Is my ranking in total order?

Question

I ask a person to fill a questionnaire to give a score to a set of items, without loss of generality, he scores items $X=\{a, b, c, d, e\}$ in a 5 point scale $[1..5]$. He is allowed to give same score to different items, a table can be gotten:

               ________________________________
          |  1      2      3      4       5    |
  .-------+-------------------------------------
  |  a    |                               x    |
  |  b    |                       x            |
  |  c    |                x                   |
  |  d    |                       x            |
  |  e    |                x                   |
  '-------+------------------------------------'

Then I define the preference relationship $\succeq$, that $x\succeq y, x,y\in X $ means $x$ is weakly preferred to $y$ (means he either strictly prefers x to y or is indifferent between the two). Thus, should my ranking be $a\succeq(b,d)\succeq(c,e)$? And can I say this ranking is in total order? To me the question is weather does the order satisfy antisymmetry?

Answer 1

It is clearly not antisymetric, since $b$ and $d$ are equally prefered but they aren't the same object

Answer 2

What you have is a total order on the set of three elements $\{\{a\}, \{b,d\},\{c,e\}\}$ that you get by "identifying" the original elements when their scores match. (You can say that more formally as the elements of the partition determined by the scoring function.)

It's not a total order on the original set since neither $b$ nor $d$ is preferred to the other.