Math 2.0

Order, lattices, ordered algebraic structures Definition

Down-closure

Builds on Partial orderUpper set

Definition

Let (P,≤)(P, \le) be a partially ordered set and let S⊆PS \subseteq P. The down-closure of SS is the set ↓S={x∈P:x≤s for some s∈S}.\downarrow S = \set{x \in P : x \le s \text{ for some } s \in S}. For x∈Px \in P, the set ↓x=↓{x}={y∈P:y≤x}\downarrow x = \downarrow \set{x} = \set{y \in P : y \le x} is the principal down-set of xx.

Remarks

In the subsets of {1,2,3}\set{1, 2, 3} ordered by inclusion, ↓{{1,2},{3}}={∅,{1},{2},{1,2},{3}}\downarrow \set{\set{1, 2}, \set{3}} = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}, \set{3}}. Each listed set is contained in {1,2}\set{1, 2} or in {3}\set{3}, and no other subset is. The set ↓S\downarrow S is a lower set that contains SS, and it lies inside each lower set that contains SS. So SS is a lower set if and only if ↓S=S\downarrow S = S. The map S↦↓SS \mapsto \downarrow S is a closure operator on the subsets of PP. Dually, the up-closure is ↑S={x∈P:s≤x for some s∈S}\uparrow S = \set{x \in P : s \le x \text{ for some } s \in S}.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionUpper set, definition, no evidence levelUpper setdefinitionDown-closure, definition, no evidence levelDown-closuredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Upper set definition
  17. Down-closure definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Upper_and_lower_sets
About this page
TagTGW5NTV3AA
URIm2://TGW5NTV3AA
Kinddefinition
WritingW0 stub
MSC06A06
DefinesDown-closure
UsesPartial order, Upper set

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by