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Exercise 34 - Chapter 1

edited June 2018 in Exercises

Let \(X\) be the set of partitions of {•, ◦, ∗}; it has five elements and an order by coarseness, as shown in the Hasse diagram Eq. (1.2) [below]. Write down every pair \((x, y)\) of elements in \(X\) such that \(x \le y\). There should be 12.

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Comments

  • 1.
    edited April 2018

    Let \(S_1, \cdots, S_5\) denote the five partitions:

    $$ \begin{eqnarray} S_1 &=& \left\{\{\bullet\}, \{\circ\}, \{\star\}\right\}\\ S_2 &=& \left\{\{\bullet\}, \{\circ, \star\}\right\}\\ S_3 &=& \left\{\{\circ\}, \{\bullet, \star\}\right\}\\ S_4 &=& \left\{\{\star\}, \{\bullet, \circ\}\right\}\\ S_5 &=& \left\{\{\bullet, \circ, \star\}\right\}\\ \end{eqnarray} $$ Then the twelve pairs are:

    $$ \begin{eqnarray} (S_1, S_1) \\ (S_1, S_2) \\ (S_1, S_3) \\ (S_1, S_4) \\ (S_1, S_5) \\ (S_2, S_2) \\ (S_2, S_5) \\ (S_3, S_3) \\ (S_3, S_5) \\ (S_4, S_4) \\ (S_4, S_5) \\ (S_5, S_5) \\ \end{eqnarray} $$

    Comment Source:Let \\(S_1, \cdots, S_5\\) denote the five partitions: $$ \begin{eqnarray} S_1 &=& \left\\{\\{\bullet\\}, \\{\circ\\}, \\{\star\\}\right\\}\\\\ S_2 &=& \left\\{\\{\bullet\\}, \\{\circ, \star\\}\right\\}\\\\ S_3 &=& \left\\{\\{\circ\\}, \\{\bullet, \star\\}\right\\}\\\\ S_4 &=& \left\\{\\{\star\\}, \\{\bullet, \circ\\}\right\\}\\\\ S_5 &=& \left\\{\\{\bullet, \circ, \star\\}\right\\}\\\\ \end{eqnarray} $$ Then the twelve pairs are: $$ \begin{eqnarray} (S_1, S_1) \\\\ (S_1, S_2) \\\\ (S_1, S_3) \\\\ (S_1, S_4) \\\\ (S_1, S_5) \\\\ (S_2, S_2) \\\\ (S_2, S_5) \\\\ (S_3, S_3) \\\\ (S_3, S_5) \\\\ (S_4, S_4) \\\\ (S_4, S_5) \\\\ (S_5, S_5) \\\\ \end{eqnarray} $$
  • 2.
    edited April 2018

    To summarize: there are 5 identity arrows; 6 drawn arrows, and; 1 composed arrow [3 equivalent composed arrows] from \( S_1 \dots S_5 \). \( 5 + 6 + 1 = 12 \)

    Comment Source:To summarize: there are 5 identity arrows; 6 drawn arrows, and; 1 composed arrow [3 equivalent composed arrows] from \\( S_1 \dots S_5 \\). \\( 5 + 6 + 1 = 12 \\)
  • 3.
    edited April 2018

    Just a minor additional explanation point that might help others. Coarser partition (as per John's lecture 10), means that more points are connected (or equivalent). Which is why in Dan's answer \(S_5\) always appears on the right, because it is the most coarse --that is, in that partition most points are equivalent. May be another way to think about 'coarsest' partition, is that it will contain the 'least' number of blocks.

    Comment Source:Just a minor additional explanation point that might help others. Coarser partition (as per John's lecture 10), means that more points are connected (or equivalent). Which is why in Dan's answer \\(S_5\\) always appears on the right, because it is the most coarse --that is, in that partition most points are equivalent. May be another way to think about 'coarsest' partition, is that it will contain the 'least' number of blocks.
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