I do not understand the definition of antisymmetric relations OK, let A be a set and let R be a binary relation on A. In my class we say that R is antisymmetric if and only if for every a, b in A, if (a, b) in R and (b, a) in R then a = b.  Fair enough, but what does "a = b" mean? Does this mean that A has some sort of a priori equivalence relation that allows me to identify a and b?
I would find this surprising, in what I've read about partially-ordered sets, I've never seen any discussion of an underlying equivalence relation.
Thanks for enlightening me!
Bob Muller

Not only an equivalence relation on a non-empty set $A$ defines a partition of $A$ in equivalence classes, the reverse is also true: every partition of $A$ corresponds to an equivalence relation.  Hence the partition which arises from the singletons of $A$ corresponds to an equivalence relation $R_{\text{singleton}}$.  The usual name for “$R_{\text{singleton}}$” is “$=$”, i.e., $aR_{\text{singleton}}b\iff a=b$. Enlightened enough?
