> **Puzzle 4.** List some interesting and important examples of posets that haven't already been listed in other comments in this thread.

A very simple example.

Let \\(P = \\{ (a,b): a < b \land (a,b) \in \mathbb{R^2} \\}\\) then following relations are posets:

1. \\((a,b)R(c,d) \iff a \le c \land b \le d \\)

2. \\(xRy \iff x \subseteq y \\)

Since this is applied course one interpretation could be that first number denotes start of some process and the second number its end. So in case of 1. a process is larger than another one if it started after another one started and ended after another one ended. While in case of 2. process is larger than another one if the latter started and ended when the first one was active.

A very simple example.

Let \\(P = \\{ (a,b): a < b \land (a,b) \in \mathbb{R^2} \\}\\) then following relations are posets:

1. \\((a,b)R(c,d) \iff a \le c \land b \le d \\)

2. \\(xRy \iff x \subseteq y \\)

Since this is applied course one interpretation could be that first number denotes start of some process and the second number its end. So in case of 1. a process is larger than another one if it started after another one started and ended after another one ended. While in case of 2. process is larger than another one if the latter started and ended when the first one was active.