Partially Ordered Sets · Partial Orders

Lesson 7

Nikolai Chukhin · Alexander S. Kulikov

An order is called linear if any two of its elements are comparable.

Problem. Mark all linear orders \((X, \preceq)\).

5 points

  1. \(X=\{1, 2, 3, 4\}\) and \(x \preceq y\), if \(x \mid y\)

  2. \(X=\mathbb{Z}\) and \(x \preceq y\), if \(x \le y\)

  3. \(X=\mathbb{Q}\) and \(x \preceq y\), if \(x \le y\)

  4. \(X=2^{\{a, b, c\}}\) (this is the set of all subsets of the set \(\{a, b, c\}\)) and \(x \preceq y\), if \(x \subseteq y\)