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A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.
In set theory, a prewellordering on a set is a preorder on (a transitive and reflexive relation on ) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation < defined by is a well-founded relation.
Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, := < is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ...
In general, two points x and y are topologically indistinguishable if and only if x ≤ y and y ≤ x, where ≤ is the specialization preorder on X. It follows that a space X is T 0 if and only if the specialization preorder ≤ on X is a partial order. There are numerous partial orders on a finite set. Each defines a unique T 0 topology.
A directed set's preorder is called a direction. The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, [2] meaning that every pair of elements is bounded below. [3] [a] Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other ...
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.For most spaces that are considered in practice, namely for all those that satisfy the T 0 separation axiom, this preorder is even a partial order (called the specialization order).
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations . The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves ...
This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or opposite order. Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order.