An order on a
direct product
of partially ordered sets
(cf.
Partially ordered set),
where the set of indices
is well-ordered (cf.
Totally well-ordered set),
defined as follows: If
,
then
if and only if either
for all
or there is an
such that
and
for all
.
A set
ordered by the lexicographic order is called the
lexicographic,
or
ordinal,
product
of the sets
.
If all the sets
coincide
(
for all
),
then their lexicographic product is called an ordinal power of
and is denoted by
.
One also says that
is ordered by the principle of
first difference
(as words are ordered in a dictionary). Thus, if
is the series of natural numbers, then
means that, for some
,
The lexicographic order is a special case of
an ordered product of partially ordered sets (see
[3]).
The lexicographic order can be defined similarly for any partially ordered set of indices
(see
[1]),
but in this case the relation on the set
is not necessarily an order in the usual sense (cf.
Order (on a set)).
A lexicographic product of finitely many well-ordered sets is
well-ordered. A lexicographic product of chains is a
chain.
For a finite
,
the lexicographic order was first considered by
G. Cantor
in the definition of a product of order types of totally ordered sets.
The lexicographic order is widely used outside mathematics, for
example in ordering words in dictionaries, reference books, etc.