Can a relation be a partial order and an equivalence at the same time? Can a relation be a partial order AND an equivalence at the same time?
For instance, if we have a set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ defined as $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}$: this relation is reflexive, anti-symmetric, symmetric, and would it be considered transitive as well? If it is considered transitive, I suppose that it is an equivalence and a partial order at the same time.

Two remarks:
(1) Equality is an equivalence relation which is also a partial order ($\leq$).
(2) An equivalence relation is never a strict partial order ($<$).
