Partial Order (poset)

A pair, a set and a relation (S, ≤) which, for any x and y in S is:

• Reflexive: x≤x
• Anti-symmetric: if x ≤ y and y ≤ x then x = y
• Transitive: if x ≤ y and y ≤ z then x ≤ z

Total Order

A set and an relation (S, ≤) which, for any x and y in S is:

• Reflexive: x≤x
• Anti-symmetric: if x≤y and y≤x then x=y
• Transitive: if x≤y and y≤z then x≤z
• Dichotomy: either x≤y or y≤x