In mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders. 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.
A set that is equipped with a preorder is called a preordered set.
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+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R- path from x to y.
Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a. (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 ~. 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. In any case, on S / ~ we can define [x] ≤ [y] if and only if x 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 "", a relation "<" can be defined as a < b if and only if (a b and not b a), or equivalently, using the equivalence relation introduced above, (a b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. 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 "", a relation "<" can be defined as a < b if and only if (a 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 b if and only if a < b or a ~ b. This is the reason for using the notation ""; "≤" 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 "" can give the same relation "<", so without more information, such as the equivalence relation, "" cannot be reconstructed from "<". Possible preorders include the following:
Example of a :
As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
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.