Ifxyandyx, we writexyand sayxandyareequivalent. We call a set with a preorder aposet.

Example1.42 (Opposite poset).Given a posetπP;∫, we may define the opposite posetπP;op∫to have the same set of elements, but withpopqif and only ifqp.

Example1.40 (Product poset).Given posetsπP;∫andπQ;∫, we may define a posetstructure on the product setPQby settingπp;q∫  πp0;q0∫if and only ifpp0andqq0. We call this theproduct poset. This is a basic example of a more generalconstruction known as the product of categories

Contrary to the definition we’ve chosen, the term poset frequently is used to meanpartiallyordered set, rather than preordered set. In category theory terminology, therequirement thatxyimpliesxyis known asskeletality. We thus call partiallyordered setsskeletal posets