In a poset$P$, a bottom is an element $\bot$ of $P$ such that $\bot \leq a$ for every element $a$. Such a bottom may not exist; if it does, then it is unique.

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

A bottom of $P$ can also be understood as a join 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 bottom is simply an initial object in that category.

The bottom of the poset of subsets or subobjects of a given set or object $A$ is called the empty subset or subobject. In a category (such as Set) with a strict initial object$\varnothing$, this will always serve as the bottom of any subobject poset.

Revised on March 15, 2012 17:25:14
by Urs Schreiber
(82.169.65.155)