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

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  1. General Commutativity Theorem
    If an operation is commutative on $2$ entities, then it is commutative on any number of them.
  2. General Associativity Theorem
    If an operation is associative on $3$ entities, then it is associative on any number of them.
  3. Element Commutes with Product of Commuting Elements
    Let $(S, \circ)$ be a semigroup.
    Let $x, y, z \in S$.
    If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$.
  4. 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
    $(2): \quad \forall a, b \in S: a \circ b \circ a = a$.
  5. Associative and Anticommutative
    Let $\circ$ be a binary operation on a set $S$.
    Let $\circ$ be both associative and anticommutative.
    $\forall x, y, z \in S: x \circ y \circ z = x \circ z$
  6. 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:
    $\forall x, y \in S: x \left[{c}\right] y = y \left[{c}\right] x$
  7. 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:
    $\forall x, y, z \in S: \left({x \left[{c}\right] y}\right) \left[{c}\right] z = x \left[{c}\right] \left({y \left[{c}\right] z}\right)$
  8. Left Operation is Idempotent
  9. Right Operation is Idempotent
  10. Left Operation is Anticommutative
  11. Right Operation is Anticommutative
  12. Left Operation is Associative
  13. Right Operation is Associative
  14. Max Operation is Commutative
  15. Min Operation is Commutative
  16. Max Operation is Associative
  17. Min Operation is Associative
  18. Max and Min are Idempotent
  19. Max and Min Operations are Distributive over Each Other


  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


  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
  12. Left Cancellable iff Left Regular Representation Injective
  13. Right Cancellable iff Right Regular Representation Injective
  14. Cancellable iff Regular Representations Injective
  15. Left Identity Element is Idempotent
  16. Right Identity Element is Idempotent
  17. Identity Element is Idempotent
  18. More than one Left Identity then no Right Identity
  19. More than one Right Identity then no Left Identity
  20. Left Operation is Associative
  21. Right Operation is Associative
  22. Left Operation is Idempotent
  23. Right Operation is Idempotent
  24. Left Operation is Anticommutative
  25. Right Operation is Anticommutative
  26. Left Operation All Elements Left Zeroes
  27. Right Operation All Elements Right Zeroes
  28. Element under Left Operation is Right Identity
  29. Element under Right Operation is Left Identity
  30. Left Operation is Right Distributive over All Operations
  31. Right Operation is Left Distributive over All Operations
  32. Left Operation is Distributive over Idempotent Operation
  33. Right Operation is Distributive over Idempotent Operation
  34. Left and Right Identity are the Same
  35. Identity is Unique
  36. Identity Property in Semigroup
  37. Identity of Monoid is Cancellable
  38. Identity of Cancellable Monoid is Identity of Submonoid
  39. Identity of Submonoid is not necessarily Identity of Monoid
  40. Set of all Self-Maps is Monoid
  41. Cancellable Elements of Monoid form Submonoid
  42. Idempotent Elements form Submonoid of Commutative Monoid


  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


  1. More than One Right Zero then No Left Zero
  2. Set System Closed under Union is Commutative Semigroup
  3. Identity of Power Set with Union
  4. Power Set with Union is Commutative Monoid
  5. Set System Closed under Intersection is Commutative Semigroup
  6. Identity of Power Set with Intersection
  7. Power Set with Intersection is Commutative Monoid
  8. Invertible Element of Associative Structure is Cancellable
  9. Right Cancellable Commutative Operation is Left Cancellable
  10. Left Cancellable Commutative Operation is Right Cancellable
  11. Commutation with Inverse in Monoid
  12. Commutation of Inverses in Monoid
  13. Inverse of Commuting Pair
  14. Product of Commuting Elements with Inverses
  15. Cancellation Laws
  16. Invertible Elements of Monoid form Subgroup of Cancellable Elements
  17. Structure Induced by Associative Operation is Associative
  18. Structure Induced by Commutative Operation is Commutative
  19. Induced Structure Identity
  20. 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


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


  1. Group is not Empty
  2. Identity is only Idempotent Element in Group
  3. Group has Latin Square Property
  4. Self-Inverse Elements Commute iff Product is Self-Inverse
  5. Power Set of Group under Induced Operation is Semigroup
  6. Inverse of Product of Subsets of Group
  7. Set Equivalence of Regular Representations
  8. All Elements Self-Inverse then Abelian
  9. Commutation Property in Group
  10. Group Element Commutes with Inverse
  11. Identity Mapping is Automorphism
  12. Identity Mapping is Automorphism/Groups
  13. Identity Mapping is Group Endomorphism
  14. Inversion Mapping is Automorphism iff Group is Abelian
  15. Mapping to Square is Endomorphism iff Abelian
  16. Induced Group Product is Homomorphism iff Commutative
  17. Induced Group Product is Homomorphism iff Commutative/Corollary
  18. Automorphism Group is Subgroup of Symmetric Group
  19. Isomorphism Preserves Commutativity
  20. Inner Automorphisms form Subgroup of Symmetric Group
  21. Opposite Group is Group
  22. Symmetric Difference on Power Set forms Abelian Group
  23. One-Step Subgroup Test
  24. Two-Step Subgroup Test
  25. Identity of Subgroup
  26. Inverse of Subgroup
  27. Intersection of Subgroups is Subgroup
  28. Elements of Group with Equal Images under Homomorphisms form Subgroup
  29. Product of Subgroup with Itself
  30. Inverse of Subgroup
  31. One-Step Subgroup Test using Subset Product
  32. Two-Step Subgroup Test using Subset Product
  33. Group Homomorphism Preserves Subgroups

Conjugacy in Groups

  1. Conjugacy is Equivalence Relation


  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


  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
  13. Zero Product with Proper Zero Divisor is with Zero Divisor
  14. Unit Not Zero Divisor
  15. Zero Divisor Product is Zero Divisor
  16. Product is Zero Divisor means Zero Divisor
  17. Ring Element is Zero Divisor iff not Cancellable
  18. Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors
  19. Idempotent Elements of Ring with No Proper Zero Divisors
  20. Division Ring has No Proper Zero Divisors
  21. Non-Zero Elements of Division Ring form Group
  22. Equivalence of Definitions of Integral Domain
  23. Null Ring and Ring Itself Subrings
  24. Subring Test
  25. Subdomain Test