In a poset$P$, a top of $P$ is an element $\top$ of $P$ such that $a \leq \top$ for every element $a$. 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 $P$ can also be understood as a meet of zero elements in $P$.

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 $A$ is always $A$ itself.

Revised on March 15, 2012 17:24:09
by Urs Schreiber
(82.169.65.155)