Preorder

For other uses, see Preorder (disambiguation).

In mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of For example, all partial orders and equivalence relations are preorders. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The name quasiorder is also common for preorders. Other notations are pre-order, quasi-order, and quasi order. Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.

1 Formal definition
2 Constructions
3 Examples of preorders
4 Number of preorders
5 Interval
6 See also

Formal definition

Consider some set P and a binary relation $\lesssim$ on P. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of Then $\lesssim$ is a preorder, or quasiorder, if it is reflexive and transitive, i. 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 whenever an element a is related to an element b e. , for all a, b and c in P, we have that:

a $\lesssim$ a (reflexivity)

if a $\lesssim$ b and b $\lesssim$ c then a $\lesssim$ c (transitivity)

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

If a preorder is also antisymmetric, that is, a $\lesssim$ b and b $\lesssim$ a 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 In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

On the other hand, if it is symmetric, that is, if a $\lesssim$ b implies b $\lesssim$ a, then it is an equivalence relation. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"

A preorder which is preserved in all contexts is called a precongruence. A precongruence which is also symmetric (i. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that e. is an equivalence relation) is a congruence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" See Congruence (geometry for the term as used in elementary geometry

Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R⁺⁼. In Mathematics, the transitive closure of a Binary relation R on a set X is the smallest Transitive relation on X In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of The transitive closure indicates path connection in R: x R⁺ y if and only if there is an R- path from x to y. In Graph theory, a path in a graph is a Sequence of vertices such that from each of its vertices there is an edge to the next vertex

Given a preorder $\lesssim$ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a $\lesssim$ b and b $\lesssim$ a. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition. )

Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Note that if the preorder is R⁺⁼, S / ~ is the set of R- cycle equivalence classes: x ∈ [y] if and only if x = y or x is in an R-cycle with y. Cycle in Graph theory and Computer science has several meanings A closed walk with repeated vertices allowed In any case, on S / ~ we can define [x] ≤ [y] if and only if x $\lesssim$ y. By the construction of ~ , this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.

Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).

For a preorder " $\lesssim$ ", a relation "<" can be defined as a < b if and only if (a $\lesssim$ b and not b $\lesssim$ a), or equivalently, using the equivalence relation introduced above, (a $\lesssim$ b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).

(Alternatively, for a preorder " $\lesssim$ ", a relation "<" can be defined as a < b if and only if (a $\lesssim$ b and a ≠ b). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before. )

Conversely we have a $\lesssim$ b if and only if a < b or a ~ b. This is the reason for using the notation " $\lesssim$ "; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.

Note that with this construction multiple preorders " $\lesssim$ " can give the same relation "<", so without more information, such as the equivalence relation, " $\lesssim$ " cannot be reconstructed from "<". Possible preorders include the following:

Define a ≤ b as a < b or a = b (i. e. , take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations $\lesssim$ and ~.
Define a $\lesssim$ b as "not b < a (i. e. , take the inverse complement of the relation), which corresponds to defining a ~ b as "neither a < b nor b < a; these relations $\lesssim$ and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a strict weak order. In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order The resulting preorder is total. In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that

Examples of preorders

Logical implication over sentences
A net is a directed preorder, that is, each pair of elements has an upper bound. In Logic and Mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement
The embedding relation for countable total orderings. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation
The graph-minor relation in graph theory. In Graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects
In computer science, subtyping relations are usually preorders. In Computer science, a subtype is a Datatype that is generally related to another datatype (the supertype) by some notion of Substitutability

Example of a total preorder:

Preference, according to common models. In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order Preference (also called " taste " or "penchant" is a concept used in the Social sciences particularly Economics.

Number of preorders

Number of n-element binary relations of different types
n	all	transitive	reflexive	preorder	partial order	total preorder	total order	equivalence relation
0	1	1	1	1	1	1	1	1
1	2	2	1	1	1	1	1	1
2	16	13	4	4	3	3	2	2
3	512	171	64	29	19	13	6	5
4	65536	3994	4096	355	219	75	24	15
OEIS	A002416	A006905	A053763	A000798	A001035	A000670	A000142	A000110

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Thus the number of preorders is the sum of the number of partial orders on every partition. For example:

for n=3:
- 1 partition of 3, giving 1 preorder
- 3 partitions of 2+1, giving 3 × 3 = 9 preorders
- 1 partition of 1+1+1, giving 19 preorders

i. e. together 29 preorders.

for n=4:
- 1 partition of 4, giving 1 preorder
- 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders
- 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders
- 1 partition of 1+1+1+1, giving 219 preorders

i. e. together 355 preorders.

Interval

For a $\lesssim$ b, the interval [a,b] is the set of points x satisfying a $\lesssim$ x and x $\lesssim$ b, also written a $\lesssim$ x $\lesssim$ b. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set It contains at least the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.

Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

Also [a,b) and (a,b] can be defined similarly.

Dictionary

-verb

(transitive) To order (goods) in advance, before they are available.

-noun

(set theory) A binary relation that is reflexive and transitive.

-adjective

(computing theory) Of a tree traversal, recursively visiting the root before the left and right subtrees.

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