1. x ≤ x,

2. If x ≤ y and y ≤ x then x = y,

3. If x ≤ y and y ≤ z then x ≤ z,

4. x ≤ y or y ≤ x, i.e. any two elements of N are comparable with respect to ≤.

We shall use the first three properties to define our first type of ordering relation:

Definition 1.1. Let R be a relation on A.

1. R is reflexive if 〈x,x〉 ∈ R for all x ∈ A.

2. R is antisymmetric if for all x,y ∈ A, 〈x,y〉 ∈ R and 〈y,x〉 ∈ R implies x = y.

3. R is transitive if for all x,y,z ∈ A, 〈x,y〉 ∈ R and 〈y,z〉 ∈ R implies 〈x,z〉 ∈ R.

4. R is a partial order on A, if R is reflexive, antisymmetric, and transitive.

Sometimes we will call a partial order on A just an order on A, or an ordering of A.

5. R is a linear order on A if R is a partial order, and xRy or yRx for all x,y ∈ A, i.e. if any two elements of A are comparable with respect to R.

If R is an ordering relation on A, then we usually write ≤ (or a similar symbol) for R, i.e. x ≤ y iff xRy. If ≤ is a partial order on A, then we call the pair hA, ≤i a partially ordered set, or just an ordered set. If ≤ furthermore is a linear order, then hA, ≤i is called a linearly ordered set or a chain.

For a finite partially ordered set 〈A, ≤〉 we can draw an order diagram by the following rules: If a ≤ b and a ≠b, then put b above a. b need not be directly

above a, but also may be shifted to one side or the other. If there is no element between a and b, you connect them by a line.

Example .1. 1. Let A = {0, 1,a, b,c}, and define 

by the diagram given in Fig. 1.1

Figure 1.1: The diamond

This diagram represents the following relation on A:

0 ≤ 0, 0 ≤ a, 0 ≤ b, 0 ≤ c, 0 ≤ 1, a ≤ a, a ≤ 1, b ≤ b, b ≤ 1, c ≤ c, c ≤ 1, 1 ≤ 1.

It is not hard to see that this is indeed a partial order on A.

2. Let A = {2, 3, 4, 5, 6}, and define R by the usual ≤ relation on N, i.e. aRb iff a ≤ b. Then R is a linear order on A.

3. Let us define another relation on N by

 (1.1)                    a/b iff a divides b.

To show that / is a partial order we have to show the three defining properties of a partial order relation:

Reflexive: Since every natural number is a divisor of itself, we have a/a for all a ∈ A.

Antisymmetric: If a divides b then we have either a = b or in the

usual ordering of N; similarly, if b divides a, then b = a or b
 a. Since   is not possible, a/b and b/a implies a = b.

Transitive: If a divides b and b divides c then a also divides c.

Thus, / is a partial order on N.

4. Let A = {x,y} and define ≤ on the power set P(A) by s ≤ t iff s is a subset of t

 This gives us the following relation:

∅ ≤ ∅, ∅ ≤ {x}, ∅ ≤ {y}, ∅ ≤ {x,y} = A, {x} ≤ {x}, {x} ≤ {x,y}