How to find a linear extension of a poset

Question

Give linear extensions of the three posets in the image. This is the image of the posets:

I am unsure how to began doing linear extensions on this so if someone could please explain linear extensions with an example of a poset.

Answer 1

Given your finite poset $P$, it is clear that the first element in your linear extension must be a minimal element (else you'll not have a linear extension). Moreover, if you picked a partial linear extension -- elements $a_1, \dots, a_k \in P$ with the property that $a_i < a_j$ in $P$ implies $i < j$ (for $1 \leq i, j \leq k$), then the next element $a_{k+1}$ in your linear extension must be a minimal element of $P \setminus \{a_1, \dots, a_k\}$.

Conversely, if you follow that recursive procedure, you'll always generate a linear extension of $P$ (exercise!). Thus, to find linear extensions of your given posets, proceed one element at a time, always picking a minimal element of the poset of the remaining not-yet-chosen elements.