Suppose throughout that
is a homogeneous binary relation on a set
(that is,
is a subset of
) and as usual, write
and say that
holds or is true if and only if
Strict weak orderings
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Preliminaries on incomparability and transitivity of incomparability
Two elements
and
of
are said to be incomparable with respect to
if neither
nor
is true.[1]
A strict partial order
is a strict weak ordering if and only if incomparability with respect to
is an equivalence relation.
Incomparability with respect to
is always a homogeneous symmetric relation on
It is reflexive if and only if
is irreflexive (meaning that
is always false), which will be assumed so that transitivity is the only property this "incomparability relation" needs in order to be an equivalence relation.
Define also an induced homogeneous relation
on
by declaring that
where importantly, this definition is not necessarily the same as:
if and only if
Two elements
are incomparable with respect to
if and only if
are equivalent with respect to
(or less verbosely,
-equivalent), which by definition means that both
are true.
The relation "are incomparable with respect to
" is thus identical to (that is, equal to) the relation "are
-equivalent" (so in particular, the former is transitive if and only if the latter is).
When
is irreflexive then the property known as "transitivity of incomparability" (defined below) is exactly the condition necessary and sufficient to guarantee that the relation "are
-equivalent" does indeed form an equivalence relation on
When this is the case, it allows any two elements
satisfying
to be identified as a single object (specifically, they are identified together in their common equivalence class).
Definition
A strict weak ordering on a set
is a strict partial order
on
for which the incomparability relation induced on
by
is a transitive relation.[1]
Explicitly, a strict weak order on
is a homogeneous relation
on
that has all four of the following properties:
- Irreflexivity: For all
it is not true that
- This condition holds if and only if the induced relation
on
is reflexive, where
is defined by declaring that
is true if and only if
is false.
- Transitivity: For all
if
then
- Asymmetry: For all
if
is true then
is false.
- Transitivity of incomparability: For all
if
is incomparable with
(meaning that neither
nor
is true) and if
is incomparable with
then
is incomparable with
- Two elements
are incomparable with respect to
if and only if they are equivalent with respect to the induced relation
(which by definition means that
are both true), where as before,
is declared to be true if and only if
is false. Thus this condition holds if and only if the symmetric relation on
defined by "
are equivalent with respect to
" is a transitive relation, meaning that whenever
are
-equivalent and also
are
-equivalent then necessarily
are
-equivalent. This can also be restated as: whenever
and also
then necessarily
Properties (1), (2), and (3) are the defining properties of a strict partial order, although there is some redundancy in this list as asymmetry (3) implies irreflexivity (1), and also because irreflexivity (1) and transitivity (2) together imply asymmetry (3).[6] The incomparability relation is always symmetric and it will be reflexive if and only if
is an irreflexive relation (which is assumed by the above definition).
Consequently, a strict partial order
is a strict weak order if and only if its induced incomparability relation is an equivalence relation.
In this case, its equivalence classes partition
and moreover, the set
of these equivalence classes can be strictly totally ordered by a binary relation, also denoted by
that is defined for all
by:
for some (or equivalently, for all) representatives
Conversely, any strict total order on a partition
of
gives rise to a strict weak ordering
on
defined by
if and only if there exists sets
in this partition such that
Not every partial order obeys the transitive law for incomparability. For instance, consider the partial order in the set
defined by the relationship
The pairs
are incomparable but
and
are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering.
For transitivity of incomparability, each of the following conditions is necessary, and for strict partial orders also sufficient:
- If
then for all
either
or both.
- If
is incomparable with
then for all
, either (
) or (
) or (
is incomparable with
and
is incomparable with
).
Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder in which any two elements are comparable.[7] A total preorder
satisfies the following properties:
- Transitivity: For all
if
then
- Strong connectedness: For all
- Which implies reflexivity: for all
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations.
The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take the converse of the complement: for a strict weak ordering
define a total preorder
by setting
whenever it is not the case that
In the other direction, to define a strict weak ordering < from a total preorder
set
whenever it is not the case that
[8]
In any preorder there is a corresponding equivalence relation where two elements
and
are defined as equivalent if
In the case of a total preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering.
A partition of a set
is a family of non-empty disjoint subsets of
that have
as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[9] and by Theodore Motzkin a list of sets.[10] An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set
are
In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the partition, and otherwise inherit the order of the sets that contain them.
Representation by functions
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For sets of sufficiently small cardinality, a fourth axiomatization is possible, based on real-valued functions. If
is any set then a real-valued function
on
induces a strict weak order on
by setting
The associated total preorder is given by setting
and the associated equivalence by setting
The relations do not change when
is replaced by
(composite function), where
is a strictly increasing real-valued function defined on at least the range of
Thus for example, a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements.[11]
If
is finite or countable, every weak order on
can be represented by a function in this way.[12] However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on
Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences.
More generally, if
is a set,
is a set with a strict weak ordering
and
is a function, then
induces a strict weak ordering on
by setting
As before, the associated total preorder is given by setting
and the associated equivalence by setting
It is not assumed here that
is an injective function, so a class of two equivalent elements on
may induce a larger class of equivalent elements on
Also,
is not assumed to be a surjective function, so a class of equivalent elements on
may induce a smaller or empty class on
However, the function
induces an injective function that maps the partition on
to that on
Thus, in the case of finite partitions, the number of classes in
is less than or equal to the number of classes on
Combinatorial enumeration
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The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an
-element set is given by the following sequence (sequence A000670 in the OEIS):
Note that S(n, k) refers to Stirling numbers of the second kind.
These numbers are also called the Fubini numbers or ordered Bell numbers.
For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one singleton set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.
The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensional faces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.
Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes, and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a Dedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies. For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that has the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[15]
Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the corresponding weak ordering.[16] In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.
For instance, for
the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon (again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon, six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie, and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.