# Higher order partial order

I'm trying to find an effective and making sense way (for programming purposes, but also from math interest) to order partial orders, means, given a set and many partial orders (trees) over it. How can those partial orders be "naturally" ordered?

Specifically, I'm interested in ordering partial orders which each one of them has a Hasse diagram of a tree. Moreover, in my case of interest, for every $a\leq b$ there exists c such that a=b-c can be defined (in some too specific way my trees are built) and $c\leq b$ (almost like "Monus").

• A partial order on a set $A$ is just a particular kind of subset of $A\times A$; thus, the set of partial orders on $A$ is naturally ordered by inclusion. – Brian M. Scott May 13 '16 at 17:08
• right, but for my specific application i have to look on a broader class as this one won't fit. it's about ordering trees, and tree is "less" than another tree in my case only in some peculiar cases – Troy McClure May 13 '16 at 17:19
• Then you should really rewrite the question to make it clear that you’re talking about partial orders whose Hasse diagrams are trees. – Brian M. Scott May 13 '16 at 17:29
• There is nothing wrong with trees not always being comparable - it is in fact normal to be so. The natural ordering on the set of trees (or order relations) is a partial order, not a total one. – Alex M. May 13 '16 at 17:38
• i edited the question. i agree it's a bit difficult to explain the specific case, maybe it's not a good question for here – Troy McClure May 13 '16 at 18:41