Hasse graph of a poset.

Question

Let $S = \{a, b, c, d, e, f\}$. The graph of a poset $(S,\lesssim)$ looks like this:

Except vertices of the graph in my textbook are represented by letters. The letters correspond to the numbers in the graph I linked to like this:

a corresponds to 12, b corresponds to 6, c corresponds to 3, d corresponds to 4, e corresponds to 5 and f corresponds to 10.

Suppose the partial order relation is $\lesssim$. That would mean $b\lesssim a$ for example. This doesn't make sense to me.

Please, elaborate. Thanks.

Answer 1

This particular relation is on the set $\{12,10,6,5,4,3\}$ and is "divides" (or "is divisible by" depending on which way you read the relations).

So $6$ divides $12$, $4$ divides $12$, $3$ divides $6$ and $5$ divides $10$ are given by the figure, and we also get $3$ divides $12$ by transitivity.

The symbol used to denote the relation does not affect what the relation is; in this case, the symbol happens to be $\lesssim$.