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Introduction Semigroups Structures Partial Operations Binary Operations Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Binary Operations Introduction Semigroups Structures

  1. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  2. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  3. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). 5. Natural language isn’t either: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  4. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). 5. Natural language isn’t either: (frequent flyer) bonus logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  5. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). 5. Natural language isn’t either: (frequent flyer) bonus � = frequent (flyer bonus) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  6. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). 5. Natural language isn’t either: (frequent flyer) bonus � = frequent (flyer bonus) Then again, inflection means a lot in language: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  7. Introduction Semigroups Structures Partial Operations Associative Operations 1. A binary operation on the set S is a function ◦ : S × S → S . 2. A binary operation ◦ : S × S → S is called associative iff for all a , b , c ∈ S we have that ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . 3. Addition of natural numbers and multiplication of natural numbers are both associative operations. 4. Division of nonzero rational numbers is not (pardon the jump). 5. Natural language isn’t either: (frequent flyer) bonus � = frequent (flyer bonus) Then again, inflection means a lot in language: “Alcohol must be consumed in the food court.” logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  8. Introduction Semigroups Structures Partial Operations Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  9. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  10. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  11. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  12. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  13. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  14. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  15. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  16. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. Example. Composition of functions is associative. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  17. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. Example. Composition of functions is associative. So if S is a set and F ( S , S ) is the set of all functions f : S → S from S to itself logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  18. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. Example. Composition of functions is associative. So if S is a set and F ( S , S ) is the set of all functions f : S → S from S to � � F ( S , S ) , ◦ itself, then is a semigroup. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  19. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ( S , ◦ ) is called a semigroup iff the operation ◦ is associative, that is, iff for all x , y , z ∈ S we have ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Example. ( N , +) and ( N , · ) are semigroups. Example. Composition of functions is associative. So if S is a set and F ( S , S ) is the set of all functions f : S → S from S to � � F ( S , S ) , ◦ itself, then is a semigroup. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  20. Introduction Semigroups Structures Partial Operations Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  21. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  22. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  23. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  24. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  25. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. ( N , +) and ( N , · ) are commutative semigroups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  26. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. ( N , +) and ( N , · ) are commutative semigroups. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  27. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. ( N , +) and ( N , · ) are commutative semigroups. Example. Composition of functions is associative, but not commutative. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  28. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. ( N , +) and ( N , · ) are commutative semigroups. Example. Composition of functions is associative, but not � � commutative. So the pair F ( S , S ) , ◦ is a non-commutative semigroup. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  29. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. Then ◦ is called commutative iff for all a , b ∈ S we have that a ◦ b = b ◦ a. A semigroup ( S , ◦ ) with commutative operation ◦ is also called a commutative semigroup . Example. ( N , +) and ( N , · ) are commutative semigroups. Example. Composition of functions is associative, but not � � commutative. So the pair F ( S , S ) , ◦ is a non-commutative semigroup. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  30. Introduction Semigroups Structures Partial Operations Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  31. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  32. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  33. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  34. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  35. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  36. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  37. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  38. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  39. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  40. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  41. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. � � F ( S , S ) , ◦ Example. has a neutral element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  42. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. � � F ( S , S ) , ◦ Example. has a neutral element. (It’s the identity function f ( s ) = s .) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  43. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S be a binary operation on S. An element e ∈ S is called a neutral element iff for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that contains a neutral element is also called a semigroup with a neutral element . Example. ( N , · ) is a semigroup with neutral element 1. Example. There is no neutral element (in N ) for addition of natural numbers. � � F ( S , S ) , ◦ Example. has a neutral element. (It’s the identity function f ( s ) = s .) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  44. Introduction Semigroups Structures Partial Operations Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  45. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  46. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  47. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  48. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  49. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . Proof. e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  50. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . Proof. e = e ◦ e ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  51. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . Proof. e = e ◦ e ′ = e ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  52. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , ◦ ) be a semigroup. Then S has at most one neutral element. That is, if e , e ′ are both elements so that for all x ∈ S we have e ◦ x = x = x ◦ e and e ′ ◦ x = x = x ◦ e ′ , then e = e ′ . Proof. e = e ◦ e ′ = e ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  53. Introduction Semigroups Structures Partial Operations Structures We Will Investigate logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  54. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ semigroups ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  55. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ semigroups N ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  56. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  57. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  58. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  59. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  60. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  61. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  62. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  63. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings fields Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  64. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings fields R , C , Z p ( p prime) Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  65. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) rings fields R , C , Z p ( p prime) Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  66. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) vector spaces rings fields R , C , Z p ( p prime) Z , Z m ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  67. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) vector spaces rings fields R , C , Z p ( p prime) Z , Z m R 5 ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  68. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) algebras vector spaces rings fields R , C , Z p ( p prime) Z , Z m R 5 ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  69. Introduction Semigroups Structures Partial Operations Structures We Will Investigate ✬ ✩ ✬ ✩ semigroups N groups Bij ( A ) algebras vector spaces rings fields R , C , Z p ( p prime) Z , Z m R 5 F ( D , R ) , R 3 ✫ ✪ ✫ ✪ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  70. Introduction Semigroups Structures Partial Operations Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  71. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  72. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  73. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. ◮ The operation ◦ called right distributive over ∗ iff for all a , b , c ∈ S we have that ( a ∗ b ) ◦ c = a ◦ c ∗ b ◦ c. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  74. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. ◮ The operation ◦ called right distributive over ∗ iff for all a , b , c ∈ S we have that ( a ∗ b ) ◦ c = a ◦ c ∗ b ◦ c. ◮ Finally, ◦ is called distributive over ∗ iff ◦ is left distributive and right distributive over ∗ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  75. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. ◮ The operation ◦ called right distributive over ∗ iff for all a , b , c ∈ S we have that ( a ∗ b ) ◦ c = a ◦ c ∗ b ◦ c. ◮ Finally, ◦ is called distributive over ∗ iff ◦ is left distributive and right distributive over ∗ . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  76. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. ◮ The operation ◦ called right distributive over ∗ iff for all a , b , c ∈ S we have that ( a ∗ b ) ◦ c = a ◦ c ∗ b ◦ c. ◮ Finally, ◦ is called distributive over ∗ iff ◦ is left distributive and right distributive over ∗ . Example. Multiplication is distributive over addition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  77. Introduction Semigroups Structures Partial Operations Definition. Let S be a set and let ◦ : S × S → S and ∗ : S × S → S be binary operations on S. ◮ The operation ◦ called left distributive over ∗ iff for all a , b , c ∈ S we have that a ◦ ( b ∗ c ) = a ◦ b ∗ a ◦ c. ◮ The operation ◦ called right distributive over ∗ iff for all a , b , c ∈ S we have that ( a ∗ b ) ◦ c = a ◦ c ∗ b ◦ c. ◮ Finally, ◦ is called distributive over ∗ iff ◦ is left distributive and right distributive over ∗ . Example. Multiplication is distributive over addition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  78. Introduction Semigroups Structures Partial Operations Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  79. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , +) be a commutative semigroup and let · be an associative binary operation that is distributive over + . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  80. Introduction Semigroups Structures Partial Operations Proposition. Let ( S , +) be a commutative semigroup and let · be an associative binary operation that is distributive over + . Then for all x , y , z , u ∈ S we have ( x + y )( z + u ) = ( xz + xu )+( yz + yu ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

  81. Introduction Semigroups Structures Partial Operations Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Binary Operations

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