This section uses an array of preference structures and their quantitative representations to illustrate our subject and provide points of departure for later sections. We first outline three factors that differentiate among various representations and contain important definitions.
Factor 1. Cardinality of X. The main distinction is among finite, countable (finite or denumerable), and uncountably infinite X.
Factor 2.
Ordering properties of 
.
The following four main categories are common.
We say that on X is:
acyclic if its transitive closure is irreflexive (we never
have 
for finite t);
a partial order if it is transitive ( 
whenever 
and 
) and irreflexive (we never have 
);
a weak order if it is a partial order for which 
is transitive;
a linear order if it is a weak order or partial order
for which is the identity relation.
Szpilrajn's theorem (1930) implies that an acyclic has a linear extension,
i.e., is included in some linear order.
If is a weak order then is an equivalence relation
(reflexive, symmetric, transitive) and the set 
of equivalence
classes in X determined by is linearly ordered by 
on defined by 
if for some
(hence for all) 
and 
.
Factor 3.
Representational uniqueness.
Suppose the quantitative structure of the representation uses only one real-valued
function u on X.
Assume that u satisfies the representation, and let U denote the set of all 
that satisfy it.
We then say that u is unique up to:
(i)
an ordinal transformation
if 
for all 
,
;
(ii)
a positive affine transformation
if there are real numbers a > 0 and b such that
v(x) = au(x)+b for all 
;
(iii)
a proportionality transformation
if there is
an 
such that v(x) = au(x) for all .
When a representation uses more than one real-valued function, the
same definitions apply to individual functions although additional
restrictions on admissible transformations usually obtain when the
functions are considered jointly.
For example, if 
and
the representation uses 
for 
,
we say that the 
are unique up to similar positive affine transformations
if another set 
of 
also satisfies the representation if and only if there is an
and 
such that
for all 
and all
.
Other distinguishing factors include special structures for X,
the presence or absence of operations like 
, and whether a
representation involves specialized properties for its real-valued functions
such as continuity or linearity.
Continuity is often associated with topological structures as described,
for example, in Fishburn (1970, 1989, 1994) and
Wakker (1989),
and it will not play a prominent role in our present
discussion, which is primarily algebraic.
The rest of the section outlines traditional topics in preference theory, where u in a representation is usually referred to as a utility function. Theorems that link utility representations to qualitative preference structures by means of assumptions or axioms for preference are noted. Most proofs are available in Fishburn (1970) or in references cited in Fishburn (1989, 1994). I include a few proof comments here to illustrate the representations.