Editing Weak ordering (section)

==Related types of ordering==

[[Semiorder]]s generalize strict weak orderings, but do not assume transitivity of incomparability.<ref>{{citation|last=Luce|first=R. Duncan|authorlink=R. Duncan Luce|mr=0078632|journal=Econometrica|pages=178–191|title=Semiorders and a theory of utility discrimination|jstor=1905751|volume=24|year=1956|issue=2|doi=10.2307/1905751|url=http://dml.cz/bitstream/handle/10338.dmlcz/128450/CzechMathJ_44-1994-1_3.pdf}}.</ref> A strict weak order that is [[Trichotomy (mathematics)|trichotomous]] is called a '''strict total order'''.<ref name="h2pi">{{citation|title=How to Prove It: A Structured Approach|first=Daniel J.|last=Velleman|publisher=Cambridge University Press|year=2006|isbn=9780521675994|page=204|url=https://books.google.com/books?id=lptwaMuAtBAC&pg=PA204}}.</ref> The total preorder which is the inverse of its complement is in this case a [[total order]].

For a strict weak order <math>\,<\,</math> another associated reflexive relation is its [[Binary relation#Operations|reflexive closure]], a (non-strict) partial order <math>\,\leq.</math> The two associated reflexive relations differ with regard to different <math>a</math> and <math>b</math> for which neither <math>a < b</math> nor <math>b < a</math>: in the total preorder corresponding to a strict weak order we get <math>a \lesssim b</math> and <math>b \lesssim a,</math> while in the partial order given by the reflexive closure we get neither <math>a \leq b</math> nor <math>b \leq a.</math> For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order.<ref name="h2pi"/> The reflexive closure of a strict weak ordering is a type of [[series-parallel partial order]].