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logo1
Introduction Semigroups Structures Partial Operations
Binary Operations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.
2. But it turns out to be inefficient. For every new example,we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient.
For every new example,we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that havecertain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation
, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions
, or, for all vector spaces, etc.5. Visualization becomes easier: Typically we will think of
one nice entity with the properties in question.6. As long as we don’t use other properties of our mental
image, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier
: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct.
This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Why Work With Abstract Entities and Binary Operations?
1. Working with examples seems more intuitive.2. But it turns out to be inefficient. For every new example,
we would need to reestablish all properties.3. It is more efficient to consider classes of objects that have
certain properties in common and then derive furtherproperties from these common properties.
4. In this fashion we obtain results that hold for all numbersystems with an associative operation, or, for allcontinuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier: Typically we will think ofone nice entity with the properties in question.
6. As long as we don’t use other properties of our mentalimage, results will be correct. This is how mathematicianscan work with entities like infinite dimensional spaces.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations
1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.
2. A binary operation ◦ : S×S → S is called associative ifffor all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).
3. Addition of natural numbers and multiplication of naturalnumbers are both associative operations.
4. Division of nonzero rational numbers is not (pardon thejump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).
3. Addition of natural numbers and multiplication of naturalnumbers are both associative operations.
4. Division of nonzero rational numbers is not (pardon thejump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers
and multiplication of naturalnumbers are both associative operations.
4. Division of nonzero rational numbers is not (pardon thejump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers
are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.
4. Division of nonzero rational numbers is not (pardon thejump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not
(pardon thejump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).
5. Natural language isn’t either:(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus
6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)
Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)Then again, inflection means a lot in language:
“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Associative Operations1. A binary operation on the set S is a function ◦ : S×S → S.2. A binary operation ◦ : S×S → S is called associative iff
for all a,b,c ∈ S we have that (a◦b)◦ c = a◦ (b◦ c).3. Addition of natural numbers and multiplication of natural
numbers are both associative operations.4. Division of nonzero rational numbers is not (pardon the
jump).5. Natural language isn’t either:
(frequent flyer) bonus 6= frequent (flyer bonus)Then again, inflection means a lot in language:“Alcohol must be consumed in the food court.”
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.
Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative
, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example.
(N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example.
Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative.
So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself
, then(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then (S,◦) is called a semigroup iff theoperation ◦ is associative, that is, iff for all x,y,z ∈ S we have(x◦ y)◦ z = x◦ (y◦ z).
Example. (N,+) and (N, ·) are semigroups.
Example. Composition of functions is associative. So if S is aset and F (S,S) is the set of all functions f : S → S from S toitself, then
(F (S,S),◦
)is a semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.
Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a.
A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example.
(N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example.
Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative.
So the pair(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. Then ◦ is called commutative iff for all a,b ∈ Swe have that a◦b = b◦a. A semigroup (S,◦) with commutativeoperation ◦ is also called a commutative semigroup.
Example. (N,+) and (N, ·) are commutative semigroups.
Example. Composition of functions is associative, but notcommutative. So the pair
(F (S,S),◦
)is a non-commutative
semigroup.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S.
An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e.
A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example.
(N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example.
There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.
(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element.
(It’s the identityfunction f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S be a binaryoperation on S. An element e ∈ S is called a neutral element ifffor all a ∈ S we have e◦a = a = a◦ e. A semigroup thatcontains a neutral element is also called a semigroup with aneutral element.
Example. (N, ·) is a semigroup with neutral element 1.
Example. There is no neutral element (in N) for addition ofnatural numbers.
Example.(F (S,S),◦
)has a neutral element. (It’s the identity
function f (s) = s.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition.
Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup.
Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element.
That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof.
e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e
= e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′
= e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,◦) be a semigroup. Then S has at most oneneutral element. That is, if e,e′ are both elements so that for allx ∈ S we have e◦x = x = x◦e and e′ ◦x = x = x◦e′, then e = e′.
Proof. e = e◦ e′ = e′.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroups'
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroups'
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%Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
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Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
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Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
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Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
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Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings fields
NBij(A)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings fields
NBij(A)
Z, Zm R, C, Zp (p prime)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings fields
NBij(A)
Z, Zm R, C, Zp (p prime)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings vector spacesfields
NBij(A)
Z, Zm R, C, Zp (p prime)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings vector spacesfields
NBij(A)
Z, Zm R5R, C, Zp (p prime)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings vector spaces
algebras
fields
NBij(A)
Z, Zm R5R, C, Zp (p prime)
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Structures We Will Investigate
semigroupsgroups
rings vector spaces
algebras
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NBij(A)
Z, Zm R5
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Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example.
Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set and let ◦ : S×S → S and∗ : S×S → S be binary operations on S.
I The operation ◦ called left distributive over ∗ iff for alla,b,c ∈ S we have that a◦ (b∗ c) = a◦b∗a◦ c.
I The operation ◦ called right distributive over ∗ iff for alla,b,c ∈ S we have that (a∗b)◦ c = a◦ c∗b◦ c.
I Finally, ◦ is called distributive over ∗ iff ◦ is leftdistributive and right distributive over ∗.
Example. Multiplication is distributive over addition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition.
Let (S,+) be a commutative semigroup and let ·be an associative binary operation that is distributive over +.Then for all x,y,z,u ∈ S we have(x+ y)(z+u) = (xz+ xu)+(yz+ yu).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,+) be a commutative semigroup and let ·be an associative binary operation that is distributive over +.
Then for all x,y,z,u ∈ S we have(x+ y)(z+u) = (xz+ xu)+(yz+ yu).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let (S,+) be a commutative semigroup and let ·be an associative binary operation that is distributive over +.Then for all x,y,z,u ∈ S we have(x+ y)(z+u) = (xz+ xu)+(yz+ yu).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set.
A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example.
Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers.
We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers
, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let S be a set. A partial (binary) operation on S isa function ◦ : A → S, where A is a subset of S×S.
Example. Subtraction of natural numbers. We can subtractsmaller numbers from larger numbers, but not the other wayround.
Let’s define subtraction more precisely.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition.
Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m.
Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof.
Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m
and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m.
Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d
and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition.
Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m.
Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.
The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let n,m ∈ N be so that n < m. Then the number dso that n+d = m is unique.
Proof. Let n,m ∈ N be so that n < m and let d, d be so thatn+d = m and n+ d = m. Then n+ d = m = n+d and henced = d.
Definition. Let n,m ∈ N be so that n < m. Then we setm−n := d, where d is the unique number so that n+d = m.The number d is also called the difference between m and n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition.
Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,x,y ∈ N be so that n < m and y < x.
Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).
2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.
3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,x,y ∈ N be so that n < m and y < x. Thenthe following hold.
1. n+ y < m+ x and (m+ x)− (n+ y) = (m−n)+(x− y).2. nx < mx and mx−nx = (m−n)x.3. If n+ x = m+ y, then m−n = x− y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof.
We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1.
n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).
Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x.
Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy.
We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x.
We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy)
=((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)
= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x
,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proof. We only prove part 1. n+ y < m+ x and(m+ x)− (n+ y) = (m−n)+(x− y).Let dmn and dxy be so that n+dmn = m and y+dxy = x. Then(m−n)+(x− y) = dmn +dxy. We must show that(n+ y)+(dmn +dxy) = m+ x. We compute
(n+ y)+(dmn +dxy) =((n+ y)+dmn
)+dxy
=(n+(y+dmn)
)+dxy
=(n+(dmn + y)
)+dxy
=((n+dmn)+ y
)+dxy
= (m+ y)+dxy
= m+(y+dxy)= m+ x,
which proves part 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition.
Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set
nd
:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq.
Then we setnd
:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set
nd
:= q
, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set
nd
:= q, and call it thequotient of n and d.
The number n is also called thenumerator and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set
nd
:= q, and call it thequotient of n and d. The number n is also called thenumerator
and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Definition. Let n,d ∈ N be so that n > d and so that there is aq ∈ N so that n = dq. Then we set
nd
:= q, and call it thequotient of n and d. The number n is also called thenumerator and the number d is called the denominator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition.
Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
logo1
Introduction Semigroups Structures Partial Operations
Proposition. Let m,n,d,e ∈ N.
1. Ifnd
andmd
both exist, then so doesm+n
dand
m+nd
=md
+nd
.
2. Ifnd
andme
both exist, then so doesmnde
andmnde
=me· n
d.
3. Ifnd
andmd
both exist and n < m, then so doesm−n
dand
m−nd
=md− n
d.
4. Ifnd
andme
both exist and ne = md, thennd
=me
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Binary Operations
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