# Weak ordering

total preorderstrict weak orderstrict weak orderingWeak orderOrdered partition of a seta strict weak orderdefines a strict weak order and a corresponding total preordernumber of total preordersorderedordered partition
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.wikipedia
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### Ranking

rankrankedrankings
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.
In mathematics, this is known as a weak order or total preorder of objects.

### Ordered Bell number

Fubini number
Weak orderings are counted by the ordered Bell numbers.
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race).

### Preorder

preordered setpreordered setsquasiorder
Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.
A preorder is total if either a ≤ b or b ≤ a for all a, b.

### Order theory

orderorder relationordering
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.
Several types of orders can be defined from numerical data on the items of the order: a total order results from attaching distinct real numbers to each item and using the numerical comparisons to order the items; instead, if distinct items are allowed to have equal numerical scores, one obtains a strict weak ordering.

### Semiorder

Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.
They generalize strict weak orderings, form a special case of partial orders and interval orders, and can be characterized among the partial orders by two forbidden four-item suborders.

### Transitive relation

transitivetransitivitytransitive property
There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets).

### Total order

totally orderedtotally ordered setlinear order
Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.
A real function of n real variables defined on a subset of R n defines a strict weak order and a corresponding total preorder on that subset.

### Binary relation

relationrelationsidentity relation
For a strict weak order "
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.

### Tie (draw)

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.
*Weak ordering, a mathematical formulation of a ranking that allows ties (such as in the outcome of a horse race)

### Partially ordered set

partial orderposetpartially ordered
Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.

### Series-parallel partial order

series compositionseries-parallel
The reflexive closure of a strict weak ordering is a type of series-parallel partial order.
They include weak orders and the reachability relationship in directed trees and directed series-parallel graphs.

### Permutohedron

permutahedrageneralized permutahedrageneralized permutohedra
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.
More generally, the faces of the permutohedron (including the permutohedron itself, but not including the empty set) are in 1-1 correspondence with the strict weak orderings on a set of n items: a face of dimension d corresponds to a strict weak ordering in which there are n − d equivalence classes.

### Partition of a set

partitionpartitionspartitioned
There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets).

### Converse relation

converseinverseinverse relation
Thus we take the inverse of the complement: for a strict weak ordering
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its converse is too.

### Associative containers

mapMap (C++)
In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering.

### Mathematics

mathematicalmathmathematician
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.

### Set (mathematics)

setsetsmathematical set
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.

### Cryptomorphism

cryptomorphicaxiomatized in two equivalent ways
There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets).

### Computer science

computer scientistcomputer sciencescomputer scientists
They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library.

### Partition refinement

They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library.

### C++ Standard Library

libstdc++standard libraryits standard library
In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering. They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library.

### Horse racing

racehorsehorse raceFlat racing
In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context) dead heats, so the outcome of a horse race may be modeled by a weak ordering.

### Photo finish

photo-finishphotoKirby Two-Eyed Camera
In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context) dead heats, so the outcome of a horse race may be modeled by a weak ordering.