A fundamental result for 
says that if X is countable then
there is a utility function 
such that
if and only if 
on X is a weak order.
In this case, u is unique up to an ordinal transformation.
Sufficiency of weak order can be seen by enumerating the
indifference classes in 
as 
, defining 
on by
noting that 
if
, and then
defining u on X by 
whenever 
.
Weak order is not generally sufficient for (1) when
is uncountable.
For example, the linear order on 
defined by 
if
or
can be represented lexicographically as 
,
where 
and
denotes lexicographic order.
But it cannot be represented by (1):
otherwise, since 
whenever 
, every interval 
would contain a different rational number and yield the
contradiction that the countable set of rational numbers is uncountable.
To obtain (1) when is uncountable, it needs to be
assumed also that includes a countable
subset that is 
-order dense in .
By definition, 
is order dense in 
if,
whenever 
for
, there is a 
such that
.
Countable order denseness is often replaced in economic discussions by a
sufficient but nonnecessary topological assumption which implies
that u in (1) can defined to be continuous in the topology used for X.
Because (1) implies that is a weak order, it cannot
hold when is acyclic or a partial order that is not also a weak order.
We can, however, continue to use u to preserve one-way in the manner
.
We can also use the same u to fully preserve, by equality, the strong
indifference relation 
on X defined by
for on X is an equivalence relation with 
and 
.
Thus, if X is countable, there is a for which
if and only if on X is acyclic.
Figure 1 illustrates on a Haase diagram for a partially ordered
set in which one point bears to a second if there is a
downward sequence of lines from the first to the second.
When 
is uncountable, (2) holds for acyclic
if , defined in the natural way on , has a
linear extension in which some countable subset is order dense.
Further discussion along this line is available in Peleg (1970)
and Sondermann (1980).
Suppose is a partially ordered set that is not necessarily
weakly ordered.
An alternative to (2) of the two-way or
if and only if variety that replaces u in (1) by
another quantitative construct may then apply if
has additional structure.
A case of this occurs when is an
interval order, i.e., when
The primary two-way representation for an interval order is
where 
with 
the set of all real intervals,
and for 
, A > B means that a > b for all 
and all 
.
One basic result (Fishburn 1970, Theorem 2.7) is: if is
countable for an interval order , then (3) holds
for a mapping I into nondegenerate closed intervals.
Other structures may require open or half-open intervals (consider
with 
), and yet others may fail for (3) because
there are not enough intervals in to accommodate the desired
representation.
Further results are in Fishburn (1985, Chapter 7),
Peris and Subiza (1995), Bosi and Isler (1995),
and other references cited there.
