Ordinal multiplication 
is the order type of
consecutive copies of a well ordered set of order type 
. Let 
and 
be well ordered sets
of order types 
and 
. On the Cartesian product 
, compare the second coordinates
first. For any 
and 
,
1. If 
,
then 
,
2. If 
,
then 
iff 
.
The resulting order type is 
. This is lexicographic
order on the reversed pairs 
. Conventions for writing the product vary among references
(Rubin 1967, Suppes 1972; Moore 1982, p. 40; Dauben 1990, p. 104; Ciesielski
1997, p. 48).
Like ordinal addition, ordinal multiplication is not commutative, but it is associative. For example,
 |
(1)
|
The product satisfies the following recursion for any ordinal number 
:
 |
(2)
|
 |
(3)
|
If 
is a limit ordinal, then 
is the supremum of the
set 
, the least ordinal
number greater than or equal to every member of that set.
In particular, 
also holds at limit ordinals.
