1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is partially ordered

  1. Jul 23, 2014 #1

    A partially ordered set, or in short, a poset, is a set A together with a relation [tex]\leq~\subseteq A\times A[/tex] which is reflexive, antisymmetric, and transitive. In other words, satisfying
    1)[itex]\forall x\in A,~x\leq x[/itex] (the relation is reflexive)
    2)[itex]\forall x,y\in A,~x\leq y~and~y\leq x\Rightarrow x=y[/itex] (the relation is antisymmetric)
    3)[itex]\forall x,y,z\in A,~x\leq y~and~y\leq z\Rightarrow x\leq z[/itex] (the relation is transitive)

    It is common to refer to a poset [itex]\left(A,\leq\right)[/itex] simply as A, with the underlying relation being implicit.



    For example, a checker-board with the relation "not further from the white end" is not a poset because two different squares can be the same distance from the white end, and so the relation is not anti-symmetric.

    But the same board with the relation "not further from the white end according to legal moves for an ordinary black piece" is a poset .

    Totally ordered set:

    A poset with the extra condition
    4) [itex]\forall x,y\in A,~x\leq y~or~y\leq x[/itex]
    is a totally ordered set.

    In other words, loosely speaking, a totally ordered set is essentially one-dimensional: it can be thought of as a line, in which any element is either one side or the other side of any other element.

    Extended explanation

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted