The partial order is reflexive and antisymmetric, which means that aRb and bRa iff a = b.
This implies that there is only one cycle starting at a and ending at a, which is a self-loop of a to itself.
For example, if $A = \{a, b, c\}$ then $R = \{(a, a), (b, b), (c, c)\}$. This means a self loop at each of a, b and c .