Algebraic Structures

(代数系)

Discrete Mathematics I

11th lecture, Dec. 12, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture11.html

Martin J. Dürst

© 2006-14 Martin J. Dürst 青山学院大学

Today's Schedule

• Summary and homework for last lecture
• Algebraic Structures
  • Groups
  • Rings, fields, latices

Summary of Last Lecture

We defined the following properties of binary relations:

1. Reflexive: ∀x∈A:xRx; ∀x∈A: (x, x) ∈ R
2. Symmetric: ∀x, y ∈A: xRy ⇔ yRx;
   ∀x, y ∈A: (x, y) ∈ R ⇔ (y, x) ∈ R
3. Antisymmetric: ∀x, y ∈A: xRy ∧ yRx ⇒ x=y
4. Transitive: ∀x, y, z ∈A: xRy ∧ yRz ⇒ xRz

A relation that is reflexive, antisymmetric, and transitive is a (partial) order relation.

A relation that is reflexive, symmetric, and transitive is an equivalence relation.

Unsubmitted Homework?

Homework submitted in paper form is listed as unsubmitted at http://moo.sw.it.aoyama.ac.jp.

Do not worry about this.

Video

Please use the video recordings!

Often they are already available on Monday.

Last Week's Homework:
Combinations of Properties of Relations

Investigate all combinations of the four properties of relations introduced in this lecture (reflexive, symmetric, antisymmetric, transitive). For each combination, give a minimal example or explain why such a combination is impossible.

Hint: There are 16 combinations. Two combinations are impossible. Two combinations need a set of four elements for a minimal example. Three combinations need a set of two elements for a minimal example. Two combinations need a set of one element for a minimal example. The other combinations need a set of three elements for a minimal example.

Hint: Use {a, b, c} for a set with three elements.

Hint: Present the 16 combinations in a table similar to the tables used in the homework of lecture 4.

Homework Solution

[都合により削除] 

Algebraic Structure

Very general view on mathematical objects

• One (or more) set(s)
• One (or more) operation(s) between the elements of the set(s)
  Condition: The results of the operation(s) also have to be an element of the set(s)
  
• One (or more) axioms
• Proof of theorems and properties from the axioms

Example of Algebraic Structure: Group

• One set (A)
• One binary operation (∘; ∀b,c∈A: b∘c∈A)
• Three axioms:
  • Associativity (∀b,c,d∈A: (b∘c)∘d = b∘(c∘d))
  • (existence of a) unit element e (∃e∈A: ∀b∈A: e∘b = b = b∘e)
  • (existence of an) inverse element b' (∀b∈A: ∃b'∈A: b∘b' = e = b'∘b)
• Note: Commutativity is not necessary

The Integers with Addition as a Group

• Set: ℤ (integers)
• Operation: + (addition)
• Associativity: ∀b,c,d∈ℤ: (b+c)+d = b+(c+d)
• Unit element: 0
• Inverse element: b' = -b

The Reals with Multiplication as a Group

• Set: ℝ-{0} (real numbers without 0)
• Operation: · (multiplication)
• Associativity: ∀b,c,d∈(ℝ-{0}): (b·c)·d = b·(c·d)
• Unit element: 1
• Inverse element: b' = 1/b

The Positive Reals with Multiplication as a Group

• Set: ℝ⁺ (positive real numbers)
• Operation: · (multiplication)
• Associativity: ∀b,c,d∈ℝ⁺: (b·c)·d = b·(c·d)
• Unit element: 1
• Inverse element: b' = 1/b

Permutations

• There are n! permutations of elements from a set S with size |S|=n
• Permutations can be seen as ordered selections
  Example: From the set {Aoyama, Sagamihara} we can create the permutations (Aoyama, Sagamihara) and (Sagamihara, Aoyama)
  Example: From the set {cat, dog, horse, cow}, we can select the permutation (dog, cow, cat, horse) (and 23 others)

Permutations as Exchanges

• Permutations can be seen as ways to exchange elements
  Example: For a touple/list with two elements, we there are two permutations:
  
  1. One permutation that keeps the same order (1, 2)
  2. One permutation that changes the order of the elements (2, 1)
• We denote such permutations by assuming we start with a touple of the first n integers ((1, 2)) and show the result of the permutation
• Example: The touple (cat, dog, horse, cow), when permuted with the permutation (2, 4, 1, 3), results in (dog, cow, cat, horse)

Composition of Permutations

• Permutations, when seen as exchanging elements, can be composed
• We use ∘ to denote composition
• Example: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
• Composition of permutations can be show by using cards
• Cut out and use the cards at permutations.svg

Symmetric Groups

• The permutations of sets of size n together with composition form a group:
  • All compositions of permutations result in another permutation
  • Permutations are associative
  • The unit element is (1, 2, 3, 4, ...)
  • Each permutation has an inverse
    Example: The inverse of (2, 4, 1, 3) is (3, 1, 4, 2)
  • Commutativity doesn't hold
    Example: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
    (1, 4, 2, 3) ∘ (2, 4, 1, 3) = (4, 3, 1, 2)
• These groups are called symmetric groups of order n

Algebraic Structures Related to Groups

• An Abelian group is a group where the operation is also commutative
  Examples:
  • (ℤ; +): Integers with addition
  • (ℝ-{0}; ·) Reals (excluding 0) with multiplication
• A semigroup is a group without unit element or inverse elements

More Algebraic Structures

• Ring: Two operations, 'addition' and 'multiplication'; forms an Abelian group under addition and a semigroup under multiplication, and addition distributes over multiplication.
  Examples: ℝ, ring of polynomials
• Field
• Lattice: Two operations; both operations are associative and commutative, and respect absorbtion laws

This Week's Homework

Deadline: December 18, 2014 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Create a table of the symmetric group of order 3. Structure the table so that it looks like a multiplication table, with a row and a column for each permutation. Use lexical order for the permutations.

Glossary

algebraic structure

代数系

group

群

group theory

群理論

symmetric group

対称群

Abelian group

アベル群、可換群

semigroup

半群

ring

環

polynomial

多項式

field

体

lattice

束

multiplication table

九九 (表)