半順序集合に関する可視化の実験

Experiments of Visualizations about Posets

(Partially Ordered Sets)

ここでは束の不等式や等式など順序集合の中のいろいろな状況を，図，特にステレオグラムによる３次元の図を使って 可視化して，直観的に理解することを試みます．
Here, we try to understand several situations in posets, e.g., the situations where some inequalities or equalities in lattice theory hold, intuitively by means of figures, especially 3D figures.

まず，分配不等式とモジュラー不等式についてやってみます．
First, We try distributive and modular inequalities.

a ∧ (b ∨ c) ≥ (a ∧ b) ∨ (a ∧ c)

c ≤ a implies a ∧ (b ∨ c) ≥ (a ∧ b) ∨ c

このページのステレオグラムは 平行法で見るように描いています．
The 3D figures of this page are drawn for viewing parallel viewing.

重要： ステレオグラムを見ることが合わない人もいると思いますので， 目が痛くなったり，気分が悪くなったりした場合は，見るのを止めて下さい．

IMPORTANT : If you feel bad viewing 3D pictures by parallel viewing, stop seeing the figures instantly.

- Visualization of Distributive and Modular Inequalities

1. c ≤ a & d ≤ a => c ∨ d ≤ a

   c も d が a 以下なら， c ∨ d も a 以下です．なぜなら， c ∨ d は c と d 以上のものの中で最小のものですから．
   You can easily check the above implication.

   

2. ≥ c & a ≥ d => a ≥ c ∨ d

   双対的にこちらの関係も言えます．
   This implication is the dual of the above.

   

3. Both a and b is greater than or equal to c and d => a ∧ b ≥ d ∨ d

   上の２つの３角形の場合を思い出しながら，矢印に従って追っていけば，左上のハッセ図のときは左下の関係が言えることがわかります．
   Follow the figures along the arrows.

   

4. Same as above in 3D Stereogram (Parallel view)

   上のことをステレオグラムで表現しています．
   This is the stereogram that represents the above relation.

   
   or simply
   

5. Same as above, but torted to a balanced shape in Stereogram (Parallel view)
   

6. Preliminary of distributive inequality (Usual 2D diagram)

   分配的な不等式

   a ∧ (b ∨ c) ≥ (a ∧ b) ∨ (a ∧ c)

   の 3D モデルを示す前に，すこし準備をしておきます．次の関係は常に成り立っていることを 確認しておいてください．これは普通の二次元の図です．
   You can easily check the following Hasse diagram.

7. Distributive Inequality in 3D stereogram (Parallel view)
   

8. Modular Inequality (Parallel view). Suppose that c ≤ a
   

9. Modular Inequality2 (Torted position) (Parallel view)
   

10. Modular Inequality3 (Torted and the element 'b' is added) (Parallel view)
    

11. Non-modular lattice and its N5 sublattice (Parallel view)
    
    Of cource, We must prove

    c < a and (a ∧ b) ∨ c < a ∧ (b ∨ c) implies
        b ∨ c, b, a ∧ b, a ∧ (b ∨ c), (a ∧ b) ∨ c are all different

    e.g.,

    b = a ∧ b => b ≤ a => (a ∧ b) ∨ c = b ∨ c and a ∧ (b ∨ c) = b ∨ c => contradiction,
    ...

12. Of cource a modular lattice has no pentagon (Parallel view)
    

13. Distributive Inequality with Full points (Parallel view)
    

- Galois Connection

1. Galois connection in usual 2D

   

2. Galois Connection in 3D (Parallel view)

   Although it would be no need to visualize the simple figure in 3D, it may be interesting.

   

3. Heyting Algebra 2D

   Bounded lattice P such that
       ∃ (a -> b) := max { x ∈ P | (x & a) ≤ b } for ∀ a, ∀ b ∈ P

   P の任意の a, b に対して，(x & a) のとき b が成立する最も一般的な（条件の弱い）論理式 x = (a -> b) が存在する論理体系 P

   Logical system P where there exists most weak condition x = (a -> b) such that for any a, b ∈ P if (x & a) then b holds

   

4. Heyting Algebra 3D (Parallel view)
   

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