Partial ordering with only reflexive relation.

Question

I have a problem understanding orders.
For know, how i understand partial order is: pair of set and relation that orders it.
From that pair, let's say, we can create some kind or 'ordered list'. For example, let's say that we have set $A=\{5, 3, 1, 2\}$ and relation $R=\{(1, 1), (2, 2), (3, 3), (5, 5), (1, 2), (1, 3), (1, 5), (2, 3), (2, 5), (3, 5)\}$, so we can, on this basis create 'ordered list' like: $$L = [1, 2, 3, 5]$$ because we are able to compare all elements with each other. But let's try to consider partial order $R = \{(1, 1), (2, 2), (3, 3), (5, 5)\}$ which is reflexive, transitive and antisymmetric, so it fulfills all properties needed to be named partial order. How do I proceed with such relation? How do I create such 'ordered list'? Or is there something not ok with my understanding of orders?

Answer 1

The definition of partially ordered set does not guarantee that all of its elements are comparable, so you can only order certain subsets(one element subsets in your case). If all elements of a partially ordered set are comparable you have a total order and the set is called totally ordered wiki article