup form Subsemigroup
 * 6) Right Cancellable Elements of Semigroup form Subsemigroup
 * 7) Cancellable Elements of Semigroup form Subsemigroup
 * 8) Left Cancellable Element is Left Cancellable in Subset
 * 9) Right Cancellable Element is Right Cancellable in Subset
 * 10) Cancellable Element is Cancellable in Subset
 * 11) Intersection of Subsemigroups


 * 1) Left Cancellable iff Left Regular Representation Injective
 * 2) Right Cancellable iff Right Regular Representation Injective
 * 3) Cancellable iff Regular Representations Injective


 * 1) Left Identity Element is Idempotent
 * 2) Right Identity Element is Idempotent
 * 3) Identity Element is Idempotent


 * 1) More than one Left Identity then no Right Identity
 * 2) More than one Right Identity then no Left Identity


 * 1) Left Operation is Associative
 * 2) Right Operation is Associative
 * 3) Left Operation is Idempotent
 * 4) Right Operation is Idempotent
 * 5) Left Operation is Anticommutative
 * 6) Right Operation is Anticommutative
 * 7) Left Operation All Elements Left Zeroes
 * 8) Right Operation All Elements Right Zeroes
 * 9) Element under Left Operation is Right Identity
 * 10) Element under Right Operation is Left Identity
 * 11) Left Operation is Right Distributive over All Operations
 * 12) Right Operation is Left Distributive over All Operations
 * 13) Left Operation is Distributive over Idempotent Operation
 * 14) Right Operation is Distributive over Idempotent Operation


 * 1) Left and Right Identity are the Same
 * 2) Identity is Unique
 * 3) Identity Property in Semigroup
 * 4) Identity of Monoid is Cancellable
 * 5) Identity of Cancellable Monoid is Identity of Submonoid
 * 6) Identity of Submonoid is not necessarily Identity of Monoid
 * 7) Set of all Self-Maps is Monoid
 * 8) Cancellable Elements of Monoid form Submonoid
 * 9) Idempotent Elements form Submonoid of Commutative Monoid

Inverses

 * 1) Left Inverse and Right Inverse is Inverse
 * 2) Inverse in Monoid is Unique
 * 3) Inverse of Inverse/General Algebraic Structure
 * 4) Inverse of Product/Monoid
 * 5) Equivalence of Definitions of Self-Inverse

Zeroes

 * 1) More than One Right Zero then No Left Zero


 * 1) Set System Closed under Union is Commutative Semigroup
 * 2) Identity of Power Set with Union
 * 3) Power Set with Union is Commutative Monoid


 * 1) Set System Closed under Intersection is Commutative Semigroup
 * 2) Identity of Power Set with Intersection
 * 3) Power Set with Intersection is Commutative Monoid
 * 4) Invertible Element of Associative Structure is Cancellable


 * 1) Right Cancellable Commutative Operation is Left Cancellable
 * 2) Left Cancellable Commutative Operation is Right Cancellable


 * 1) Commutation with Inverse in Monoid
 * 2) Commutation of Inverses in Monoid
 * 3) Inverse of Commuting Pair
 * 4) Product of Commuting Elements with Inverses
 * 5) Cancellation Laws
 * 6) Invertible Elements of Monoid form Subgroup of Cancellable Elements
 * 7) Structure Induced by Associative Operation is Associative
 * 8) Structure Induced by Commutative Operation is Commutative
 * 9) Induced Structure Identity
 * 10) Induced Structure Inverse

Quotient Structures

 * 1) Trivial Relation is Universally Congruent
 * 2) Equivalence Relation is Congruence for Constant Operation
 * 3) Equivalence Relation is Congruence for Left Operation
 * 4) Equivalence Relation is Congruence for Right Operation
 * 5) Quotient Structure is Well-Defined

Cartesian Product

 * 1) External Direct Product Closure
 * 2) External Direct Product Associativity
 * 3) External Direct Product of Semigroups
 * 4) External Direct Product Commutativity
 * 5) External Direct Product Identity
 * 6) External Direct Product Inverses
 * 7) External Direct Product Distributivity

Homomorphisms

 * 1) Morphism Property Preserves Closure
 * 2) Composite of Homomorphisms is Homomorphism/Algebraic Structure
 * 3) Morphism Property Preserves Cancellability
 * 4) Quotient Mapping on Structure is Canonical Epimorphism


 * 1) Restriction of Homomorphism to Image is Epimorphism
 * 2) Epimorphism Preserves Associativity
 * 3) Epimorphism Preserves Semigroups
 * 4) Homomorphism Preserves Subsemigroups
 * 5) Epimorphism Preserves Commutativity
 * 6) Epimorphism Preserves Identity
 * 7) Epimorphism Preserves Inverses
 * 8) Epimorphism Preserves Distributivity
 * 9) Homomorphism with Cancellable Codomain Preserves Identity
 * 10) Group Homomorphism Preserves Identity
 * 11) Group Homomorphism of Product with Inverse
 * 12) Homomorphism to Group Preserves Identity
 * 13) Homomorphism with Identity Preserves Inverses
 * 14) Homomorphism of External Direct Products
 * 15) Monomorphism Image Isomorphic to Domain
 * 16) Inverse of Algebraic Structure Isomorphism is Isomorphism
 * 17) Isomorphism is Equivalence Relation

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


 * 1) Group Homomorphism Preserves Subgroups

Conjugacy in Groups

 * 1) Conjugacy is Equivalence Relation

Kernel

 * 1) Kernel of Group Homomorphism is Subgroup
 * 2) Kernel is Trivial iff Monomorphism/Group
 * 3) Commutative Semigroup is Entropic Structure
 * 4) Abelian Group Induces Entropic Structure

Rings

 * 1) Ring is not Empty
 * 2) Cancellable Semiring with Unity is Additive Semiring
 * 3) Ring Product with Zero
 * 4) Product with Ring Negative
 * 5) Product of Ring Negatives
 * 6) Trivial Ring is Commutative Ring
 * 7) Null Ring is Trivial Ring
 * 8) Null Ring iff Zero and Unity Coincide
 * 9) Unity is Unit
 * 10) Group of Units is Group
 * 11) Unity and Negative form Subgroup of Units
 * 12) Negative of Product Inverse


 * 1) Zero Product with Proper Zero Divisor is with Zero Divisor
 * 2) Unit Not Zero Divisor
 * 3) Zero Divisor Product is Zero Divisor
 * 4) Product is Zero Divisor means Zero Divisor


 * 1) Ring Element is Zero Divisor iff not Cancellable
 * 2) Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors
 * 3) Idempotent Elements of Ring with No Proper Zero Divisors


 * 1) Division Ring has No Proper Zero Divisors
 * 2) Non-Zero Elements of Division Ring form Group
 * 3) Equivalence of Definitions of Integral Domain
 * 4) Null Ring and Ring Itself Subrings


 * 1) Subring Test
 * 2) Subdomain Test