# Weak ordering

## Mathematical ranking of a set / From Wikipedia, the free encyclopedia

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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]}

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Y indicates that the column's property is always true 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 Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ |

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]}