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upper bound in nLab

Contents

Context

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

(0,1)-category theory: logic, order theory

(0,1)-category

relation between preorders and (0,1)-categories
proset, partially ordered set (directed set, total order, linear order)
- top, true,
- bottom, false
- monotone function
- implication
- filter, interval
lattice, semilattice
- meet, logical conjunction, and
- join, logical disjunction, or
- compact element
lattice of subobjects
complete lattice, algebraic lattice
distributive lattice, completely distributive lattice, canonical extension
hyperdoctrine
- first-order, Boolean, coherent, tripos

(0,1)-topos

Heyting algebra
- regular element
- Boolean algebra
frame, locale

Theorems

Stone duality

Definitions
Related concepts

Definitions

Definition

For (S,≤)(S, \leq) a preorder and A↪SA \hookrightarrow S a subset, an upper bound of AA in SS is an element xx of SS such that y≤xy \leq x whenever y∈Ay \in A.

Dually (in the opposite poset):

Definition

For (S,≤)(S, \leq) a preorder and A↪SA \hookrightarrow S a subset, a lower bound of AA in SS is an element xx of SS such that x≤yx \leq y whenever y∈Ay \in A.

Zorn's lemma

Last revised on September 27, 2012 at 23:09:15. See the history of this page for a list of all contributions to it.