Binary Operations - University of Southern Mississippi...logo1 Introduction Semigroups Structures...

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

Transcript of Binary Operations - University of Southern Mississippi...logo1 Introduction Semigroups Structures...

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Introduction Semigroups Structures Partial Operations

Binary Operations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.

2. But it turns out to be inefficient. For every new example,we would need to reestablish all properties.

3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient.

For every new example,we would need to reestablish all properties.

3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.

3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation

, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions

, or, for all vector spaces, etc.5. Visualization becomes easier: Typically we will think of

one nice entity with the properties in question.6. As long as we don’t use other properties of our mental

image, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier

: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct.

This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,

we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have

certain properties in common and then derive furtherproperties from these common properties.

4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.

5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.

6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.

2. A binary operation ◦ : S×S → S is called associative ifffor all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).

3. Addition of natural numbers and multiplication of naturalnumbers are both associative operations.

4. Division of nonzero rational numbers is not (pardon thejump).

5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 naturalnumbers are both associative operations.

4. Division of nonzero rational numbers is not (pardon thejump).

5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 naturalnumbers are both associative operations.

4. Division of nonzero rational numbers is not (pardon thejump).

5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 thejump).

5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 thejump).

5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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

6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)

Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)Then again, inflection means a lot in language:

“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Associative Operations1. 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 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition.

Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.

Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself

, then(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ 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 aset and F (S,S) is the set of all functions f : S → S from S toitself, then

(F (S,S),◦

)is a semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition.

Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.

Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a.

A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example.

(N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example.

Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative.

So the pair(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.

Example. (N,+) and (N, ·) are commutative semigroups.

Example. Composition of functions is associative, but notcommutative. So the pair

(F (S,S),◦

)is a non-commutative

semigroup.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition.

Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.

An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e.

A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example.

(N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example.

There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.

(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element.

(It’s the identityfunction f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.

Example. (N, ·) is a semigroup with neutral element 1.

Example. There is no neutral element (in N) for addition ofnatural numbers.

Example.(F (S,S),◦

)has a neutral element. (It’s the identity

function f (s) = s.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition.

Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup.

Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element.

That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof.

e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e

= e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′

= e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.

Proof. e = e◦ e′ = e′.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups'

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$

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Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups'

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%Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups N'

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups N'

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

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Binary Operations

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Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

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Binary Operations

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Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

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Binary Operations

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Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings

NBij(A)

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Binary Operations

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Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings

NBij(A)

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings

NBij(A)

Z, Zm

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$

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%Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings fields

NBij(A)

Z, Zm

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$

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings fields

NBij(A)

Z, Zm R, C, Zp (p prime)

'

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%Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings fields

NBij(A)

Z, Zm R, C, Zp (p prime)

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings vector spacesfields

NBij(A)

Z, Zm R, C, Zp (p prime)

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings vector spacesfields

NBij(A)

Z, Zm R5R, C, Zp (p prime)

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings vector spaces

algebras

fields

NBij(A)

Z, Zm R5R, C, Zp (p prime)

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Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroupsgroups

rings vector spaces

algebras

fields

NBij(A)

Z, Zm R5

F (D,R), R3

R, C, Zp (p prime)

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Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example.

Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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.

I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.

I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.

I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.

Example. Multiplication is distributive over addition.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

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).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition.

Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set.

A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example.

Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers.

We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers

, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.

Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.

Let’s define subtraction more precisely.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition.

Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m.

Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof.

Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m

and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m.

Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d

and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition.

Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m.

Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.

The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.

Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.

Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition.

Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,x,y ∈ N be so that n < m and y < x.

Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).

2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.

3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.

1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof.

We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1.

n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).

Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x.

Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy.

We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x.

We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy)

=((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)

= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x

,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute

(n+ y)+(dmn +dxy) =((n+ y)+dmn

)+dxy

=(n+(y+dmn)

)+dxy

=(n+(dmn + y)

)+dxy

=((n+dmn)+ y

)+dxy

= (m+ y)+dxy

= m+(y+dxy)= m+ x,

which proves part 1.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition.

Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set

nd

:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq.

Then we setnd

:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set

nd

:= q

, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set

nd

:= q, and call it thequotient of n and d.

The number n is also called thenumerator and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set

nd

:= q, and call it thequotient of n and d. The number n is also called thenumerator

and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set

nd

:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition.

Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations

logo1

Introduction Semigroups Structures Partial Operations

Proposition. Let m,n,d,e ∈ N.

1. Ifnd

andmd

both exist, then so doesm+n

dand

m+nd

=md

+nd

.

2. Ifnd

andme

both exist, then so doesmnde

andmnde

=me· n

d.

3. Ifnd

andmd

both exist and n < m, then so doesm−n

dand

m−nd

=md− n

d.

4. Ifnd

andme

both exist and ne = md, thennd

=me

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Binary Operations