情報数学 I

第九回 (2013年12月 6日)

半順序から群へ
From Half-Orders to Groups

Martin J. Dürst

http://www.sw.it.aoyama.ac.jp/2013/Math1/lecture9.html

© 2005-13 Martin J. Dürst 青山学院大学

今日の予定

• 前回の残り (推移閉包)、復習と宿題
• 半順序、順序関係、全順序
• 代数系、群、対称群

前回の復習

A の中の関係 R に対して次の性質がありうる:

1. 反射的 (reflexive): xRx; x∈A ⇒ (x, x) ∈ R
2. 対称的 (symmetric): xRy ⇒ yRx ; (x, y) ∈ R ⇒ (y, x) ∈ R
3. 反対称的 (antisymmetric): xRy ∧ yRx ⇒ x=y
4. 推移的 (transitive): xRy ∧ yRz ⇒ xRz

前回の宿題: 関係の性質の組み合わせ

関係の性質 (反射的、対称的、推移的、反対称的) の有無の組合せを一つ一つ調べ、できるだけ小さい具体例 (例えば {a, b, c} の中の関係とか {a, b, c, d} の中の関係として) を作りなさい。

Partial Order

• If a relation is reflexive, antisymmetric, and transitive, then it is called a partial order relation
• This is also often just called an order relation
• The set on which the relation is defined is called a partially ordered set or just an ordered set
• The symbol ≤ is often used for order relations
• For any order relation ≤, the order relation ≥ and the relations > and < are also defined
• In any order relation, two elements x and y can be in any of four mutually exclusive relationships:
  x < y, x = y, x > y, or there is no relationship between x and y

Examples of Order Relations

• The "divisible by" relation on the set of integers ≥1, or a subset thereof
• The "subset" relation on a set of sets

Some examples need a careful definition:

• The relation on a set of tasks, where some tasks need be done before or at the same time as others
• The relation "stronger or equal than" in a Tennis tournament, defined by (the transitive closure of) the tournament results

Hasse Diagram

An order relation can be represented by a Hasse diagram.

How to convert a directed graph of an order relation to a Hasse diagram:

1. Remove arrows that indicate reflexivity
2. Rearange the vertices of the graph so that all arrows point upwards (or downwards)
3. Remove the arrows that can be reconstructed using transitive closure
4. Remove the arrowheads

Example 1: The relation "divisible by" on the set {10, 5, 3, 2, 1}

Example 2: The relation "needs to be done before (or at the same time as)" for the above tasks 1. to 4.

Equivalence Relations and Order Relations in Matrix Representation

• The elements in a set A are not ordered.
• Therefore, we can exchange (permute) the rows and the columns in the matrix representation of a relation if and only if we use the same permutation for both rows and columns.
• A relation on a set A is an equivalence relation if and only if we can permute the rows and columns so that we obtain the following:
  
  • The areas of 1s form squares.
  • The centers of the squares are on the (main) diagonal of the matrix.
  • The squares don't overlap.
  • The entries on the (main) diagonal are all 1.
• A relation on a set A is an order relation if and only if we can permute the rows and columns so that we obtain the following:
  • All entries below the (main) diagonal [or above] are 0.
  • All entries on the (main) diagonal are 1.

全順序

ある集合 A の全ての元 a と b に対して

a≥b 又はb≥a が成り立つ場合、

≥ は全順序関係 (total order relation)

(あるいは線形順序関係 (linear order relation))

例: 整数、実数などの≥; 日付や時間; 辞書での単語の順番

代数系

(algebraic system)

• 集合が一つ (以上)
• その集合の元の間の演算が一つ以上
  (結果もまたその集合の元)
• 公理が一つ以上
• 公理から演算の性質などが証明可能

代数系の例: 群

(group)

演算は二項演算「∘」一つ。公理:

• 結合律
• 単位元の存在
• 逆元 (inverse element) の存在

例: (Z; +), (R-{0}; ×), 対称群 (symmetric group)

対称群

• n 次対称群は n 個の要素の置き換えすべて
• 対称群の要素は 1..n の順列で表す
  例: (2, 4, 1, 3) は (猫、犬、馬、牛) を
  (犬、牛、猫、馬) に置き換える
• 対称群の二次演算は置き換えの連結
  例: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
• 単位元は「何もしない」置き換え
  例: (1, 2, 3, 4)
• 逆元: (2, 4, 1, 3) の逆元は (3, 1, 4, 2)
• 結合律は成立するが、交換律は成り立たない:
  例: (2, 4, 1, 3) ∘ (1, 4, 2, 3) ≠
  (1, 4, 2, 3) ∘ (2, 4, 1, 3)

代数系の例

• 可換群 (アベル群、Abelian group):
  群に交換律を追
  例: (Z; +), (R-{0}; ×)
• 半群 (semigroup): 結合律だけ、単位元と逆元なし
• 環 (ring): 加法と乗法という二つの演算があり、加法に対して可換群、乗法に対して半群、乗法は加法に対して分配可能 (例: 多項式環)
• 体 (field)
• 束 (lattice): 二つの演算に対して結合律と交換律、そしてお互いに吸収律が成り立つ

Glossary

partial order

半順序

partial order relation

半順序関係

order relation

順序関係

partially ordered set

半順序集合

ordered set

順序集合

mutually exclusive

相互排他的な

Hasse diagram

ハッセ図

vertex (plural vertices)

(グラフの) 節、頂点

reconstruct

復元する

square

正方形

overlap

重なる、重複する