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partially ordered
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Definition/Summary
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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. |
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Recent forum threads on partially ordered
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Breakdown
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Mathematics
> Foundations
>> Set Theory
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Extended explanation
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Antisymmetry:
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. |
Commentary
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