In a poset , a top of is an element of such that for every element . 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 can also be understood as a meet of zero elements in .
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 is always itself.
Revised on March 15, 2012 17:24:09
by Urs Schreiber