Binary Operations Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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