Question

Refine the relation on a strict poset that is not maximal

Answer 1

mainly find R prime which is I think the negation of R. I'll type more details in a bit ....

Answer 2

Set X
\[P \subset X \times X\] strict partial order
\[L \subset X \times X\] linear order (it defines a strict partial order)

Suppose we have a set
\[P=[(a,b)(a,c)]\]

When we enlarge P TO L we have \[L = (a,b)(b,c)(a,c) \] and that's a transitive linear order. So, this set is partially ordered by containment.

\[P ( X \times X ) = \bar P ( X \times X)\]
the left side is the relations... on the right side we have strictly partial orderings on X
Given

\[x= [a,b,c]\] we find \[P ( X \times X )

\bar P ( X \times X)\]

The result is that we have too much



Q: WHen is a strict partial order in \[\bar P ( X \times X) \] maximal?
A: If it's linear

So show if R is a strict partial order on X, and R is not linear then there exists a strict partial \[R' \] and