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logo1 Introduction Semigroups Structures Partial Operations

Binary Operations

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Why Work With Abstract Entities and Binary Operations?

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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 further properties from these common properties.

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.

• 5. Visualization becomes easier

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.

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

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.

• 5. Visualization becomes easier: Typically we will think of
• ne 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.

Bernd Schr¨

• der

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 further properties from these common properties.

• 4. In this fashion we obtain results that hold for all number

systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.

• 5. Visualization becomes easier: Typically we will think of
• ne 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 mathematicians can work with entities like infinite dimensional spaces.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Associative Operations

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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).

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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).

Bernd Schr¨

• der

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:

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 = frequent (flyer bonus)

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

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 binary
• peration on S.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ is associative

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ is associative, that is, iff for all x,y,z ∈ S we have

(x◦y)◦z = x◦(y◦z).

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ is associative, that is, iff for all x,y,z ∈ S we have

(x◦y)◦z = x◦(y◦z). Example.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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, then

• F(S,S),◦
• is a semigroup.

Bernd Schr¨

• der

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 binary
• peration on S. Then (S,◦) is called a semigroup iff the
• peration ◦ 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, then

• F(S,S),◦
• is a semigroup.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

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 binary
• peration on S.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ is also called a commutative semigroup.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ is also called a commutative semigroup.

Example.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ is also called a commutative semigroup.
• Example. (N,+) and (N,·) are commutative semigroups.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ is also called a commutative semigroup.
• Example. (N,+) and (N,·) are commutative semigroups.

Example.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ is also called a commutative semigroup.
• Example. (N,+) and (N,·) are commutative semigroups.
• Example. Composition of functions is associative, but not

commutative.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ 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.

Bernd Schr¨

• der

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 binary
• peration on S. Then ◦ is called commutative iff for all a,b ∈ S

we have that a◦b = b◦a. A semigroup (S,◦) with commutative

• peration ◦ 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.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

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 binary
• peration on S.

Bernd Schr¨

• der

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 binary
• peration on S. An element e ∈ S is called a neutral element iff

for all a ∈ S we have e◦a = a = a◦e.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

Bernd Schr¨

• der

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 binary
• peration 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.

• F(S,S),◦
• has a neutral element.

Bernd Schr¨

• der

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 binary
• peration 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.

• F(S,S),◦
• has a neutral element. (It’s the identity

function f(s) = s.)

Bernd Schr¨

• der

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 binary
• peration 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.

• F(S,S),◦
• has a neutral element. (It’s the identity

function f(s) = s.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

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

Bernd Schr¨

• der

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 one

neutral element.

Bernd Schr¨

• der

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 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′.

Bernd Schr¨

• der

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 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.

Bernd Schr¨

• der

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 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

Bernd Schr¨

• der

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 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′

Bernd Schr¨

• der

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 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′.

Bernd Schr¨

• der

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 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′.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups

✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups N

✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups N

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups N

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups N Bij(A)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups N Bij(A)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings N Bij(A)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings N Bij(A) Z, Zm

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings N Bij(A) Z, Zm

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings fields N Bij(A) Z, Zm

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings fields N Bij(A) Z, Zm R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings fields N Bij(A) Z, Zm R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings vector spaces fields N Bij(A) Z, Zm R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings vector spaces fields N Bij(A) Z, Zm R5 R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings vector spaces algebras fields N Bij(A) Z, Zm R5 R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Structures We Will Investigate

semigroups groups rings vector spaces algebras fields N Bij(A) Z, Zm R5 F(D,R), R3 R, C, Zp (p prime)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

◮ The operation ◦ called left distributive over ∗ iff for all

a,b,c ∈ S we have that a◦(b∗c) = a◦b∗a◦c.

Bernd Schr¨

• der

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.

◮ 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.

Bernd Schr¨

• der

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.

◮ 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 ∗.

Bernd Schr¨

• der

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.

◮ 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.

Bernd Schr¨

• der

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.

◮ 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.

Bernd Schr¨

• der

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.

◮ 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.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proposition.

Bernd Schr¨

• der

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 +.

Bernd Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Definition. Let S be a set.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S. Example.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

• Example. Subtraction of natural numbers.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

• Example. Subtraction of natural numbers. We can subtract

smaller numbers from larger numbers

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

• Example. Subtraction of natural numbers. We can subtract

smaller numbers from larger numbers, but not the other way round.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

• Example. Subtraction of natural numbers. We can subtract

smaller numbers from larger numbers, but not the other way round.

Bernd Schr¨

• der

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 is

a function ◦ : A → S, where A is a subset of S×S.

• Example. Subtraction of natural numbers. We can subtract

smaller numbers from larger numbers, but not the other way round. Let’s define subtraction more precisely.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proposition.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

Bernd Schr¨

• der

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 d

so that n+d = m is unique. Proof.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d. Definition.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d.

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

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d.

• Definition. Let n,m ∈ N be so that n < m. Then we set

m−n := d, where d is the unique number so that n+d = m.

Bernd Schr¨

• der

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 d

so that n+d = m is unique.

• Proof. Let n,m ∈ N be so that n < m and let d, ˜

d be so that n+d = m and n+ ˜ d = m. Then n+ ˜ d = m = n+d and hence d = ˜ d.

• Definition. Let n,m ∈ N be so that n < m. Then we set

m−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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proposition.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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. Then

the following hold.

Bernd Schr¨

• der

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. Then

the following hold.

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

Bernd Schr¨

• der

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. Then

the 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.

Bernd Schr¨

• der

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. Then

the 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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proof. We only prove part 1.

Bernd Schr¨

• der

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).

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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.

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 Schr¨

• der

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 Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Definition.

Bernd Schr¨

• der

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 a

q ∈ N so that n = dq.

Bernd Schr¨

• der

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 a

q ∈ N so that n = dq. Then we set n d := q

Bernd Schr¨

• der

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 a

q ∈ N so that n = dq. Then we set n d := q, and call it the quotient of n and d.

Bernd Schr¨

• der

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 a

q ∈ N so that n = dq. Then we set n d := q, and call it the quotient of n and d. The number n is also called the numerator

Bernd Schr¨

• der

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 a

q ∈ N so that n = dq. Then we set n d := q, and call it the quotient of n and d. The number n is also called the numerator and the number d is called the denominator.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proposition. Let m,n,d,e ∈ N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proposition. Let m,n,d,e ∈ N.
• 1. If n

d and m d both exist, then so does m+n d and m+n d = m d + n d.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proposition. Let m,n,d,e ∈ N.
• 1. If n

d and m d both exist, then so does m+n d and m+n d = m d + n d.

• 2. If n

d and m e both exist, then so does mn de and mn de = m e · n d.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proposition. Let m,n,d,e ∈ N.
• 1. If n

d and m d both exist, then so does m+n d and m+n d = m d + n d.

• 2. If n

d and m e both exist, then so does mn de and mn de = m e · n d.

• 3. If n

d and m d both exist and n < m, then so does m−n d and m−n d = m d − n d.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations

logo1 Introduction Semigroups Structures Partial Operations

• Proposition. Let m,n,d,e ∈ N.
• 1. If n

d and m d both exist, then so does m+n d and m+n d = m d + n d.

• 2. If n

d and m e both exist, then so does mn de and mn de = m e · n d.

• 3. If n

d and m d both exist and n < m, then so does m−n d and m−n d = m d − n d.

• 4. If n

d and m e both exist and ne = md, then n d = m e .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Binary Operations