Content deleted Content added
Tag: Reverted |
Reverted 4 edits by Gabriel Wang 12333 (talk): Links must be avoided in headings |
||
Line 170:
If the count is made only [[up to]] isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... {{OEIS|A000112}} is obtained.
==
A poset <math>P^*=(X^*, \leq^*)</math> is called a '''subposet''' of another poset <math>P=(X, \leq)</math> provided that <math>X^*</math> is a [[subset]] of <math>X</math> and <math>\leq^*</math> is a subset of <math>\leq</math>. The latter condition is equivalent to the requirement that for any <math>x</math> and <math>y</math> in <math>X^*</math> (and thus also in <math>X</math>), if <math>x \leq^* y</math> then <math>x \leq y</math>.
Line 180:
In [[computer science]], algorithms for finding linear extensions of partial orders (represented as the [[reachability]] orders of [[directed acyclic graph]]s) are called [[topological sorting]].
== In
{{main|Posetal category}}
Every poset (and every [[Preorder|preordered set]]) may be considered as a [[Category (mathematics)|category]] where, for objects <math>x</math> and <math>y,</math> there is at most one [[morphism]] from <math>x</math> to <math>y.</math> More explicitly, let {{nowrap|1=hom(''x'', ''y'') = {{mset|(''x'', ''y'')}}}} if {{nowrap|''x'' ≤ ''y''}} (and otherwise the [[empty set]]) and <math>(y, z) \circ (x, y) = (x, z).</math> Such categories are sometimes called ''[[Posetal category|posetal]]''.
Line 186:
Posets are [[Equivalence of categories|equivalent]] to one another if and only if they are [[Isomorphism of categories|isomorphic]]. In a poset, the smallest element, if it exists, is an [[initial object]], and the largest element, if it exists, is a [[terminal object]]. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is [[isomorphism-closed]].
== Partial orders in
{{Main|Partially ordered space}}
| |||