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 logic

A\phantom{A}\inA\phantom{A}element relation
A\phantom{A}:\,:A\phantom{A}typing relation
A\phantom{A}\vdashA\phantom{A}A\phantom{A}entailment / sequentA\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A}true / topA\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A}false / bottomA\phantom{A}
A\phantom{A}\LeftrightarrowA\phantom{A}logical equivalence
A\phantom{A}\neqA\phantom{A}negation of equality / apartnessA\phantom{A}
A\phantom{A}\notinA\phantom{A}negation of element relation A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}negation of negationA\phantom{A}
A\phantom{A}\existsA\phantom{A}existential quantificationA\phantom{A}
A\phantom{A}\forallA\phantom{A}universal quantificationA\phantom{A}
A\phantom{A}\wedgeA\phantom{A}logical conjunction
A\phantom{A}\veeA\phantom{A}logical disjunction
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}multiplicative disjunctionA\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.