Weak ordering

Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY indicates that the column's property is always true for the row's term (at the very left), while indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

A weak order on the set where and are of equal rank, is ranked below them, and is ranked above them.
I) representation as a strict weak order where is shown as an arrow from to ;
II) representation as a total preorder , shown using arrows;
III) representation as an ordered partition, with the sets of the partition as dotted ellipses and the total order on these sets shown with arrows.
The 13 possible strict weak orderings on a set of three elements The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy.

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 orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders.[1]

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 (strictly 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). In many cases another representation called a preferential arrangement based on a utility function is also possible.

Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library.[2]

  1. ^ Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–256, ISBN 9781420099836.
  2. ^ Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN 9780132977739.