Understanding Semigroups: Definition, Properties, and Structures, Lecture notes of Number Theory

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Binary Operations

Bernd Schr¨oder

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Why Work With Abstract Entities and Binary Operations?

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Why Work With Abstract Entities and Binary Operations?

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

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.
2. But it turns out to be inefficient. For every new example, we would need to reestablish all properties.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.
2. But it turns out to be inefficient. For every new example, we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that have certain properties in common and then derive further properties from these common properties.
4. In this fashion we obtain results that hold for all number systems with an associative operation

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.
2. But it turns out to be inefficient. For every new example, we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that have certain properties in common and then derive further properties from these common properties.
4. In this fashion we obtain results that hold for all number systems with an associative operation, or, for all continuous functions

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Why Work With Abstract Entities and Binary Operations?

1. Working with examples seems more intuitive.
2. But it turns out to be inefficient. For every new example, we would need to reestablish all properties.
3. It is more efficient to consider classes of objects that have certain properties in common and then derive further properties from these common properties.
4. In this fashion we obtain results that hold for all number systems with an associative operation, or, for all continuous functions, or, for all vector spaces, etc.
5. Visualization becomes easier

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

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 Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

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. Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Associative Operations

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Associative Operations

1. A binary operation on the set S is a function ◦ : S × S → S.
2. A binary operation ◦ : S × S → S is called associative iff for all a, b, c ∈ S we have that (a ◦ b) ◦ c = a ◦ (b ◦ c).

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Associative Operations

1. A binary operation on the set S is a function ◦ : S × S → S.
2. A binary operation ◦ : S × S → S is called associative iff for all a, b, c ∈ S we have that (a ◦ b) ◦ c = a ◦ (b ◦ c).
3. Addition of natural numbers

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Associative Operations

1. A binary operation on the set S is a function ◦ : S × S → S.
2. A binary operation ◦ : S × S → S is called associative iff for all a, b, c ∈ S we have that (a ◦ b) ◦ c = a ◦ (b ◦ c).
3. Addition of natural numbers and multiplication of natural numbers are both associative operations.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Associative Operations

1. A binary operation on the set S is a function ◦ : S × S → S.
2. A binary operation ◦ : S × S → S is called associative iff for all a, b, c ∈ S we have that (a ◦ b) ◦ c = a ◦ (b ◦ c).
3. Addition of natural numbers and multiplication of natural numbers are both associative operations.
4. Division of nonzero rational numbers is not

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science