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

Operations

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

   and:

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

5. Associative and Anticommutative

   Let $\circ$ be a binary operation on a set $S$.

   Let $\circ$ be both associative and anticommutative.

   Then:

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

6. Constant Operation is Commutative

   Let $S$ be a set.

   Let $x \sqbrk c y = c$ be a constant operation on $S$.

   Then $\sqbrk c$ is a commutative operation:

   $\forall x, y \in S: x \sqbrk c y = y \sqbrk c x$

7. Constant Operation is Associative

   Let $S$ be a set.

   Let $x \sqbrk c y = c$ be a constant operation on $S$.

   Then $\sqbrk c$ is an associative operation:

   $\forall x, y, z \in S: \paren {x \sqbrk c y} \sqbrk c z = x \sqbrk c \paren {y \sqbrk c z}$

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

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 Relation is Compatible with 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. All Elements of Left Operation are Left Zeroes
27. All Elements of Right Operation are 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 under Composition forms Monoid
41. Cancellable Elements of Monoid form Submonoid
42. 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
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. Pointwise Inverse in Induced Structure

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 of Ringoids is Ringoid

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 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

Groups

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 Structure of Group 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

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
13. Zero Product with Proper Zero Divisor is with Zero Divisor
14. Unit of Ring is 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 are Subrings
24. Subring Test
25. Subdomain Test