Comparing ideals in posets

Question

Consider a partially ordered set $P$, and two upper sets $U_1$, $U_2$ in this poset.

What are some natural ways to measure how equal these two upper sets are? This question arise naturally in the study of proteins which upper closed sets.

Of course, all measures on sub-graph similarity in graphs can be applied here, but perhaps using that these are upper-closed sets in a poset, make some constructions more natural.

What similarity means, is of course depending on the similarity measuring function, but say $sim(U_1,U_1)$ should be big, and $sim(U_1,U_2)$ is small if $U_1$ and $U_2$ if the overlap between $U_1$ and $U_2$ is small.

Answer 1

I am unsure of exactly what you are looking for, so I will post something that sounds reasonable. If $P$ is a poset, then the collection of all lower sets (and upper sets as well) forms a frame. Furthermore, every frame is a Heyting algebra, so every frame has an $\rightarrow$ operation. In a Heyting algebra, one may define an "if and only if" operation by letting $x\leftrightarrow y=(x\rightarrow y)\wedge(y\rightarrow x)$.

If $P$ is a poset, then let $L$ denote the frame of all lower sets. Then $A\rightarrow B$ is the largest lower set with $(A\rightarrow B)\cap A\subseteq B$. In particular, $x\in A\rightarrow B$ if and only if $\downarrow x\cap A\subseteq B$. Thus, $x\in A\leftrightarrow B$ if and only if $\downarrow x\cap A=\downarrow x\cap B$.