Preorder (proset) is a binary relation (\leq) that is reflexive and transitive. (Preorder is isomorphic to Thin Category.)

Partial order (poset) is a preorder s.t. x,y.  xyyy    x=y\forall x,y.\; x \leq y \land y \leq y \implies x = y. i.e., reflexive, transitive, and antisymmetric.

Total order (linear order, loset) is like partial order, but with Connex property (i.e., defined between any two elements)