nLab
bottom

Context

(0,1)(0,1)-Category theory

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Theorems

Contents

Idea

In a poset PP, a bottom is a least element: an element \bot of PP such that a\bot \leq a for every element aa. 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 PP can also be understood as a join 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 bottom is simply an initial object in that category.

The bottom of the poset of subsets or subobjects of a given set or object AA 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.

basic symbols used in

A\phantom{A}symbolA\phantom{A}A\phantom{A}meaningA\phantom{A}
A\phantom{A}\inA\phantom{A}
A\phantom{A}:\,:A\phantom{A}
A\phantom{A}==A\phantom{A}
A\phantom{A}\vdashA\phantom{A}A\phantom{A} / A\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A} / A\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A} / A\phantom{A}
A\phantom{A}\RightarrowA\phantom{A}
A\phantom{A}\LeftrightarrowA\phantom{A}
A\phantom{A}¬\notA\phantom{A}
A\phantom{A}\neqA\phantom{A} of / A\phantom{A}
A\phantom{A}\notinA\phantom{A} of A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}A\phantom{A}
A\phantom{A}\existsA\phantom{A}A\phantom{A}
A\phantom{A}\forallA\phantom{A}A\phantom{A}
A\phantom{A}\wedgeA\phantom{A}
A\phantom{A}\veeA\phantom{A}
A\phantom{A}\otimesA\phantom{A}A\phantom{A}A\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}A\phantom{A}

Last revised on July 3, 2018 at 03:05:31. See the history of this page for a list of all contributions to it.