Introduction
Semigroups
Structures
Partial Operations
Binary Operations
Bernd Schröder
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Why Work With Abstract Entities and Binary Operations?
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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
one nice entity with the properties in question.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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
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.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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
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 mathematicians
can work with entities like infinite dimensional spaces.
logo1
Bernd Schröder
Binary Operations
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Associative Operations
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Associative Operations
1. A binary operation on the set S is a function ◦ : S × S → S.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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)
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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:
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Example. Composition of functions is associative.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is a
set and F (S, S) is the set of all functions f : S → S from S to
itself
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is a
set and F (S, S) is the set
of all functions f : S → S from S to
itself, then F (S, S), ◦ is a semigroup.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then (S, ◦) is called a semigroup iff the
operation ◦ is associative, that is, iff for all x, y, z ∈ S we have
(x ◦ y) ◦ z = x ◦ (y ◦ z).
Example. (N, +) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is a
set and F (S, S) is the set
of all functions f : S → S from S to
itself, then F (S, S), ◦ is a semigroup.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example. (N, +) and (N, ·) are commutative semigroups.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example. (N, +) and (N, ·) are commutative semigroups.
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example. (N, +) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but not
commutative.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example. (N, +) and (N, ·) are commutative semigroups.
Example. Composition of functions is
associative, but not
commutative. So the pair F (S, S), ◦ is a non-commutative
semigroup.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. Then ◦ is called commutative iff for all a, b ∈ S
we have that a ◦ b = b ◦ a. A semigroup (S, ◦) with commutative
operation ◦ is also called a commutative semigroup.
Example. (N, +) and (N, ·) are commutative semigroups.
Example. Composition of functions is
associative, but not
commutative. So the pair F (S, S), ◦ is a non-commutative
semigroup.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Example.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Example. F (S, S), ◦ has a neutral element.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Example. F (S, S), ◦ has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set and let ◦ : S × S → S be a binary
operation on S. An element e ∈ S is called a neutral element iff
for all a ∈ S we have e ◦ a = a = a ◦ e. A semigroup that
contains a neutral element is also called a semigroup with a
neutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition of
natural numbers.
Example. F (S, S), ◦ has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Proof.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Proof. e
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Proof. e = e ◦ e0
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Proof. e = e ◦ e0 = e0 .
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, ◦) be a semigroup. Then S has at most one
neutral element. That is, if e, e0 are both elements so that for all
x ∈ S we have e ◦ x = x = x ◦ e and e0 ◦ x = x = x ◦ e0 , then e = e0 .
Proof. e = e ◦ e0 = e0 .
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
$
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semigroups
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
$
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semigroups
&
Bernd Schröder
Binary Operations
N
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
&
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Bernd Schröder
Binary Operations
N
$
$
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logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
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groups
&
&
Bernd Schröder
Binary Operations
N
$
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logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
&
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Bernd Schröder
Binary Operations
N
Bij(A)
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
&
&
Bernd Schröder
Binary Operations
N
Bij(A)
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logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
N
Bij(A)
$
$
rings
&
&
Bernd Schröder
Binary Operations
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
N
Bij(A)
$
$
rings
Z, Zm
&
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Bernd Schröder
Binary Operations
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
N
Bij(A)
$
$
rings
Z, Zm
&
&
Bernd Schröder
Binary Operations
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
rings
$
$
N
Bij(A)
fields
Z, Zm
&
&
Bernd Schröder
Binary Operations
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
rings
fields
Z, Zm
R, C, Zp (p prime)
&
&
Bernd Schröder
Binary Operations
$
$
N
Bij(A)
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
rings
fields
Z, Zm
R, C, Zp (p prime)
&
&
Bernd Schröder
Binary Operations
$
$
N
Bij(A)
%
%
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
rings
fields
Z, Zm
R, C, Zp (p prime)
&
&
Bernd Schröder
Binary Operations
$
$
N
Bij(A)
vector spaces
%
%
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
rings
fields
Z, Zm
R, C, Zp (p prime)
&
&
Bernd Schröder
Binary Operations
$
$
N
Bij(A)
vector spaces
R5
%
%
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
$
$
N
Bij(A)
algebras
rings
fields
Z, Zm
R, C, Zp (p prime)
&
&
Bernd Schröder
Binary Operations
vector spaces
R5
%
%
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Structures We Will Investigate
'
semigroups
'
groups
$
$
N
Bij(A)
algebras
rings
fields
Z, Zm
R, C, Zp (p prime)
vector spaces
R5
F (D, R), R3
&
&
Bernd Schröder
Binary Operations
%
%
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
I The operation ◦ called right distributive over ∗ iff for all
a, b, c ∈ S we have that (a ∗ b) ◦ c = a ◦ c ∗ b ◦ c.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
I The operation ◦ called right distributive over ∗ iff for all
a, b, c ∈ S we have that (a ∗ b) ◦ c = a ◦ c ∗ b ◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is left
distributive and right distributive over ∗.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
I The operation ◦ called right distributive over ∗ iff for all
a, b, c ∈ S we have that (a ∗ b) ◦ c = a ◦ c ∗ b ◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is left
distributive and right distributive over ∗.
Example.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
I The operation ◦ called right distributive over ∗ iff for all
a, b, c ∈ S we have that (a ∗ b) ◦ c = a ◦ c ∗ b ◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is left
distributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 all
a, b, c ∈ S we have that a ◦ (b ∗ c) = a ◦ b ∗ a ◦ c.
I The operation ◦ called right distributive over ∗ iff for all
a, b, c ∈ S we have that (a ∗ b) ◦ c = a ◦ c ∗ b ◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is left
distributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let (S, +) be a commutative semigroup and let ·
be an associative binary operation that is distributive over +.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let S be a set.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let n, m ∈ N be so that n < m.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
Definition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
Definition. Let n, m ∈ N be so that n < m.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
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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, d̃ be so that
n + d = m and n + d̃ = m. Then n + d̃ = m = n + d and hence
d = d̃.
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.
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Louisiana Tech University, College of Engineering and Science
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Partial Operations
Proposition.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, x, y ∈ N be so that n < m and y < x.
Bernd Schröder
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Louisiana Tech University, College of Engineering and Science
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Partial Operations
Proposition. Let m, n, x, y ∈ N be so that n < m and y < x. Then
the following hold.
Bernd Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
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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 Schröder
Binary Operations
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Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
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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.
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Proof.
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Partial Operations
Proof. We only prove part 1.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let n, d ∈ N be so that n > d and so that there is a
n
q ∈ N so that n = dq. Then we set := q
d
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let n, d ∈ N be so that n > d and so that there is a
n
q ∈ N so that n = dq. Then we set := q, and call it the
d
quotient of n and d.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let n, d ∈ N be so that n > d and so that there is a
n
q ∈ N so that n = dq. Then we set := q, and call it the
d
quotient of n and d. The number n is also called the
numerator
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Definition. Let n, d ∈ N be so that n > d and so that there is a
n
q ∈ N so that n = dq. Then we set := q, and call it the
d
quotient of n and d. The number n is also called the
numerator and the number d is called the denominator.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, d, e ∈ N.
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, d, e ∈ N.
n
m
m+n
1. If and both exist, then so does
and
d
d
d
m+n m n
= + .
d
d d
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, d, e ∈ N.
n
m
m+n
1. If and both exist, then so does
and
d
d
d
m+n m n
= + .
d
d d
n
m
mn
mn m n
2. If and both exist, then so does
and
= · .
d
e
de
de
e d
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, d, e ∈ N.
n
m
m+n
1. If and both exist, then so does
and
d
d
d
m+n m n
= + .
d
d d
n
m
mn
mn m n
2. If and both exist, then so does
and
= · .
d
e
de
de
e d
n
m
m−n
3. If and both exist and n < m, then so does
and
d
d
d
m−n m n
= − .
d
d d
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science
Introduction
Semigroups
Structures
Partial Operations
Proposition. Let m, n, d, e ∈ N.
n
m
m+n
1. If and both exist, then so does
and
d
d
d
m+n m n
= + .
d
d d
n
m
mn
mn m n
2. If and both exist, then so does
and
= · .
d
e
de
de
e d
n
m
m−n
3. If and both exist and n < m, then so does
and
d
d
d
m−n m n
= − .
d
d d
n
m
n m
4. If and both exist and ne = md, then = .
d
e
d
e
Bernd Schröder
Binary Operations
logo1
Louisiana Tech University, College of Engineering and Science