Ordinal Arithmetic - Multiplication

Multiplication

The Cartesian product, S×T, of two well-ordered sets S and T can be well-ordered by a variant of lexicographical order which puts the least significant position first. Effectively, each element of T is replaced by a disjoint copy of S. The order-type of the Cartesian product is the ordinal which results from multiplying the order-types of S and T. Again, this operation is associative and generalizes the multiplication of natural numbers.

Here is ω·2:

0₀ < 1₀ < 2₀ < 3₀ < ... < 0₁ < 1₁ < 2₁ < 3₁ < ...

and we see: ω·2 = ω + ω. But 2·ω looks like this:

0₀ < 1₀ < 0₁ < 1₁ < 0₂ < 1₂ < 0₃ < 1₃ < ...

and after relabeling, this looks just like ω and so we get 2·ω = ω ≠ ω·2. Hence multiplication of ordinals is not commutative.

Distributivity partially holds for ordinal arithmetic: R(S + T) = RS + RT. However, the other distributive law (T + U)R = TR + UR is not generally true: (1 + 1) ·ω = 2·ω = ω while 1·ω + 1·ω = ω + ω which is different. Therefore, the ordinal numbers do not form a ring.

The definition of multiplication can also be given inductively (the following induction is on β):

• α·0 = 0,
• α·(β + 1) = (α·β) + α,
• and if δ is limit then α·δ is the limit of the α·β for all β < δ.

The main properties of the product are:

• α·0 = 0·α = 0.

• One is a multiplicative identity α·1 = 1·α = α.

• Multiplication is associative (α·β)·γ = α·(β·γ).

• Multiplication is strictly increasing and continuous in the right argument: (α < β and γ > 0) γ·α < γ·β

• In the left argument, do not have the same as in the right argument. For example, 1 < 2 but 1·ω = 2·ω = ω. Instead one gets α ≤ β α·γ ≤ β·γ.

• There is a left cancellation law: If α > 0 and α·β = α·γ, then β = γ.

• Right cancellation does not work e.g. 1·ω = 2·ω = ω but 1 and 2 are different

• α·β = 0 α = 0 or β = 0.

• Distributive law on the left: α·(β + γ) = α·β + α·γ

• No distributive law on the right: e.g. (ω + 1)·2 = ω + 1 + ω + 1 = ω + ω + 1 = ω·2 + 1 which is not ω·2 + 2.

• Left division with remainder: for all α and β, if β > 0, then there are unique γ and δ such that α = β·γ + δ and δ < β. (This does not however mean the ordinals are a Euclidean domain, since they are not even a ring, and the Euclidean "norm" is ordinal-valued.)

• Right division does not work: there is no α such that α·ω ≤ ωω ≤ (α + 1)·ω.