Total (linear) order

Parent: basic-orders

Total order

A partial-order that is total: any two elements are comparable (a≤b or b≤a). No incomparability — the whole set is one chain.

Its Hasse diagram is just a line (every element above or below every other):

Examples: ≤ on ℤ, ℚ, ℝ; lexicographic / dictionary order; ASCII order on characters.

Facts: every finite total order of size n is isomorphic to {1,…,n}; a total order plus well-foundedness is a well-order (the ordinals). The strict version (<) satisfies trichotomy: for any a,b, exactly one of a<b, a=b, b<a — see strict-order.