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Encyclopedia > Preorder

In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. The name quasiorder is also a common expression for preorders. Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...


Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i.e., for all a, b and c in P, we have that: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

aa (reflexivity)
if ab and bc then ac (transitivity)

A set that is equipped with a preorder is called a preordered set.


If a preorder is also antisymmetric, that is, ab and ba implies a = b, then it is a partial order. In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...


A partial order on a set T can be constructed from any preorder on set S by associating members of T with "equivalent" members of S. Formally, one defines an equivalence relation ~ over S such that a ~ b if and only if ab and ba. Now let T be the quotient set S / ~, i.e., the set of all equivalence classes of ~. T can easily be ordered by defining [x] ≤ [y] if and only if xy. By the construction of ~ this definition is independent from the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...


A preorder which is also a congruence relationship (i.e. it is preserved in all contexts), is called a precongruence. As an abstract term, congruence means similarity between objects. ...


Examples of preorders

In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that results from a subgraph of G by zero or more edge contractions. ... A labeled graph with 6 vertices and 7 edges. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ... ...

See also


  Results from FactBites:
 
Preorder - Wikipedia, the free encyclopedia (367 words)
If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.
A preorder which is also a congruence relationship (i.e.
A net is a directed preorder, that is, each pair of elements has an upper bound.
  More results at FactBites »

 
 

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