I'm having a difficulty understanding the concept of "antichains".
An antichain in a poset $P$ is a subset $C \subseteq P$ such that no 2 elements are comparable
I understand the definition, but I guess I'm having difficulty applying it to a real situation. For example, let's say I have the following set:
$A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ .
Here, let's define: $x\preceq y \Longleftrightarrow x$ divides $y$.
We can then find the following chains:
$(1,2,4,8), \quad (1,3,9),\quad (1,5,10),\quad (1,2,10),\quad (1,2,6),\quad (1,3,6)$
What would the antichains be (if any)? My guess:
$(1,5,6), \quad (1,5,9), \quad (1,7,10), \quad (1,7,9), \quad (1,3,10), \quad (1,5,7), \quad (1,3,8), \quad (1,3,4)$
etc. etc.
Am I on the right track or no? Is there a way to determine the number of antichains without figuring them all out yourself?
If this is a bad example, can someone give me a simple example?
Your example is not bad, however there is a classic example of a Lattice, which has a large number of antichains - it is a Power Set. Sorry to say, the exact number of antichains is not known even in this classic case - it's famous and long-standing Dedekind Problem.