I am considering a set $C$ and want to impose a binary relation $\prec$ on it that is
- transitive, i.e. if $a\prec b\land b\prec c$ then $a\prec c\ \forall a,b,c\in C$
- irreflexive, i.e. $\neg a \prec a \ \forall a \in C$
- total, i.e. $a \prec b \lor b \prec a\ \forall a,b\in C$ edit: this should read $a \prec b \lor b \prec a \lor a \sim b\ \forall a,b\in C$ to be consistent, where $\sim$ denotes equivalence of $a$ and $b$
and by consulting wikipedia (http://en.wikipedia.org/wiki/Strict_weak_ordering#Total_preorders) it appears that this binary relation is a strict, total preorder. However, Fishburn (http://www.jstor.org/stable/168680) seems to imply that it is sufficient to consider a strict, total order because he assumes that for $a,b\in C$, $\neg a \prec b \land \neg b \prec a$ implies equivalence and not equality.
I have not had much luck finding classic literature on the topic and would be very thankful for suggestions as well as a clarification of whether antisymmetry of a strict total order implies equivalence or equality.