In the beginning of 1.2: "a poset - a preordered set" with emphasis on letters "p" and "o" of preordered. I find it a little strange. Maybe the authors meant a "partially ordered set" instead. This mnemonic makes it easier to forget about antisymmetry requirement for posets. Calling a partially ordered set a "poset" and a preordered set a "preset" makes more sense and it's more obvious where the names come from.
I read further and now I understand it is not a mistake. The author made a deliberate decision to use the name "poset" for presets "skeletal posets" for posets, "partial order" for a preorder and "preorder" for a partial order.