## Monday, March 25, 2013

### Establishing a weak order from a partial order

Here is the partial ordering relation of the power set algebra of a set of size two:
[[1 1 1 1]
[0 1 0 1]
[0 0 1 1]
[0 0 0 1]]

From this partial order we can get a weak order of the vertices [#{0} #{1 2} #{3}] based upon the upper and lower bounds. The lower bound is #{0} and the upper bound is #{3}. This produces the following weak order:
[[1 1 1 1]
[0 1 1 1]
[0 1 1 1]
[0 0 0 1]]

Only weak orders themselves are equivalent to their underlying weak ordering relation. Topological sorting is related to weak ordering.