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 \((X, \preceq)\).

This problem can only be submitted at Cogniterra.

  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 \(x \subseteq y\)