In a poset PP, a top of PP is an element \top of PP such that aa \leq \top for every element aa. Such a top may not exist; if it does, then it is unique.

In a proset, a top may be defined similarly, but it need not be unique. (However, it is still unique up the natural equivalence in the proset.)

A top of PP can also be understood as a meet of zero elements in PP.

A poset that has both top and bottom is called bounded.

As a poset is a special kind of category, a top is simply a terminal object in that category.

The top of the poset of subsets or subobjects of a given set or object AA is always AA itself.

Revised on March 15, 2012 17:24:09 by Urs Schreiber (