Total Order

A total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set, which satisfies \forall a, b, c \in S:
# Reflexivity: a \leq a 
# Transitivity: a \leq b and b \leq c \implies a \leq c 
# Anti-symmetry: a \leq b and b \leq a \implies a = b 
# Connectedness: a \leq b or b \leq a 
Note that (1) follows from (4), but is often included in the definition for comparison to that of partial orders.