Linear orders

Linear orders, discrete linear orders, dense linear orders, graded orders, complete orders, the back-and-forth construction, universality

A linear order is an order relation ≤ that satisfies the linearity condition: for any a and b, either a ≤ b or b ≤ a. In other words, an order is linear when every pair of elements are comparable. Linear orders are also sometimes called total orders and of course the partial order terminology for an order relation was adopted in distinction to that. The numerous familiar examples of linear orders would include the natural numbers ⟨ℕ, ≤⟩, the integers ⟨ℤ, ≤⟩, the rational numbers ⟨ℚ, ≤⟩, the real numbers ⟨ℝ, ≤⟩ and more.

Discrete linear orders

A discrete linear order is a linear order in which every node has an immediate successor (unless it is already largest) and an immediate predecessor (unless it is already least). Just like a ladder.

More specifically, we say in any order that b succeeds a in an order when a < b and it is an immediate successor of a when also it is least amongst the successors of a. If b is merely minimal amongst the successors of a, then we say that b covers a. Similarly, we say that a precedes b when a < b and it is an immediate predecessor if it is largest amongst the predecessors. An element may have many covers in a partial order, but in a linear order, covers are the same as immediate successors. In any order, immediate successors and predecessors are unique when they exist.

Examples of discrete linear orders would include the natural numbers ⟨ℕ, ≤⟩ and the integers ⟨ℤ, ≤⟩.

Can you imagine more examples? Perhaps one might think that any two points in a discrete linear order must have only finitely many points between them. Is that right?