1 / 8

Partial Orderings

Partial Orderings. Partial Orderings. A relation R on a set S is called a partial ordering if it is: r eflexive antisymmetric transitive A set S together with a partial ordering R is called a partially ordered set , or poset , and is denoted by ( S , R ).

xiang
Télécharger la présentation

Partial Orderings

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Partial Orderings

  2. Partial Orderings • A relation R on a set S is called a partial ordering if it is: • reflexive • antisymmetric • transitive • A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S,R). • Example: “” is a partial ordering on the set of integers • reflexive: a  a for every integer a • anti-symmetric: If a  b and b  a then a = b • transitive: a  b and b  c implies a  c • Therefore “” is a partial ordering on the set of integers and (Z, ) is a poset.

  3. Comparable/Incomparable Elements • Let “≼” denote any relation in a poset (e.g. ) • The elements a and b of a poset (S, ≼) are: • comparable if either a≼b or b≼a • incomparable if neither a≼b nor b≼a • Example: Consider the poset (Z+,│), where “a│b” denotes “a divides b” • 3 and 9 are comparable because 3│9 • 5 and 7 are not comparable because nether 5⫮7 nor 7⫮5

  4. Partial and Total Orders • If some elements in a poset(S, ≼) are incomparable, then it is partially ordered • ≼ is a partial order • If every two elements of a poset (S, ≼) are comparable, then it is totally ordered or linearly ordered • ≼ is a total (or linear) order • Examples: • (Z+,│) is not totally ordered because some integers are incomparable • (Z, ≤) is totally ordered because any two integers are comparable (a ≤ b or b ≤ a)

  5. Hasse Diagrams • Graphical representation of a poset • It eliminates all implied edges (reflexive, transitive) • Arranges all edges to point up (implied arrow heads) • Algorithm: • Start with the digraph of the partial order • Remove the loops at each vertex (reflexive) • Remove all edges that must be present because of the transitivity • Arrange each edge so that all arrows point up • Remove all arrowheads

  6. 1 2 3 1 2 3 1 2 3 3 2 1 3 2 1 Constructing Hasse Diagrams • Example: Construct the Hasse diagram for ({1,2,3},)

  7. h j g f d e b c a Maximal and minimal Elements • Let (S, ≼) be a poset • a is maximal in (S, ≼) if there is no bS such that a≼b • a is minimal in (S, ≼) if there is no bS such that b≼a • a is the greatest elementof (S, ≼) if b≼a for all bS • a is the leastelement of (S, ≼) if a≼b for all bS • greatest and least must be unique • Example: • Maximal: h,j • Minimal: a • Greatest element: None • Least element: a

  8. h j g f d e b c a Upper and Lower Bounds • Let A be a subset of (S, ≼) • If uS such that a≼u for all aA, then u is an upper bound of A • If x is an upper bound of A and x≼z whenever z is an upper bound of A, then x is the least upper boundof A (must be unique) • Analogous for lower bound and greatest upper bound • Example: let A be {a,b,c} • Upper bounds of A: e,f,j,h • Least upper bound of A: e • Lower bound of A: a • Greatest lower bound of A: a

More Related