In a poset , a bottom is a least element: an element of such that for every element . Such a bottom may not exist; if it does, then it is unique.
A bottom of can also be understood as a join of zero elements in .
A poset that has both top and bottom is called bounded.
The bottom of the poset of subsets or subobjects of a given set or object is called the empty subset or subobject. In a category (such as Set) with a strict initial object , this will always serve as the bottom of any subobject poset.