I was wondering about how to represent a partial order which includes multiple instance of the same element. For example a partial order could include these two nodes: "(a ⋁ b) ⋀ a" and "(a ⋀ b) ⋁ b", and also include different combinations and sub-parts of these.
Linguistically you might take the following sentence:
"John went to see his friend, who owed him 10 dollars.", or:
"a" went to see "b", who owed "a" 10 dollars.
Represented as: (a ⋁ b) ⋀ a
The fundamental idea is that John is mentioned twice, and his friend only once. I wish to represent this fundamental idea as a partial ordered set (or perhaps pre-order), in order to be able to compare it against other such possibilities which might include "John" and "the friend".
⋁ and ⋀ might be meet and join, or alternatively they might be unison and intersection.